id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-12600 | Pythagorean Triangle from Sum of Reciprocals of Consecutive Same Parity Integers | Let $a, b \in \Z_{>0}$ be (strictly) positive integers such that they are consecutively of the same parity.
Let $\dfrac p q = \dfrac 1 a + \dfrac 1 b$.
Then $p$ and $q$ are the legs of a Pythagorean triangle. | Let $a$ and $b$ both be even.
Then:
{{begin-eqn}}
{{eqn | l = a
| r = 2 n
| c =
}}
{{eqn | l = b
| r = 2 \paren {n + 1}
| c = for some $n \in \Z_{>0}$
}}
{{eqn | ll= \leadsto
| l = \dfrac 1 a + \dfrac 1 b
| r = \dfrac 1 {2 n} + \dfrac 1 {2 \paren {n + 1} }
| c =
}}
{{eqn | r ... | Let $a, b \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]] such that they are consecutively of the same [[Definition:Parity of Integer|parity]].
Let $\dfrac p q = \dfrac 1 a + \dfrac 1 b$.
Then $p$ and $q$ are the [[Definition:Leg of Right Triangle|legs]] of a [[Definition:Pythag... | Let $a$ and $b$ both be [[Definition:Even Integer|even]].
Then:
{{begin-eqn}}
{{eqn | l = a
| r = 2 n
| c =
}}
{{eqn | l = b
| r = 2 \paren {n + 1}
| c = for some $n \in \Z_{>0}$
}}
{{eqn | ll= \leadsto
| l = \dfrac 1 a + \dfrac 1 b
| r = \dfrac 1 {2 n} + \dfrac 1 {2 \paren {n + 1}... | Pythagorean Triangle from Sum of Reciprocals of Consecutive Same Parity Integers | https://proofwiki.org/wiki/Pythagorean_Triangle_from_Sum_of_Reciprocals_of_Consecutive_Same_Parity_Integers | https://proofwiki.org/wiki/Pythagorean_Triangle_from_Sum_of_Reciprocals_of_Consecutive_Same_Parity_Integers | [
"Pythagorean Triangles"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Parity of Integer",
"Definition:Triangle (Geometry)/Right-Angled/Legs",
"Definition:Pythagorean Triangle"
] | [
"Definition:Even Integer",
"Solutions of Pythagorean Equation",
"Definition:Generator for Pythagorean Triple",
"Definition:Pythagorean Triple/Primitive",
"Definition:Triangle (Geometry)/Right-Angled/Legs",
"Definition:Primitive Pythagorean Triangle",
"Definition:Odd Integer",
"Solutions of Pythagorean... |
proofwiki-12601 | Legs of Pythagorean Triangle used as Generator for another Pythagorean Triangle | Let $a$ and $b$ be the legs of a Pythagorean triangle $P_1$.
Let $\tuple {a, b}$ be used as the generator for a new Pythagorean triangle $P_2$.
Then the hypotenuse of $P_2$ is the square of the hypotenuse of $P_1$. | By Pythagoras's Theorem, the square of the hypotenuse of $P_1$ is $a^2 + b^2$.
By Solutions of Pythagorean Equation, the sides of $P_2$ can be expressed as $\tuple {2 a b, a^2 - b^2, a^2 + b^2}$, where the hypotenuse is $a^2 + b^2$.
{{qed}} | Let $a$ and $b$ be the [[Definition:Leg of Right Triangle|legs]] of a [[Definition:Pythagorean Triangle|Pythagorean triangle]] $P_1$.
Let $\tuple {a, b}$ be used as the [[Definition:Generator for Pythagorean Triple|generator]] for a new [[Definition:Pythagorean Triangle|Pythagorean triangle]] $P_2$.
Then the [[Defini... | By [[Pythagoras's Theorem]], the [[Definition:Square (Algebra)|square]] of the [[Definition:Hypotenuse|hypotenuse]] of $P_1$ is $a^2 + b^2$.
By [[Solutions of Pythagorean Equation]], the [[Definition:Side of Polygon|sides]] of $P_2$ can be expressed as $\tuple {2 a b, a^2 - b^2, a^2 + b^2}$, where the [[Definition:Hyp... | Legs of Pythagorean Triangle used as Generator for another Pythagorean Triangle | https://proofwiki.org/wiki/Legs_of_Pythagorean_Triangle_used_as_Generator_for_another_Pythagorean_Triangle | https://proofwiki.org/wiki/Legs_of_Pythagorean_Triangle_used_as_Generator_for_another_Pythagorean_Triangle | [
"Pythagorean Triangles",
"Legs of Pythagorean Triangle used as Generator for another Pythagorean Triangle"
] | [
"Definition:Triangle (Geometry)/Right-Angled/Legs",
"Definition:Pythagorean Triangle",
"Definition:Generator for Pythagorean Triple",
"Definition:Pythagorean Triangle",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Square/Function",
"Definition:Triangle (Geometry)/Right-Angled/Hy... | [
"Pythagoras's Theorem",
"Definition:Square/Function",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Solutions of Pythagorean Equation",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse"
] |
proofwiki-12602 | Rational Number plus Irrational Number is Irrational | Rational number plus irrational number is irrational.
That is, let $x \in \Q$, $y \in \R \setminus \Q$ and $x + y = z$.
Then $z \in \R \setminus \Q$. | {{AimForCont}} $z \in \Q$.
By definition of rational numbers:
:$\exists a, b \in \Z, b \ne 0: x = \dfrac a b$
:$\exists c, d \in \Z, d \ne 0: z = \dfrac c d$
Then:
{{begin-eqn}}
{{eqn | l = x + y
| r = z
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac a b + y
| r = \dfrac c d
| c =
}}
{{eqn |... | [[Definition:Rational Number|Rational number]] plus [[Definition:Irrational Number|irrational number]] is [[Definition:Irrational Number|irrational]].
That is, let $x \in \Q$, $y \in \R \setminus \Q$ and $x + y = z$.
Then $z \in \R \setminus \Q$. | {{AimForCont}} $z \in \Q$.
By definition of [[Definition:Rational Number|rational numbers]]:
:$\exists a, b \in \Z, b \ne 0: x = \dfrac a b$
:$\exists c, d \in \Z, d \ne 0: z = \dfrac c d$
Then:
{{begin-eqn}}
{{eqn | l = x + y
| r = z
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac a b + y
| r =... | Rational Number plus Irrational Number is Irrational | https://proofwiki.org/wiki/Rational_Number_plus_Irrational_Number_is_Irrational | https://proofwiki.org/wiki/Rational_Number_plus_Irrational_Number_is_Irrational | [
"Rational Numbers",
"Irrational Numbers"
] | [
"Definition:Rational Number",
"Definition:Irrational Number",
"Definition:Irrational Number"
] | [
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Irrational Number",
"Category:Rational Numbers",
"Category:Irrational Numbers"
] |
proofwiki-12603 | Simple Variable End Point Problem/Endpoints on Curves | Let $y$, $F$, $\phi$ and $\psi$ be smooth real functions.
Let $J = J \sqbrk y$ be a functional of the form:
:$\ds J \sqbrk y = \int_{x_0}^{x_1} \map F {x, y, y'} \rd x$
Let $P_0$, $P_1$ be the endpoints of the curve $y$.
Suppose $P_0$, $P_1$ lie on curves $y = \map {\phi} x$, $y = \map {\psi} x$.
Then the extremum of $... | By general variation of integral functional with $n = 1$:
:$\ds \delta J \sqbrk{y; h} = \int_{x_0}^{x_1} \intlimits {\paren {F_y - \dfrac \d {\d x} F_{y'} } \map h x + F_{y'} \delta y} {x \mathop = x_0} {x \mathop = x_1} + \bigintlimits {\paren {F - y'F_{y'} } \delta x} {x \mathop = x_0} {x \mathop = x_1}$
Since the cu... | Let $y$, $F$, $\phi$ and $\psi$ be [[Definition:Smooth Real Function|smooth real functions]].
Let $J = J \sqbrk y$ be a [[Definition:Real Functional|functional]] of the form:
:$\ds J \sqbrk y = \int_{x_0}^{x_1} \map F {x, y, y'} \rd x$
Let $P_0$, $P_1$ be [[Definition:Endpoints of Directed Smooth Curve|the endpoints... | By [[General Variation of Integral Functional/Dependent on N Functions|general variation of integral functional]] with $n = 1$:
:$\ds \delta J \sqbrk{y; h} = \int_{x_0}^{x_1} \intlimits {\paren {F_y - \dfrac \d {\d x} F_{y'} } \map h x + F_{y'} \delta y} {x \mathop = x_0} {x \mathop = x_1} + \bigintlimits {\paren {F -... | Simple Variable End Point Problem/Endpoints on Curves | https://proofwiki.org/wiki/Simple_Variable_End_Point_Problem/Endpoints_on_Curves | https://proofwiki.org/wiki/Simple_Variable_End_Point_Problem/Endpoints_on_Curves | [
"Calculus of Variations"
] | [
"Definition:Smooth Real Function",
"Definition:Functional/Real",
"Definition:Directed Smooth Curve/Endpoints",
"Definition:Line/Curve",
"Definition:Extremum/Functional",
"Definition:Directed Smooth Curve",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Transversality Conditions"
] | [
"General Variation of Integral Functional/Dependent on N Functions",
"Taylor's Theorem"
] |
proofwiki-12604 | Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set | Let $\struct {S, \preceq}$ be an up-complete ordered set.
Let $T = \struct {S, \preceq, \tau}$ be a relational structure with the Scott topology.
Let $x \in S$.
Let $x^\preceq$ denote the lower closure of $x$.
Then $x^\preceq$ is topologically closed. | By Lower Closure of Element is Closed under Directed Suprema:
:$x^\preceq$ is closed under directed suprema.
By Lower Closure of Singleton:
:$\set x^\preceq = x^\preceq$
By Lower Closure is Lower Section:
:$x^\preceq$ is a lower section.
Thus by Closed Set iff Lower and Closed under Directed Suprema in Scott Topologica... | Let $\struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered set]].
Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Relational Structure with Topology|relational structure]] with the [[Definition:Scott Topology|Scott topology]].
Let $x \in S$.
Let $x^\preceq$ denot... | By [[Lower Closure of Element is Closed under Directed Suprema]]:
:$x^\preceq$ is [[Definition:Closed under Directed Suprema|closed under directed suprema]].
By [[Lower Closure of Singleton]]:
:$\set x^\preceq = x^\preceq$
By [[Lower Closure is Lower Section]]:
:$x^\preceq$ is a [[Definition:Lower Section|lower secti... | Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set | https://proofwiki.org/wiki/Lower_Closure_of_Element_is_Topologically_Closed_in_Scott_Topological_Ordered_Set | https://proofwiki.org/wiki/Lower_Closure_of_Element_is_Topologically_Closed_in_Scott_Topological_Ordered_Set | [
"Topological Order Theory",
"Closed Sets"
] | [
"Definition:Up-Complete",
"Definition:Ordered Set",
"Definition:Relational Structure with Topology",
"Definition:Scott Topology",
"Definition:Lower Closure/Element",
"Definition:Closed Set/Topology"
] | [
"Lower Closure of Element is Closed under Directed Suprema",
"Definition:Closed under Directed Suprema",
"Lower Closure of Singleton",
"Lower Closure is Lower Section",
"Definition:Lower Section",
"Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set",
"Definition:Clos... |
proofwiki-12605 | Complement of Lower Closure of Element is Open in Scott Topological Ordered Set | Let $T = \struct {S, \preceq, \tau}$ be a relational structure with Scott topology
where $\struct {S, \preceq}$ is an up-complete ordered set.
Let $x \in S$.
Then $\relcomp S {x^\preceq}$ is topologically open,
where
:$x^\preceq$ denotes the lower closure of $x$,
:$\relcomp S {x^\preceq}$ denotes the relative complemen... | By Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set:
:$x^\preceq$ is closed.
By definition of closed set:
:$\relcomp S {x^\preceq} \in \tau$
Thus by definition:
:$\relcomp S {x^\preceq}$ is a open set.
{{qed}} | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Relational Structure with Topology|relational structure with]] [[Definition:Scott Topology|Scott topology]]
where $\struct {S, \preceq}$ is an [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered set]].
Let $x \in S$.
Then $\relcomp S {x^\pre... | By [[Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set]]:
:$x^\preceq$ is [[Definition:Closed Set (Topology)|closed]].
By definition of [[Definition:Closed Set (Topology)|closed set]]:
:$\relcomp S {x^\preceq} \in \tau$
Thus by definition:
:$\relcomp S {x^\preceq}$ is a [[Definition:Op... | Complement of Lower Closure of Element is Open in Scott Topological Ordered Set | https://proofwiki.org/wiki/Complement_of_Lower_Closure_of_Element_is_Open_in_Scott_Topological_Ordered_Set | https://proofwiki.org/wiki/Complement_of_Lower_Closure_of_Element_is_Open_in_Scott_Topological_Ordered_Set | [
"Topological Order Theory",
"Open Sets"
] | [
"Definition:Relational Structure with Topology",
"Definition:Scott Topology",
"Definition:Up-Complete",
"Definition:Ordered Set",
"Definition:Open Set/Topology",
"Definition:Lower Closure/Element",
"Definition:Relative Complement"
] | [
"Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology"
] |
proofwiki-12606 | Open iff Upper and with Property (S) in Scott Topological Lattice | Let $T = \struct {S, \preceq, \tau}$ be an up-complete topological lattice.
Let $A$ be a subset of $S$.
Then $A$ is open {{iff}} $A$ is upper and with property (S). | === Sufficient Condition ===
Let $A$ be open.
Thus by definition of Scott topology:
:$A$ is an upper section.
Let $D$ be a directed subset of $S$ such that
:$\sup D \in A$
By definition of Scott topology:
:$A$ is inaccessible by directed suprema.
By definition of inaccessible by directed suprema:
:$A \cap D \ne \O$
By ... | Let $T = \struct {S, \preceq, \tau}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Topological Lattice|topological lattice]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Then $A$ is [[Definition:Open Set (Topology)|open]] {{iff}} $A$ is [[Definition:Upper Section|upper]] and with [[Definition:Prope... | === Sufficient Condition ===
Let $A$ be [[Definition:Open Set (Topology)|open]].
Thus by definition of [[Definition:Scott Topology|Scott topology]]:
:$A$ is an [[Definition:Upper Section|upper section]].
Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
:$\sup D \in A$
By definition of [[... | Open iff Upper and with Property (S) in Scott Topological Lattice | https://proofwiki.org/wiki/Open_iff_Upper_and_with_Property_(S)_in_Scott_Topological_Lattice | https://proofwiki.org/wiki/Open_iff_Upper_and_with_Property_(S)_in_Scott_Topological_Lattice | [
"Topological Order Theory",
"Open Sets"
] | [
"Definition:Up-Complete",
"Definition:Topological Lattice",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Upper Section",
"Definition:Property (S)"
] | [
"Definition:Open Set/Topology",
"Definition:Scott Topology",
"Definition:Upper Section",
"Definition:Directed Subset",
"Definition:Scott Topology",
"Definition:Inaccessible by Directed Suprema",
"Definition:Inaccessible by Directed Suprema",
"Definition:Non-Empty Set",
"Definition:Set Intersection",... |
proofwiki-12607 | Necessary Condition for Integral Functional to have Extremum for given Function/Non-differentiable at Intermediate Point | Let $y, F$ be real functions.
Let $y$ be continuously differentiable for $x \in \hointr a c \cap \hointl c b$ and satisfy:
:$\map y a = A$
:$\map y b = B$
Let $J\sqbrk y$ be a functional of the form
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
Then the functional $J$ has a weak extremum if $y$ satisfies the fo... | Rewrite $J \sqbrk y$ as a sum of two functionals:
{{begin-eqn}}
{{eqn | l = J \sqbrk y
| r = \int_a^b \map F {x, y, y'} \rd x
| c =
}}
{{eqn | r = \int_a^c \map F {x, y, y'} \rd x + \int_c^b \map F {x, y, y'} \rd x
}}
{{eqn | r = J_1 \sqbrk y + J_2 \sqbrk y
}}
{{end-eqn}}
Recall that end points $x = a,x = b... | Let $y, F$ be [[Definition:Real Function|real functions]].
Let $y$ be [[Definition:Continuously Differentiable|continuously differentiable]] for $x \in \hointr a c \cap \hointl c b$ and satisfy:
:$\map y a = A$
:$\map y b = B$
Let $J\sqbrk y$ be a [[Definition:Real Functional|functional]] of the form
:$\ds J \sqbr... | Rewrite $J \sqbrk y$ as a sum of two functionals:
{{begin-eqn}}
{{eqn | l = J \sqbrk y
| r = \int_a^b \map F {x, y, y'} \rd x
| c =
}}
{{eqn | r = \int_a^c \map F {x, y, y'} \rd x + \int_c^b \map F {x, y, y'} \rd x
}}
{{eqn | r = J_1 \sqbrk y + J_2 \sqbrk y
}}
{{end-eqn}}
Recall that [[Definition:Endpoint... | Necessary Condition for Integral Functional to have Extremum for given Function/Non-differentiable at Intermediate Point | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_Function/Non-differentiable_at_Intermediate_Point | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_Function/Non-differentiable_at_Intermediate_Point | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Continuously Differentiable",
"Definition:Functional/Real",
"Definition:Weak Extremum",
"Definition:Limit of Real Function/Left",
"Definition:Limit of Real Function/Right"
] | [
"Definition:Real Interval/Endpoints",
"General Variation of Integral Functional/Dependent on N Functions"
] |
proofwiki-12608 | Euler's Equation for Vanishing Variation in Canonical Variables | {{refactor|There are a number of pages linking here with the presentation of the link set as "momenta". This is going to need a definition of its own, but it is not clear what that is from looking at this page.|level = advanced}}
Consider the following system of differential equations:
:<nowiki>$(1): \quad \begin {case... | Find the full differential of Hamiltonian:
{{begin-eqn}}
{{eqn | l = \rd H
| r = -\rd F + \rd {\sum_{i \mathop = 1}^n y_i' p_i}
| c = {{Defof|Hamiltonian}}
}}
{{eqn | r = -\rd F + \sum_{i \mathop = 1}^n \paren {\rd {y_i'} p_i + y_i' \rd p_i}
| c = Full differential of a product
}}
{{eqn | r = -\frac {... | {{refactor|There are a number of pages linking here with the presentation of the link set as "momenta". This is going to need a definition of its own, but it is not clear what that is from looking at this page.|level = advanced}}
Consider the following [[Definition:System of Differential Equations|system of differenti... | Find the full differential of [[Definition:Hamiltonian|Hamiltonian]]:
{{begin-eqn}}
{{eqn | l = \rd H
| r = -\rd F + \rd {\sum_{i \mathop = 1}^n y_i' p_i}
| c = {{Defof|Hamiltonian}}
}}
{{eqn | r = -\rd F + \sum_{i \mathop = 1}^n \paren {\rd {y_i'} p_i + y_i' \rd p_i}
| c = Full differential of a [[D... | Euler's Equation for Vanishing Variation in Canonical Variables | https://proofwiki.org/wiki/Euler's_Equation_for_Vanishing_Variation_in_Canonical_Variables | https://proofwiki.org/wiki/Euler's_Equation_for_Vanishing_Variation_in_Canonical_Variables | [
"Calculus of Variations"
] | [
"Definition:Differential Equation/System",
"Definition:Coordinate System",
"Definition:Canonical Variable",
"Definition:Differential Equation/System"
] | [
"Definition:Hamiltonian",
"Definition:Multiplication"
] |
proofwiki-12609 | Relational Structure with Topology of Subsets with Property (S) is Topological Space | Let $T = \struct {S, \preceq, \tau}$ be a relational structure with topology
where
:$\struct {S, \preceq}$ is an up-complete ordered set
:$\tau$ is the set of all subsets of $S$ with property (S).
Then $\struct {S, \tau}$ is topological space. | We will prove that
:$S$ has property (S).
Let $D$ be a directed subset of $S$ such that
:$\sup D \in S$
By definition of non-empty set:
:$\exists y: y \in D$
Thus $y \in D$.
Thus by definition of subset:
:$\forall x \in D: y \preceq x \implies x \in S$
{{qed|lemma}}
Then:
$(\text O 3): \quad S \in \tau$
We will prove t... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Relational Structure with Topology|relational structure with topology]]
where
:$\struct {S, \preceq}$ is an [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered set]]
:$\tau$ is the [[Definition:Set of Sets|set]] of all [[Definition:Subset|subse... | We will prove that
:$S$ has [[Definition:Property (S)|property (S)]].
Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
:$\sup D \in S$
By definition of [[Definition:Non-Empty Set|non-empty set]]:
:$\exists y: y \in D$
Thus $y \in D$.
Thus by definition of [[Definition:Subset|subset]]:
:$... | Relational Structure with Topology of Subsets with Property (S) is Topological Space | https://proofwiki.org/wiki/Relational_Structure_with_Topology_of_Subsets_with_Property_(S)_is_Topological_Space | https://proofwiki.org/wiki/Relational_Structure_with_Topology_of_Subsets_with_Property_(S)_is_Topological_Space | [
"Topological Order Theory",
"Topology"
] | [
"Definition:Relational Structure with Topology",
"Definition:Up-Complete",
"Definition:Ordered Set",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Property (S)",
"Definition:Topological Space"
] | [
"Definition:Property (S)",
"Definition:Directed Subset",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Property (S)",
"Definition:Directed Subset",
"Definition:Set Union/Set of Sets",
"Definition:Subset",
"Definition:Property (S)",
"Definition:Property (S)",
"Definition:Set Union/... |
proofwiki-12610 | Subgroup of Real Numbers is Discrete or Dense | Let $G$ be a subgroup of the additive group of real numbers.
Then one of the following holds:
:$G$ is dense in $\R$.
:$G$ is discrete and there exists $a \in \R$ such that $G = a \Z$, that is, $G$ is cyclic. | If $G$ is trivial, then $G$ is discrete and cyclic.
Let $G$ be non-trivial.
Because $x \in G \iff -x \in G$, $G$ has a strictly positive element.
Thus $G \cap \R_{>0}$ is non-empty.
We have that $G \cap \R_{>0}$ is bounded below by $0$.
Hence, by the Continuum Property, $G \cap \R_{>0}$ admits an infimum.
So, let $a = ... | Let $G$ be a [[Definition:Subgroup|subgroup]] of the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Then one of the following holds:
:$G$ is [[Definition:Everywhere Dense|dense]] in $\R$.
:$G$ is [[Definition:Discrete Subgroup of Real Numbers|discrete]] and there exists $a \in \R$ such ... | If $G$ is [[Definition:Trivial Group|trivial]], then $G$ is [[Definition:Discrete Subgroup of Real Numbers|discrete]] and [[Definition:Cyclic Group|cyclic]].
Let $G$ be [[Definition:Non-Trivial Group|non-trivial]].
Because $x \in G \iff -x \in G$, $G$ has a [[Definition:Strictly Positive Real Number|strictly positiv... | Subgroup of Real Numbers is Discrete or Dense | https://proofwiki.org/wiki/Subgroup_of_Real_Numbers_is_Discrete_or_Dense | https://proofwiki.org/wiki/Subgroup_of_Real_Numbers_is_Discrete_or_Dense | [
"Group Theory",
"Topological Groups",
"Real Numbers"
] | [
"Definition:Subgroup",
"Definition:Additive Group of Real Numbers",
"Definition:Everywhere Dense",
"Definition:Discrete Subgroup/Real Numbers",
"Definition:Cyclic Group"
] | [
"Definition:Trivial Group",
"Definition:Discrete Subgroup/Real Numbers",
"Definition:Cyclic Group",
"Definition:Non-Trivial Group",
"Definition:Strictly Positive/Real Number",
"Definition:Non-Empty Set",
"Definition:Bounded Below Set/Real Numbers",
"Continuum Property",
"Definition:Infimum of Set/Re... |
proofwiki-12611 | Discrete Subgroup of Real Numbers is Closed | Let $G$ be a subgroup of the additive group of real numbers.
Let $G$ be discrete.
Then $G$ is closed. | By Subgroup of Real Numbers is Discrete or Dense, there exists $a \in \R$ such that $G = a \Z$.
If $a = 0$, then $G$ is closed.
Let $a > 0$.
Then:
:$\ds \R \setminus G = \bigcup_{z \mathop \in \Z} \openint {a z} {a z + a}$
{{explain|Why?}}
By Union of Open Sets of Metric Space is Open, $\R\setminus G$ is open.
Thus $G$... | Let $G$ be a [[Definition:Subgroup|subgroup]] of the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $G$ be [[Definition:Discrete Subgroup of Real Numbers|discrete]].
Then $G$ is [[Definition:Closed Set (Real Analysis)|closed]]. | By [[Subgroup of Real Numbers is Discrete or Dense]], there exists $a \in \R$ such that $G = a \Z$.
If $a = 0$, then $G$ is [[Definition:Closed Set (Real Analysis)|closed]].
Let $a > 0$.
Then:
:$\ds \R \setminus G = \bigcup_{z \mathop \in \Z} \openint {a z} {a z + a}$
{{explain|Why?}}
By [[Union of Open Sets of Me... | Discrete Subgroup of Real Numbers is Closed | https://proofwiki.org/wiki/Discrete_Subgroup_of_Real_Numbers_is_Closed | https://proofwiki.org/wiki/Discrete_Subgroup_of_Real_Numbers_is_Closed | [
"Topological Groups",
"Real Numbers"
] | [
"Definition:Subgroup",
"Definition:Additive Group of Real Numbers",
"Definition:Discrete Subgroup/Real Numbers",
"Definition:Closed Set/Real Analysis"
] | [
"Subgroup of Real Numbers is Discrete or Dense",
"Definition:Closed Set/Real Analysis",
"Union of Open Sets of Metric Space is Open",
"Definition:Open Set/Real Analysis",
"Definition:Closed Set/Real Analysis"
] |
proofwiki-12612 | Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation | Let $\Phi = \map {\Phi} {x, \family {y_i}_{1 \mathop \le i \mathop \le n}, \family {p_i}_{1 \mathop \le i \mathop \le n} }$ be a real function.
Let $H$ be Hamiltonian.
Then a necessary and sufficient condition for $\Phi$ to be the first integral of Euler's Equations is
:$\dfrac {\partial \Phi} {\partial x} + \sqbrk{\Ph... | {{begin-eqn}}
{{eqn | l = \dfrac {\d \Phi} {\d x}
| r = \frac {\partial\Phi} {\partial x} + \sum_{i \mathop = 1}^n \frac {\partial \Phi} {\partial y_i} \frac {\partial y_i} {\partial x} + \sum_{i \mathop = 1}^n \frac {\partial \Phi} {\partial p_i} \frac{\partial p_i} {\partial x}
}}
{{eqn | r = \frac {\partial \P... | Let $\Phi = \map {\Phi} {x, \family {y_i}_{1 \mathop \le i \mathop \le n}, \family {p_i}_{1 \mathop \le i \mathop \le n} }$ be a [[Definition:Real Function|real function]].
Let $H$ be [[Definition:Hamiltonian|Hamiltonian]].
Then [[Definition:Biconditional/Semantics of Biconditional/Necessary and Sufficient|a necessar... | {{begin-eqn}}
{{eqn | l = \dfrac {\d \Phi} {\d x}
| r = \frac {\partial\Phi} {\partial x} + \sum_{i \mathop = 1}^n \frac {\partial \Phi} {\partial y_i} \frac {\partial y_i} {\partial x} + \sum_{i \mathop = 1}^n \frac {\partial \Phi} {\partial p_i} \frac{\partial p_i} {\partial x}
}}
{{eqn | r = \frac {\partial \P... | Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation | https://proofwiki.org/wiki/Conditions_for_Function_to_be_First_Integral_of_Euler's_Equations_for_Vanishing_Variation | https://proofwiki.org/wiki/Conditions_for_Function_to_be_First_Integral_of_Euler's_Equations_for_Vanishing_Variation | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Hamiltonian",
"Definition:Biconditional/Semantics of Biconditional/Necessary and Sufficient",
"Definition:First Integral of System of Differential Equations",
"Definition:Euler's Equation for Vanishing Variation"
] | [
"Definition:First Integral of System of Differential Equations"
] |
proofwiki-12613 | Coarser Between Generator Set and Filter is Generator Set of Filter | Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $F$ be a filter on $L$.
Let $G$ be a generator set of $F$.
Let $A$ be a subset of $S$ such that
:$G$ is coarser than $A$ and $A$ is coarser than $F$.
Then $A$ is generator set of $F$. | By definition of generator set of filter:
:$F = \paren {\map {\operatorname {fininfs} } G}^\succeq$
where
:$\map {\operatorname {fininfs} } G$ denotes the finite infima set of $G$,
:$A^\succeq$ denotes the upper closure of $A$.
By Finite Infima Set of Coarser Subset is Coarser than Finite Infima Set:
:$\map {\operatorn... | Let $L = \struct {S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $F$ be a [[Definition:Filter|filter]] on $L$.
Let $G$ be a [[Definition:Generator Set of Filter|generator set]] of $F$.
Let $A$ be a [[Definition:Subset|subset]] of $S$ such that
:$G$ is [[Definition:Coarser Subset (Ord... | By definition of [[Definition:Generator Set of Filter|generator set of filter]]:
:$F = \paren {\map {\operatorname {fininfs} } G}^\succeq$
where
:$\map {\operatorname {fininfs} } G$ denotes the [[Definition:Finite Infima Set|finite infima set]] of $G$,
:$A^\succeq$ denotes the [[Definition:Upper Closure of Subset|upper... | Coarser Between Generator Set and Filter is Generator Set of Filter | https://proofwiki.org/wiki/Coarser_Between_Generator_Set_and_Filter_is_Generator_Set_of_Filter | https://proofwiki.org/wiki/Coarser_Between_Generator_Set_and_Filter_is_Generator_Set_of_Filter | [
"Join and Meet Semilattices"
] | [
"Definition:Meet Semilattice",
"Definition:Filter",
"Definition:Generator Set of Filter",
"Definition:Subset",
"Definition:Coarser Subset (Order Theory)",
"Definition:Coarser Subset (Order Theory)",
"Definition:Generator Set of Filter"
] | [
"Definition:Generator Set of Filter",
"Definition:Finite Infima Set",
"Definition:Upper Closure/Set",
"Finite Infima Set of Coarser Subset is Coarser than Finite Infima Set",
"Definition:Coarser Subset (Order Theory)",
"Upper Closure of Coarser Subset is Subset of Upper Closure",
"Definition:Filter in O... |
proofwiki-12614 | Borsuk-Ulam Theorem | Let $n$ be a positive integer.
Let $f: \mathbb S^n \to \R^n$ be a continuous mapping from an $n$-sphere to $\R^n$.
Then there exists $x \in \mathbb S^n$ such that $\map f x = \map f {-x}$. | {{ProofWanted}}
{{Namedfor|Karol Borsuk|name2 = Stanisław Marcin Ulam|cat = Borsuk|cat2 = Ulam}}
Category:Algebraic Topology
i7z9kevixtqhdfvgow1fbiagjqjis94 | Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $f: \mathbb S^n \to \R^n$ be a [[Definition:Continuous Mapping (Topology)|continuous mapping]] from an [[Definition:Sphere (Topology)|$n$-sphere]] to $\R^n$.
Then there exists $x \in \mathbb S^n$ such that $\map f x = \map f {-x}$. | {{ProofWanted}}
{{Namedfor|Karol Borsuk|name2 = Stanisław Marcin Ulam|cat = Borsuk|cat2 = Ulam}}
[[Category:Algebraic Topology]]
i7z9kevixtqhdfvgow1fbiagjqjis94 | Borsuk-Ulam Theorem | https://proofwiki.org/wiki/Borsuk-Ulam_Theorem | https://proofwiki.org/wiki/Borsuk-Ulam_Theorem | [
"Algebraic Topology"
] | [
"Definition:Positive/Integer",
"Definition:Continuous Mapping (Topology)",
"Definition:Sphere/Topology"
] | [
"Category:Algebraic Topology"
] |
proofwiki-12615 | Gershgorin Circle Theorem | Let $n$ be a positive integer.
Let $A = \sqbrk {a_{i j} }$ be a complex square matrix of order $n$.
Let $\lambda$ be an eigenvalue of $A$.
Then there exists $i \in \set {1, 2, \ldots, n}$ such that:
:$\lambda \in \map {\mathbb D} {a_{i i}, R_i}$
where:
:$\ds R_i = \sum_{j \mathop \ne i} \cmod {a_{ i j} }$
:$\map {\math... | {{ProofWanted}}
{{Namedfor|Semyon Aranovich Gershgorin|cat = Gershgorin}} | Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $A = \sqbrk {a_{i j} }$ be a [[Definition:Complex Number|complex]] [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order]] $n$.
Let $\lambda$ be an [[Definition:Eigenvalue|eigenvalue]] of $A$.
Then there exists $i \... | {{ProofWanted}}
{{Namedfor|Semyon Aranovich Gershgorin|cat = Gershgorin}} | Gershgorin Circle Theorem | https://proofwiki.org/wiki/Gershgorin_Circle_Theorem | https://proofwiki.org/wiki/Gershgorin_Circle_Theorem | [
"Linear Algebra"
] | [
"Definition:Positive/Integer",
"Definition:Complex Number",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Eigenvalue",
"Definition:Complex Disk",
"Definition:Disk/Center",
"Definition:Disk/Radius"
] | [] |
proofwiki-12616 | Homotopic Paths Implies Homotopic Composition | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $f_1, f_2, g_1, g_2: \left[{0 \,.\,.\, 1}\right] \to S$ be paths in $T$.
Let $f_1$ be homotopic to $f_2$ and $g_1$ be homotopic to $g_2$.
Then the concatenated paths $f_1 * g_1$ and $f_2 * g_2$ are homotopic. | Let $F: \left[{0 \,.\,.\, 1}\right] \times \left[{0 \,.\,.\, 1}\right] \to S$ be a homotopy between $f_1$ and $f_2$.
Let $G: \left[{0 \,.\,.\, 1}\right] \times \left[{0 \,.\,.\, 1}\right] \to S$ be a homotopy between $g_1$ and $g_2$.
Define $H: \left[{0 \,.\,.\, 1}\right] \times \left[{0 \,.\,.\, 1}\right] \to S$ by:
:... | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $f_1, f_2, g_1, g_2: \left[{0 \,.\,.\, 1}\right] \to S$ be [[Definition:Path (Topology)|paths]] in $T$.
Let $f_1$ be [[Definition:Path-Homotopic|homotopic]] to $f_2$ and $g_1$ be [[Definition:Path-Homotopic|homotopic]] to $g... | Let $F: \left[{0 \,.\,.\, 1}\right] \times \left[{0 \,.\,.\, 1}\right] \to S$ be a [[Definition:Path Homotopy|homotopy]] between $f_1$ and $f_2$.
Let $G: \left[{0 \,.\,.\, 1}\right] \times \left[{0 \,.\,.\, 1}\right] \to S$ be a [[Definition:Path Homotopy|homotopy]] between $g_1$ and $g_2$.
Define $H: \left[{0 \,.\,.... | Homotopic Paths Implies Homotopic Composition | https://proofwiki.org/wiki/Homotopic_Paths_Implies_Homotopic_Composition | https://proofwiki.org/wiki/Homotopic_Paths_Implies_Homotopic_Composition | [
"Homotopy Theory"
] | [
"Definition:Topological Space",
"Definition:Path (Topology)",
"Definition:Homotopy/Path",
"Definition:Homotopy/Path",
"Definition:Concatenation of Paths",
"Definition:Homotopy/Path"
] | [
"Definition:Homotopy/Path/Path Homotopy",
"Definition:Homotopy/Path/Path Homotopy",
"Pasting Lemma/Finite Union of Closed Sets",
"Definition:Continuous Mapping (Topology)",
"Definition:Concatenation of Paths",
"Definition:Homotopy/Path/Path Homotopy"
] |
proofwiki-12617 | Finite Infima Set of Coarser Subset is Coarser than Finite Infima Set | Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $A, B$ be subsets of $S$ such that
:$A$ is coarser than $B$.
Then $\map {\operatorname{fininfs} } A$ is coarser than $\map {\operatorname{fininfs} } B$
where $\map {\operatorname{fininfs} } B$ denotes the finite infima set of $B$. | Let $x \in \map {\operatorname{fininfs} } A$
By definition of finite infima set:
:$\exists Y \in \map {\operatorname {Fin} } A: x = \inf Y$ and $Y$ admits an infimum,
where $\map {\operatorname {Fin} } A$ denotes the set of all finite subsets of $A$.
By definition of coarser subset:
:$\forall y \in Y: \exists z \in B: ... | Let $L = \struct {S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$ such that
:$A$ is [[Definition:Coarser Subset (Order Theory)|coarser]] than $B$.
Then $\map {\operatorname{fininfs} } A$ is [[Definition:Coarser Subset (Order Theory)|coars... | Let $x \in \map {\operatorname{fininfs} } A$
By definition of [[Definition:Finite Infima Set|finite infima set]]:
:$\exists Y \in \map {\operatorname {Fin} } A: x = \inf Y$ and $Y$ admits an [[Definition:Infimum of Set|infimum]],
where $\map {\operatorname {Fin} } A$ denotes the [[Definition:Set of Sets|set]] of all [... | Finite Infima Set of Coarser Subset is Coarser than Finite Infima Set | https://proofwiki.org/wiki/Finite_Infima_Set_of_Coarser_Subset_is_Coarser_than_Finite_Infima_Set | https://proofwiki.org/wiki/Finite_Infima_Set_of_Coarser_Subset_is_Coarser_than_Finite_Infima_Set | [
"Order Theory"
] | [
"Definition:Meet Semilattice",
"Definition:Subset",
"Definition:Coarser Subset (Order Theory)",
"Definition:Coarser Subset (Order Theory)",
"Definition:Finite Infima Set"
] | [
"Definition:Finite Infima Set",
"Definition:Infimum of Set",
"Definition:Set of Sets",
"Definition:Finite Subset",
"Definition:Coarser Subset (Order Theory)",
"Axiom:Axiom of Choice",
"Definition:Infimum of Set",
"Image of Empty Set is Empty Set/Corollary 1",
"Definition:Infimum of Set",
"Definiti... |
proofwiki-12618 | Approximations to Equilateral Triangles by Heronian Triangles | The sequence of best approximations to an equilateral triangle by a Heronian triangle begins:
:The $\tuple {3, 4, 5}$ triangle, with area $6$
:The $\tuple {13, 14, 15}$ triangle, with area $84$, where $14 = 4^2 - 2$
:The $\tuple {193, 194, 195}$ triangle, where $194 = 14^2 - 2$
:The $\tuple {37 \, 633, 37 \, 634, 37 \,... | Suppose a triangle with side lengths $\tuple {a - 1, a, a + 1}$ is Heronian.
By Heron's Formula, the area of this triangle is:
{{begin-eqn}}
{{eqn | o =
| r = \sqrt {s \paren {s - a + 1} \paren {s - a} \paren {s - a - 1} }
}}
{{eqn | r = \sqrt {\frac 3 2 a \paren {\frac 1 2 a + 1} \paren {\frac 1 2 a} \paren {\fr... | The [[Definition:Sequence|sequence]] of best approximations to an [[Definition:Equilateral Triangle|equilateral triangle]] by a [[Definition:Heronian Triangle|Heronian triangle]] begins:
:The [[Pythagorean Triangle/Examples/3-4-5|$\tuple {3, 4, 5}$ triangle]], with [[Definition:Area|area]] $6$
:The $\tuple {13, 14, 1... | Suppose a [[Definition:Triangle (Geometry)|triangle]] with [[Definition:Side of Polygon|side]] [[Definition:Length of Line|lengths]] $\tuple {a - 1, a, a + 1}$ is [[Definition:Heronian Triangle|Heronian]].
By [[Heron's Formula]], the [[Definition:Area|area]] of this [[Definition:Triangle (Geometry)|triangle]] is:
{{... | Approximations to Equilateral Triangles by Heronian Triangles | https://proofwiki.org/wiki/Approximations_to_Equilateral_Triangles_by_Heronian_Triangles | https://proofwiki.org/wiki/Approximations_to_Equilateral_Triangles_by_Heronian_Triangles | [
"Fleenor-Heronian Triangles",
"Equilateral Triangles"
] | [
"Definition:Sequence",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Heronian Triangle",
"Pythagorean Triangle/Examples/3-4-5",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Heronian Triangle",
"Heron's Formula",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Even Integer",
"Definition:Integer",
"Definition:Pell's Equation",
"Definition:Inte... |
proofwiki-12619 | Legendre Transform is Involution | The Legendre transform is an Involution. | Let $\map f x$ be a strictly convex real function.
Let $p = \map {f'} x$.
By definition of the Legendre transform, the transformed real function is of the form:
:$\map {f^*} p = - \map f {\map x p} + p \map x p$
By Legendre Transform of Strictly Convex Real Function is Strictly Convex, $f^*$ is strictly convex.
Let $t ... | The [[Definition:Legendre Transform|Legendre transform]] is an [[Definition:Involution (Mapping)|Involution]]. | Let $\map f x$ be a [[Definition:Strictly Convex Real Function|strictly convex real function]].
Let $p = \map {f'} x$.
By definition of [[Definition:Legendre Transform|the Legendre transform]], the transformed [[Definition:Real Function|real function]] is of the form:
:$\map {f^*} p = - \map f {\map x p} + p \map x ... | Legendre Transform is Involution | https://proofwiki.org/wiki/Legendre_Transform_is_Involution | https://proofwiki.org/wiki/Legendre_Transform_is_Involution | [
"Calculus of Variations"
] | [
"Definition:Legendre Transform",
"Definition:Involution (Mapping)"
] | [
"Definition:Strictly Convex Real Function",
"Definition:Legendre Transform",
"Definition:Real Function",
"Legendre Transform of Strictly Convex Real Function is Strictly Convex",
"Definition:Strictly Convex Real Function",
"Definition:Independent Variable/Real Function",
"Definition:Function",
"Defini... |
proofwiki-12620 | Upper Closure of Coarser Subset is Subset of Upper Closure | Let $L = \left({S, \preceq}\right)$ be a preordered set.
Let $A, B$ be subsets of $S$ such that
:$A$ is coarser than $B$.
Then $A^\succeq \subseteq B^\succeq$ | Let $x \in A^\succeq$
By definition of upper closure of subset:
:$\exists y \in A: y \preceq x$
By definition of coarser subset:
:$\exists z \in B: z \preceq y$
By definition of transitivity:
:$z \preceq x$
Thus by definition of upper closure of subset:
:$x \in B^\succeq$
{{qed}} | Let $L = \left({S, \preceq}\right)$ be a [[Definition:Preordered Set|preordered set]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$ such that
:$A$ is [[Definition:Coarser Subset (Order Theory)|coarser]] than $B$.
Then $A^\succeq \subseteq B^\succeq$ | Let $x \in A^\succeq$
By definition of [[Definition:Upper Closure of Subset|upper closure of subset]]:
:$\exists y \in A: y \preceq x$
By definition of [[Definition:Coarser Subset (Order Theory)|coarser subset]]:
:$\exists z \in B: z \preceq y$
By definition of [[Definition:Transitivity|transitivity]]:
:$z \preceq x... | Upper Closure of Coarser Subset is Subset of Upper Closure | https://proofwiki.org/wiki/Upper_Closure_of_Coarser_Subset_is_Subset_of_Upper_Closure | https://proofwiki.org/wiki/Upper_Closure_of_Coarser_Subset_is_Subset_of_Upper_Closure | [
"Preorder Theory",
"Upper Closures"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Subset",
"Definition:Coarser Subset (Order Theory)"
] | [
"Definition:Upper Closure/Set",
"Definition:Coarser Subset (Order Theory)",
"Definition:Transitive",
"Definition:Upper Closure/Set"
] |
proofwiki-12621 | Set Coarser than Upper Section is Subset | Let $\struct {S, \preceq}$ be a preordered set.
Let $A, B$ be subsets of $S$ such that
:$A$ is coarser than $B$
and
:$B$ is an upper section.
Then:
:$A \subseteq B$ | Let $x \in A$.
By definition of coarser subset:
:$\exists y \in B: y \preceq x$
Thus by definition of upper section:
:$x \in B$
{{qed}} | Let $\struct {S, \preceq}$ be a [[Definition:Preordered Set|preordered set]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$ such that
:$A$ is [[Definition:Coarser Subset (Order Theory)|coarser]] than $B$
and
:$B$ is an [[Definition:Upper Section|upper section]].
Then:
:$A \subseteq B$ | Let $x \in A$.
By definition of [[Definition:Coarser Subset (Order Theory)|coarser subset]]:
:$\exists y \in B: y \preceq x$
Thus by definition of [[Definition:Upper Section|upper section]]:
:$x \in B$
{{qed}} | Set Coarser than Upper Section is Subset | https://proofwiki.org/wiki/Set_Coarser_than_Upper_Section_is_Subset | https://proofwiki.org/wiki/Set_Coarser_than_Upper_Section_is_Subset | [
"Preorder Theory",
"Upper Sections"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Subset",
"Definition:Coarser Subset (Order Theory)",
"Definition:Upper Section"
] | [
"Definition:Coarser Subset (Order Theory)",
"Definition:Upper Section"
] |
proofwiki-12622 | Sum of 4 Unit Fractions that equals 1 | There are $14$ ways to represent $1$ as the sum of exactly $4$ unit fractions. | Let:
:$1 = \dfrac 1 a + \dfrac 1 b + \dfrac 1 c + \dfrac 1 d$
where:
:$a \le b \le c \le d$
and:
:$a \ge 2$ | There are $14$ ways to represent $1$ as the sum of exactly $4$ [[Definition:Unit Fraction|unit fractions]]. | Let:
:$1 = \dfrac 1 a + \dfrac 1 b + \dfrac 1 c + \dfrac 1 d$
where:
:$a \le b \le c \le d$
and:
:$a \ge 2$ | Sum of 4 Unit Fractions that equals 1 | https://proofwiki.org/wiki/Sum_of_4_Unit_Fractions_that_equals_1 | https://proofwiki.org/wiki/Sum_of_4_Unit_Fractions_that_equals_1 | [
"1",
"Unit Fractions",
"Recreational Mathematics"
] | [
"Definition:Unit Fraction"
] | [] |
proofwiki-12623 | Conditions for Function to be Maximum of its Legendre Transform Two-variable Equivalent | Let $x, p \in \R$.
Let $\map f x$ be a strictly convex real function.
Let $f^*$ be a Legendre transformed $f$.
Let $\map g {x, p} = - \map {f^*} p + x p$
Then:
:$\ds \map f x = \max_p \paren {-\map {f^*} p + x p}$
where $\ds \max_p$ maximises the function with respect to a variable $p$. | Function $g$ acquires an extremum along $p$, when its first derivative along $p$ vanishes:
{{begin-eqn}}
{{eqn | l = \frac {\partial g} {\partial p}
| r = -\frac {\partial f^*} {\partial p} + x
}}
{{eqn | r = 0
| c = Extremum condition
}}
{{eqn | ll= \leadsto
| l = \map { {f^*}'} p
| r = x
}}
... | Let $x, p \in \R$.
Let $\map f x$ be a [[Definition:Strictly Convex Real Function|strictly convex real function]].
Let $f^*$ be a [[Definition:Legendre Transform|Legendre transformed]] $f$.
Let $\map g {x, p} = - \map {f^*} p + x p$
Then:
:$\ds \map f x = \max_p \paren {-\map {f^*} p + x p}$
where $\ds \max_p$ [[D... | [[Definition:Function|Function]] $g$ acquires an extremum along $p$, when its first [[Definition:Derivative|derivative]] along $p$ vanishes:
{{begin-eqn}}
{{eqn | l = \frac {\partial g} {\partial p}
| r = -\frac {\partial f^*} {\partial p} + x
}}
{{eqn | r = 0
| c = Extremum condition
}}
{{eqn | ll= \lea... | Conditions for Function to be Maximum of its Legendre Transform Two-variable Equivalent | https://proofwiki.org/wiki/Conditions_for_Function_to_be_Maximum_of_its_Legendre_Transform_Two-variable_Equivalent | https://proofwiki.org/wiki/Conditions_for_Function_to_be_Maximum_of_its_Legendre_Transform_Two-variable_Equivalent | [
"Calculus of Variations"
] | [
"Definition:Strictly Convex Real Function",
"Definition:Legendre Transform",
"Definition:Max Operation",
"Definition:Function",
"Definition:Variable"
] | [
"Definition:Function",
"Definition:Derivative",
"Definition:Derivative",
"Legendre Transform of Strictly Convex Real Function is Strictly Convex",
"Real Function is Strictly Convex iff Derivative is Strictly Increasing",
"Definition:Derivative",
"Definition:Legendre Transform",
"Definition:Legendre Tr... |
proofwiki-12624 | Set is Coarser than Image of Mapping of Infima | Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $f, g:\N \to S$ be mappings such that:
:$\forall n \in \N: \map g n = \inf \set {\map f m: m \in \N \land m \le n}$
Then $f \sqbrk \N$ is coarser than $g \sqbrk \N$
where $f \sqbrk \N$ denotes the image of mapping $f$. | Let $x \in f \sqbrk \N$.
By definition of image of mapping:
:$\exists n \in \N: x = \map f n$
By definition of $g$:
:$\map g n = \inf \set {\map f m: m \in \N \land m \le n}$
By definition of reflexivity:
:$n \le n$
Then
:$\map f n \in \set {\map f m: m \in \N \land m \le n}$
By definitions of infimum and lower bound:
... | Let $\struct {S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $f, g:\N \to S$ be [[Definition:Mapping|mappings]] such that:
:$\forall n \in \N: \map g n = \inf \set {\map f m: m \in \N \land m \le n}$
Then $f \sqbrk \N$ is [[Definition:Coarser Subset (Order Theory)|coarser]] than $g \... | Let $x \in f \sqbrk \N$.
By definition of [[Definition:Image of Mapping|image of mapping]]:
:$\exists n \in \N: x = \map f n$
By definition of $g$:
:$\map g n = \inf \set {\map f m: m \in \N \land m \le n}$
By definition of [[Definition:Reflexivity|reflexivity]]:
:$n \le n$
Then
:$\map f n \in \set {\map f m: m \in... | Set is Coarser than Image of Mapping of Infima | https://proofwiki.org/wiki/Set_is_Coarser_than_Image_of_Mapping_of_Infima | https://proofwiki.org/wiki/Set_is_Coarser_than_Image_of_Mapping_of_Infima | [
"Join and Meet Semilattices"
] | [
"Definition:Meet Semilattice",
"Definition:Mapping",
"Definition:Coarser Subset (Order Theory)",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Reflexivity",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Definition:Image (Set Theory)/Mapping/Mapping"
] |
proofwiki-12625 | Solutions of Ramanujan-Nagell Equation | Integer solutions to the Ramanujan-Nagell equation:
:$x^2 + 7 = 2^n$
exist for only $5$ values of $n$:
:$3, 4, 5, 7, 15$
{{OEIS|A060728}}
The corresponding values of $x$ are:
:$1, 3, 5, 11, 181$
{{OEIS|A038198}} | By direct implementation:
{{begin-eqn}}
{{eqn | n = 1
| l = 1^2 + 7
| r = 1 + 7
| c =
}}
{{eqn | r = 8
| c =
}}
{{eqn | r = 2^3
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 2
| l = 3^2 + 7
| r = 9 + 7
| c =
}}
{{eqn | r = 16
| c =
}}
{{eqn | r = 2^4
| c... | [[Definition:Integer|Integer]] solutions to the [[Definition:Ramanujan-Nagell Equation|Ramanujan-Nagell equation]]:
:$x^2 + 7 = 2^n$
exist for only $5$ values of $n$:
:$3, 4, 5, 7, 15$
{{OEIS|A060728}}
The corresponding values of $x$ are:
:$1, 3, 5, 11, 181$
{{OEIS|A038198}} | By direct implementation:
{{begin-eqn}}
{{eqn | n = 1
| l = 1^2 + 7
| r = 1 + 7
| c =
}}
{{eqn | r = 8
| c =
}}
{{eqn | r = 2^3
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 2
| l = 3^2 + 7
| r = 9 + 7
| c =
}}
{{eqn | r = 16
| c =
}}
{{eqn | r = 2^4
|... | Solutions of Ramanujan-Nagell Equation | https://proofwiki.org/wiki/Solutions_of_Ramanujan-Nagell_Equation | https://proofwiki.org/wiki/Solutions_of_Ramanujan-Nagell_Equation | [
"Ramanujan-Nagell Equation",
"15"
] | [
"Definition:Integer",
"Definition:Ramanujan-Nagell Equation"
] | [] |
proofwiki-12626 | Five Ramanujan-Nagell Numbers | There exist exactly $5$ Ramanujan-Nagell numbers: positive integers of the form $2^m - 1$ which are triangular:
:$0, 1, 3, 15, 4095$
{{OEIS|A076046}} | Consider the numbers of the form $2^m - 1$ which are triangular:
{{begin-eqn}}
{{eqn | l = 2^m - 1
| r = \frac {r \paren {r + 1} } 2
| c = Closed Form for Triangular Numbers
}}
{{eqn | ll= \leadstoandfrom
| l = 8 \paren {2^m - 1}
| r = 4 r \paren {r + 1}
| c =
}}
{{eqn | ll= \leadstoandfr... | There exist exactly $5$ [[Definition:Ramanujan-Nagell Number|Ramanujan-Nagell numbers]]: [[Definition:Positive Integer|positive integers]] of the form $2^m - 1$ which are [[Definition:Triangular Number|triangular]]:
:$0, 1, 3, 15, 4095$
{{OEIS|A076046}} | Consider the numbers of the form $2^m - 1$ which are [[Definition:Triangular Number|triangular]]:
{{begin-eqn}}
{{eqn | l = 2^m - 1
| r = \frac {r \paren {r + 1} } 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | ll= \leadstoandfrom
| l = 8 \paren {2^m - 1}
| r = 4 r \paren {r + 1}
... | Five Ramanujan-Nagell Numbers | https://proofwiki.org/wiki/Five_Ramanujan-Nagell_Numbers | https://proofwiki.org/wiki/Five_Ramanujan-Nagell_Numbers | [
"Ramanujan-Nagell Equation",
"5"
] | [
"Definition:Ramanujan-Nagell Number",
"Definition:Positive/Integer",
"Definition:Triangular Number"
] | [
"Definition:Triangular Number",
"Closed Form for Triangular Numbers",
"Solutions of Ramanujan-Nagell Equation",
"Definition:Triangular Number",
"Definition:Ramanujan-Nagell Number",
"Category:Ramanujan-Nagell Equation",
"Category:5"
] |
proofwiki-12627 | Conditions for Functional to be Extremum of Two-variable Functional over Canonical Variable p | Let $y = \map y x$ and $\map F {x, y, y'}$ be real functions.
Let $\dfrac {\partial^2 F} {\partial {y'}^2} \ne 0$.
Let $\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
Let $\ds J \sqbrk {y, p} = \int_a^b \paren {-\map H {x, y, p} + p y'} \rd x$, where $H$ is the Hamiltonian of $J \sqbrk y$.
Then $\ds J \sqbrk y = \b... | Euler's equation for $J \sqbrk {y, p}$:
{{begin-eqn}}
{{eqn | l = \frac {\delta J \sqbrk{y, p} } {\delta p}
| r = \frac {\partial} {\partial p} \paren {-\map H {x, y, p} + p y'}
| c = Depends only on $p$ and not its derivatives
}}
{{eqn | r = -\frac {\partial H} {\partial p} + y'
}}
{{eqn | r = 0
}}
{{eqn |... | Let $y = \map y x$ and $\map F {x, y, y'}$ be [[Definition:Real Function|real functions]].
Let $\dfrac {\partial^2 F} {\partial {y'}^2} \ne 0$.
Let $\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
Let $\ds J \sqbrk {y, p} = \int_a^b \paren {-\map H {x, y, p} + p y'} \rd x$, where $H$ is [[Definition:Hamiltonian|t... | [[Vanishing First Variational Derivative implies Euler's Equation for Vanishing Variation|Euler's equation]] for $J \sqbrk {y, p}$:
{{begin-eqn}}
{{eqn | l = \frac {\delta J \sqbrk{y, p} } {\delta p}
| r = \frac {\partial} {\partial p} \paren {-\map H {x, y, p} + p y'}
| c = Depends only on $p$ and not its... | Conditions for Functional to be Extremum of Two-variable Functional over Canonical Variable p | https://proofwiki.org/wiki/Conditions_for_Functional_to_be_Extremum_of_Two-variable_Functional_over_Canonical_Variable_p | https://proofwiki.org/wiki/Conditions_for_Functional_to_be_Extremum_of_Two-variable_Functional_over_Canonical_Variable_p | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Hamiltonian"
] | [
"Vanishing First Variational Derivative implies Euler's Equation for Vanishing Variation",
"Definition:Derivative",
"Definition:Functional/Real"
] |
proofwiki-12628 | Image of Mapping of Infima is Generator Set of Filter | Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $f, g:\N \to S$ be mappings such that:
:$\forall n \in \N: \map g n = \inf \set {\map f m: m \in \N \land m \le n}$
Let $F$ be a filter such that
:$f \sqbrk \N$ is generator set of $F$,
where $f \sqbrk \N$ denotes the image of $f$.
Then $g \sqbrk \N$ is gene... | By Set is Coarser than Image of Mapping of Infima:
:$f \sqbrk \N$ is coarser than $g \sqbrk \N$.
By definition of generator set of filter:
:$F = \paren {\map {\operatorname{fininfs} } {f \sqbrk N} }^\succeq$
where
:$\map {\operatorname{fininfs} } {f \sqbrk \N}$ denotes the finite infima set of $f \sqbrk \N$,
:for subse... | Let $\struct {S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $f, g:\N \to S$ be [[Definition:Mapping|mappings]] such that:
:$\forall n \in \N: \map g n = \inf \set {\map f m: m \in \N \land m \le n}$
Let $F$ be a [[Definition:Filter|filter]] such that
:$f \sqbrk \N$ is [[Definition:Ge... | By [[Set is Coarser than Image of Mapping of Infima]]:
:$f \sqbrk \N$ is [[Definition:Coarser Subset (Order Theory)|coarser]] than $g \sqbrk \N$.
By definition of [[Definition:Generator Set of Filter|generator set of filter]]:
:$F = \paren {\map {\operatorname{fininfs} } {f \sqbrk N} }^\succeq$
where
:$\map {\operator... | Image of Mapping of Infima is Generator Set of Filter | https://proofwiki.org/wiki/Image_of_Mapping_of_Infima_is_Generator_Set_of_Filter | https://proofwiki.org/wiki/Image_of_Mapping_of_Infima_is_Generator_Set_of_Filter | [
"Join and Meet Semilattices"
] | [
"Definition:Meet Semilattice",
"Definition:Mapping",
"Definition:Filter",
"Definition:Generator Set of Filter",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Generator Set of Filter"
] | [
"Set is Coarser than Image of Mapping of Infima",
"Definition:Coarser Subset (Order Theory)",
"Definition:Generator Set of Filter",
"Definition:Finite Infima Set",
"Definition:Subset",
"Definition:Upper Closure/Set",
"Finite Infima Set and Upper Closure is Smallest Filter",
"Definition:Image (Set Theo... |
proofwiki-12629 | Subset of Set is Coarser than Set | Let $\left({S, \preceq}\right)$ be a preordered set.
Let $A, B$ be subset of $S$ such that
:$A \subseteq B$
Then $A$ is coarser than $B$. | Let $x \in A$.
By definition of subset:
:$x \in B$
By definition of reflexivity:
:$x \preceq x$
Thus
:$\exists y \in B: y \preceq x$
{{qed}} | Let $\left({S, \preceq}\right)$ be a [[Definition:Preordered Set|preordered set]].
Let $A, B$ be [[Definition:Subset|subset]] of $S$ such that
:$A \subseteq B$
Then $A$ is [[Definition:Coarser Subset (Order Theory)|coarser]] than $B$. | Let $x \in A$.
By definition of [[Definition:Subset|subset]]:
:$x \in B$
By definition of [[Definition:Reflexivity|reflexivity]]:
:$x \preceq x$
Thus
:$\exists y \in B: y \preceq x$
{{qed}} | Subset of Set is Coarser than Set | https://proofwiki.org/wiki/Subset_of_Set_is_Coarser_than_Set | https://proofwiki.org/wiki/Subset_of_Set_is_Coarser_than_Set | [
"Preorder Theory"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Subset",
"Definition:Coarser Subset (Order Theory)"
] | [
"Definition:Subset",
"Definition:Reflexivity"
] |
proofwiki-12630 | Existence of Product of Three Distinct Primes between n and 2n | Let $n \in \Z$ be an integer such that $n > 15$.
Then between $n$ and $2 n$ there exists at least one integer which is the product of $3$ distinct prime numbers. | Let $16 \le n \le 29$.
Then:
:$n < 30 < 2 n$
and we have:
:$30 = 2 \times 3 \times 5$
which is a product of $3$ distinct primes.
Hence the result holds for $n$ in that range.
Let $n \ge 30$.
Then by the Division Theorem:
:$\exists q, r \in \N: n = 6 q + r$, $0 \le r < 6$, $q \ge 5$
By Bertrand-Chebyshev Theorem, there ... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n > 15$.
Then between $n$ and $2 n$ there exists at least one [[Definition:Integer|integer]] which is the [[Definition:Integer Multiplication|product]] of $3$ [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime numbers]]. | Let $16 \le n \le 29$.
Then:
:$n < 30 < 2 n$
and we have:
:$30 = 2 \times 3 \times 5$
which is a [[Definition:Integer Multiplication|product]] of $3$ [[Definition:Distinct|distinct]] [[Definition:Prime Number|primes]].
Hence the result holds for $n$ in that range.
Let $n \ge 30$.
Then by the [[Division Theorem]]:... | Existence of Product of Three Distinct Primes between n and 2n | https://proofwiki.org/wiki/Existence_of_Product_of_Three_Distinct_Primes_between_n_and_2n | https://proofwiki.org/wiki/Existence_of_Product_of_Three_Distinct_Primes_between_n_and_2n | [
"Prime Numbers",
"15"
] | [
"Definition:Integer",
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Distinct",
"Definition:Prime Number"
] | [
"Definition:Multiplication/Integers",
"Definition:Distinct",
"Definition:Prime Number",
"Division Theorem",
"Bertrand-Chebyshev Theorem",
"Definition:Prime Number"
] |
proofwiki-12631 | Second Column and Diagonal of Pascal's Triangle consist of Triangular Numbers | The $2$nd column and $2$nd diagonal of Pascal's triangle consists of the set of triangular numbers. | Recall Pascal's triangle:
{{:Definition:Pascal's Triangle}}
By definition, the entry in row $n$ and column $m$ contains the binomial coefficient $\dbinom n m$.
Thus the $2$nd column contains all the elements of the form $\dbinom n 2$.
The $m$th diagonal consists of the elements in column $n - m$.
Thus the $m$th diagona... | The $2$nd [[Definition:Column of Pascal's Triangle|column]] and $2$nd [[Definition:Diagonal of Pascal's Triangle|diagonal]] of [[Definition:Pascal's Triangle|Pascal's triangle]] consists of the [[Definition:Set|set]] of [[Definition:Triangular Number|triangular numbers]]. | Recall [[Definition:Pascal's Triangle|Pascal's triangle]]:
{{:Definition:Pascal's Triangle}}
By definition, the entry in [[Definition:Row of Pascal's Triangle|row]] $n$ and [[Definition:Column of Pascal's Triangle|column]] $m$ contains the [[Definition:Binomial Coefficient|binomial coefficient]] $\dbinom n m$.
Thus t... | Second Column and Diagonal of Pascal's Triangle consist of Triangular Numbers | https://proofwiki.org/wiki/Second_Column_and_Diagonal_of_Pascal's_Triangle_consist_of_Triangular_Numbers | https://proofwiki.org/wiki/Second_Column_and_Diagonal_of_Pascal's_Triangle_consist_of_Triangular_Numbers | [
"Pascal's Triangle",
"Triangular Numbers"
] | [
"Definition:Pascal's Triangle/Column",
"Definition:Pascal's Triangle/Diagonal",
"Definition:Pascal's Triangle",
"Definition:Set",
"Definition:Triangular Number"
] | [
"Definition:Pascal's Triangle",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle/Column",
"Definition:Binomial Coefficient",
"Definition:Pascal's Triangle/Column",
"Definition:Pascal's Triangle/Diagonal",
"Definition:Pascal's Triangle/Column",
"Definition:Pascal's Triangle/Diagonal",
... |
proofwiki-12632 | Complement of Element is Irreducible implies Element is Meet Irreducible | Let $T = \struct {S, \tau}$ be a topological space.
Let $P = \struct {\tau, \preceq}$ be an ordered set
where $\mathord \preceq = \mathord \subseteq \cap \paren {\tau \times \tau}$
Let $A \in \tau$.
Then $\relcomp S A$ is irreducible implies
$A$ is meet irreducible in $P$
where $\relcomp S A$ denotes the relative compl... | Assume that
:$\relcomp S A$ is irreducible.
Let $x, y \in \tau$ such that
:$A = x \wedge y$
By definition of topological space:
:$x \cap y \in \tau$
By Meet in Inclusion Ordered Set:
:$x \wedge y = x \cap y$
By De Morgan's Laws: Complement of Intersection:
:$\relcomp S A = \relcomp S x \cup \relcomp S y$
By definition:... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $P = \struct {\tau, \preceq}$ be an [[Definition:Ordered Set|ordered set]]
where $\mathord \preceq = \mathord \subseteq \cap \paren {\tau \times \tau}$
Let $A \in \tau$.
Then $\relcomp S A$ is [[Definition:Irreducible Subset (T... | Assume that
:$\relcomp S A$ is [[Definition:Irreducible Subset (Topology)|irreducible]].
Let $x, y \in \tau$ such that
:$A = x \wedge y$
By definition of [[Definition:Topological Space|topological space]]:
:$x \cap y \in \tau$
By [[Meet in Inclusion Ordered Set]]:
:$x \wedge y = x \cap y$
By [[De Morgan's Laws (Set... | Complement of Element is Irreducible implies Element is Meet Irreducible | https://proofwiki.org/wiki/Complement_of_Element_is_Irreducible_implies_Element_is_Meet_Irreducible | https://proofwiki.org/wiki/Complement_of_Element_is_Irreducible_implies_Element_is_Meet_Irreducible | [
"Topology",
"Meet Irreducible Elements"
] | [
"Definition:Topological Space",
"Definition:Ordered Set",
"Definition:Irreducible Subset (Topology)",
"Definition:Conditional",
"Definition:Meet Irreducible Element",
"Definition:Relative Complement"
] | [
"Definition:Irreducible Subset (Topology)",
"Definition:Topological Space",
"Meet in Inclusion Ordered Set",
"De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection",
"Definition:Closed Set/Topology",
"Definition:Irreducible Subset (Topology)",
"Relative Complement of Relative Comp... |
proofwiki-12633 | Pentagonal Number as Sum of Triangular Numbers | Let $P_n$ be the $n$th pentagonal number.
Then:
:$P_n = T_n + 2 T_{n - 1}$
where $T_n$ is the $n$th triangular number. | {{begin-eqn}}
{{eqn | l = T_n + 2 T_{n - 1}
| r = \frac {n \paren {n + 1} } 2 + 2 \frac {\paren {n - 1} n} 2
| c = Closed Form for Triangular Numbers
}}
{{eqn | r = \frac {n^2 + n + 2 \paren {n^2 - n} } 2
| c =
}}
{{eqn | r = \frac {3 n^2 - n} 2
| c =
}}
{{eqn | r = \frac {3 n \paren {n - 1} }... | Let $P_n$ be the $n$th [[Definition:Pentagonal Number|pentagonal number]].
Then:
:$P_n = T_n + 2 T_{n - 1}$
where $T_n$ is the $n$th [[Definition:Triangular Number|triangular number]]. | {{begin-eqn}}
{{eqn | l = T_n + 2 T_{n - 1}
| r = \frac {n \paren {n + 1} } 2 + 2 \frac {\paren {n - 1} n} 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | r = \frac {n^2 + n + 2 \paren {n^2 - n} } 2
| c =
}}
{{eqn | r = \frac {3 n^2 - n} 2
| c =
}}
{{eqn | r = \frac {3 n \paren {n - ... | Pentagonal Number as Sum of Triangular Numbers | https://proofwiki.org/wiki/Pentagonal_Number_as_Sum_of_Triangular_Numbers | https://proofwiki.org/wiki/Pentagonal_Number_as_Sum_of_Triangular_Numbers | [
"Triangular Numbers",
"Pentagonal Numbers"
] | [
"Definition:Pentagonal Number",
"Definition:Triangular Number"
] | [
"Closed Form for Triangular Numbers",
"Closed Form for Pentagonal Numbers"
] |
proofwiki-12634 | Product of Two Triangular Numbers to make Square | Let $T_n$ be a triangular number.
Then there is an infinite number of $m \in \Z_{>0}$ such that $T_n \times T_m$ is a square number. | Since $n^2 < n \paren {n + 1} < \paren {n + 1}^2$, $n \paren {n + 1}$ cannot be a square number.
Thus there are infinitely many distinct integer solutions to Pell's equation:
:$x^2 - n \paren {n + 1} y^2 = 1$
and for each solution:
{{begin-eqn}}
{{eqn | l = T_n T_{x^2 - 1}
| r = \frac {n \paren {n + 1} } 2 \times... | Let $T_n$ be a [[Definition:Triangular Number|triangular number]].
Then there is an [[Definition:Infinite Set|infinite number]] of $m \in \Z_{>0}$ such that $T_n \times T_m$ is a [[Definition:Square Number|square number]]. | Since $n^2 < n \paren {n + 1} < \paren {n + 1}^2$, $n \paren {n + 1}$ cannot be a [[Definition:Square Number|square number]].
Thus there are [[Definition:Infinite Set|infinitely many]] [[Definition:Distinct|distinct]] [[Definition:Integer|integer]] solutions to [[Definition:Pell's Equation|Pell's equation]]:
:$x^2 - n... | Product of Two Triangular Numbers to make Square | https://proofwiki.org/wiki/Product_of_Two_Triangular_Numbers_to_make_Square | https://proofwiki.org/wiki/Product_of_Two_Triangular_Numbers_to_make_Square | [
"Product of Two Triangular Numbers to make Square",
"Triangular Numbers",
"Square Numbers"
] | [
"Definition:Triangular Number",
"Definition:Infinite Set",
"Definition:Square Number"
] | [
"Definition:Square Number",
"Definition:Infinite Set",
"Definition:Distinct",
"Definition:Integer",
"Definition:Pell's Equation",
"Closed Form for Triangular Numbers"
] |
proofwiki-12635 | Conditions for Integral Functionals to have same Euler's Equations | Let $\mathbf y$ be a real $n$-dimensional vector-valued function.
Let $\map F {x, \mathbf y, \mathbf y'}$, $\map \Phi {x, \mathbf y}$ be real functions.
Let $\Phi$ be twice differentiable.
Let:
{{begin-eqn}}
{{eqn | l = \Psi
| r = \frac {\d \Phi} {\d x}
}}
{{eqn | r = \frac {\partial \Phi} {\partial x} + \sum_{i... | According to Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions:
Euler's Equations for functional $J_1$ are:
:$\ds F_{\mathbf y} - \frac \d {\d x} F_{\mathbf y'} = 0$
Equivalently, for $J_2$ we have
{{begin-eqn}}
{{eqn | l = \paren {F_{\mathbf y} + \Psi_{\mathbf y}... | Let $\mathbf y$ be a real $n$-dimensional [[Definition:Vector-Valued Function|vector-valued function]].
Let $\map F {x, \mathbf y, \mathbf y'}$, $\map \Phi {x, \mathbf y}$ be [[Definition:Real Function|real functions]].
Let $\Phi$ be [[Definition:Differentiability Class|twice differentiable]].
Let:
{{begin-eqn}}
{{... | According to [[Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions]]:
[[Definition:Euler's Equation for Vanishing Variation|Euler's Equations]] for [[Definition:Real Functional|functional]] $J_1$ are:
:$\ds F_{\mathbf y} - \frac \d {\d x} F_{\mathbf y'} = 0$
Equi... | Conditions for Integral Functionals to have same Euler's Equations | https://proofwiki.org/wiki/Conditions_for_Integral_Functionals_to_have_same_Euler's_Equations | https://proofwiki.org/wiki/Conditions_for_Integral_Functionals_to_have_same_Euler's_Equations | [
"Calculus of Variations"
] | [
"Definition:Vector-Valued Function",
"Definition:Real Function",
"Definition:Differentiability Class",
"Definition:Euler's Equation for Vanishing Variation"
] | [
"Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Functional/Real",
"Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions",
"Defini... |
proofwiki-12636 | Square of Odd Number as Difference between Triangular Numbers | Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$\exists a, b \in \Z_{\ge 0}: \paren {2 n + 1}^2 = T_a - T_b$
where:
:$T_a$ and $T_b$ are triangular numbers
:$T_a$ and $T_b$ are coprime.
That is, the square of every odd number is the difference between two coprime triangular numbers. | {{begin-eqn}}
{{eqn | l = T_a - T_b
| r = \dfrac {a^2 + a} 2 - \dfrac {b^2 + b} 2
| c = Closed Form for Triangular Numbers
}}
{{end-eqn}}
Let $a = 3b + 1$
{{begin-eqn}}
{{eqn | l = T_{3 b + 1} - T_b
| r = \dfrac {\paren {3 b + 1}^2 + 3 b + 1} 2 - \dfrac {b^2 + b} 2
| c =
}}
{{eqn | r = \dfrac {\... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$\exists a, b \in \Z_{\ge 0}: \paren {2 n + 1}^2 = T_a - T_b$
where:
:$T_a$ and $T_b$ are [[Definition:Triangular Number|triangular numbers]]
:$T_a$ and $T_b$ are [[Definition:Coprime Integers|coprime]].
That is, the [[Definition:Sq... | {{begin-eqn}}
{{eqn | l = T_a - T_b
| r = \dfrac {a^2 + a} 2 - \dfrac {b^2 + b} 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{end-eqn}}
Let $a = 3b + 1$
{{begin-eqn}}
{{eqn | l = T_{3 b + 1} - T_b
| r = \dfrac {\paren {3 b + 1}^2 + 3 b + 1} 2 - \dfrac {b^2 + b} 2
| c =
}}
{{eqn | r = \df... | Square of Odd Number as Difference between Triangular Numbers | https://proofwiki.org/wiki/Square_of_Odd_Number_as_Difference_between_Triangular_Numbers | https://proofwiki.org/wiki/Square_of_Odd_Number_as_Difference_between_Triangular_Numbers | [
"Triangular Numbers",
"Square Numbers",
"Coprime Integers"
] | [
"Definition:Positive/Integer",
"Definition:Triangular Number",
"Definition:Coprime/Integers",
"Definition:Square/Function",
"Definition:Odd Integer",
"Definition:Coprime/Integers",
"Definition:Triangular Number"
] | [
"Closed Form for Triangular Numbers",
"Definition:Square/Function",
"Definition:Odd Integer",
"Bézout's Identity",
"Definition:Triangular Number",
"Definition:Coprime/Integers"
] |
proofwiki-12637 | Meet in Inclusion Ordered Set | Let $P = \struct {X, \subseteq}$ be an inclusion ordered set.
Let $A, B \in X$ such that
:$A \cap B \in X$
Then $A \wedge B = A \cap B$ | By Intersection is Subset:
:$A \cap B \subseteq A$ and $A \cap B \subseteq B$
By definition:
:$A \cap B$ is a lower bound for $\set {A, B}$
We will prove that
:$\forall C \in X: C$ is a lower bound for $\set {A, B} \implies C \subseteq A \cap B$
Let $C \in X$ such that:
:$C$ is a lower bound for $\set {A, B}$.
By defin... | Let $P = \struct {X, \subseteq}$ be an [[Definition:Subset|inclusion]] [[Definition:Ordered Set|ordered set]].
Let $A, B \in X$ such that
:$A \cap B \in X$
Then $A \wedge B = A \cap B$ | By [[Intersection is Subset]]:
:$A \cap B \subseteq A$ and $A \cap B \subseteq B$
By definition:
:$A \cap B$ is a [[Definition:Lower Bound of Set|lower bound]] for $\set {A, B}$
We will prove that
:$\forall C \in X: C$ is a [[Definition:Lower Bound of Set|lower bound]] for $\set {A, B} \implies C \subseteq A \cap B$
... | Meet in Inclusion Ordered Set | https://proofwiki.org/wiki/Meet_in_Inclusion_Ordered_Set | https://proofwiki.org/wiki/Meet_in_Inclusion_Ordered_Set | [
"Join and Meet"
] | [
"Definition:Subset",
"Definition:Ordered Set"
] | [
"Intersection is Subset",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Intersection is Largest Subset",
"Definition:Infimum of Set",
"Definition:Meet (Order Theory)"
] |
proofwiki-12638 | Element is Meet Irreducible iff Complement of Element is Irreducible | Let $T = \struct {S, \tau}$ be a topological space.
Let $P = \struct {\tau, \preceq}$ be an ordered set
where $\mathord \preceq = \mathord \subseteq \cap \paren {\tau \times \tau}$
Let $A \in \tau$ such that
:$A \ne \top_P$
where $\top_P$ denotes the greatest element in $P$.
Then $A$ is meet irreducible in $P$ {{iff}} ... | === Sufficient Condition ===
Let $A$ be meet irreducible in $P$.
By Top in Ordered Set of Topology:
:$A \ne S$
By Relative Complement of Empty Set and Relative Complement of Relative Complement:
:$\relcomp S A \ne \O$
Thus by definition:
:$\relcomp S A$ is non-empty.
Thus by definition:
:$\relcomp S A$ is closed.
Let $... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $P = \struct {\tau, \preceq}$ be an [[Definition:Ordered Set|ordered set]]
where $\mathord \preceq = \mathord \subseteq \cap \paren {\tau \times \tau}$
Let $A \in \tau$ such that
:$A \ne \top_P$
where $\top_P$ denotes the [[Defin... | === Sufficient Condition ===
Let $A$ be [[Definition:Meet Irreducible Element|meet irreducible]] in $P$.
By [[Top in Ordered Set of Topology]]:
:$A \ne S$
By [[Relative Complement of Empty Set]] and [[Relative Complement of Relative Complement]]:
:$\relcomp S A \ne \O$
Thus by definition:
:$\relcomp S A$ is [[Defin... | Element is Meet Irreducible iff Complement of Element is Irreducible | https://proofwiki.org/wiki/Element_is_Meet_Irreducible_iff_Complement_of_Element_is_Irreducible | https://proofwiki.org/wiki/Element_is_Meet_Irreducible_iff_Complement_of_Element_is_Irreducible | [
"Topology",
"Meet Irreducible Elements"
] | [
"Definition:Topological Space",
"Definition:Ordered Set",
"Definition:Greatest Element",
"Definition:Meet Irreducible Element",
"Definition:Irreducible Subset (Topology)",
"Definition:Relative Complement"
] | [
"Definition:Meet Irreducible Element",
"Top in Ordered Set of Topology",
"Relative Complement of Empty Set",
"Relative Complement of Relative Complement",
"Definition:Non-Empty Set",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Subset",
"De Morgan's Laws (Set Theor... |
proofwiki-12639 | Square of Triangular Number equals Sum of Sequence of Cubes | :$\ds \sum_{i \mathop = 1}^n i^3 = {T_n}^2$
where $T_n$ denotes the $n$th triangular number. | {{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^n i^3
| r = \frac {n^2 \paren {n + 1}^2} 4
| c = Sum of Sequence of Cubes
}}
{{eqn | r = \paren {\frac {n \paren {n + 1} } 2}^2
| c =
}}
{{eqn | r = {T_n}^2
| c = Closed Form for Triangular Numbers
}}
{{end-eqn}}
{{qed}} | :$\ds \sum_{i \mathop = 1}^n i^3 = {T_n}^2$
where $T_n$ denotes the $n$th [[Definition:Triangular Number|triangular number]]. | {{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^n i^3
| r = \frac {n^2 \paren {n + 1}^2} 4
| c = [[Sum of Sequence of Cubes]]
}}
{{eqn | r = \paren {\frac {n \paren {n + 1} } 2}^2
| c =
}}
{{eqn | r = {T_n}^2
| c = [[Closed Form for Triangular Numbers]]
}}
{{end-eqn}}
{{qed}} | Square of Triangular Number equals Sum of Sequence of Cubes/Proof 1 | https://proofwiki.org/wiki/Square_of_Triangular_Number_equals_Sum_of_Sequence_of_Cubes | https://proofwiki.org/wiki/Square_of_Triangular_Number_equals_Sum_of_Sequence_of_Cubes/Proof_1 | [
"Triangular Numbers",
"Cube Numbers",
"Square of Triangular Number equals Sum of Sequence of Cubes"
] | [
"Definition:Triangular Number"
] | [
"Sum of Sequence of Cubes",
"Closed Form for Triangular Numbers"
] |
proofwiki-12640 | Square of Triangular Number equals Sum of Sequence of Cubes | :$\ds \sum_{i \mathop = 1}^n i^3 = {T_n}^2$
where $T_n$ denotes the $n$th triangular number. | The proof proceeds by induction.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$\ds \sum_{i \mathop = 1}^n i^3 = {T_n}^2$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^1 i^3
| r = 1^3
| c =
}}
{{eqn | r = 1
| c =
}}
{{eqn | r =... | :$\ds \sum_{i \mathop = 1}^n i^3 = {T_n}^2$
where $T_n$ denotes the $n$th [[Definition:Triangular Number|triangular number]]. | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{i \mathop = 1}^n i^3 = {T_n}^2$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^1 i^3
... | Square of Triangular Number equals Sum of Sequence of Cubes/Proof 2 | https://proofwiki.org/wiki/Square_of_Triangular_Number_equals_Sum_of_Sequence_of_Cubes | https://proofwiki.org/wiki/Square_of_Triangular_Number_equals_Sum_of_Sequence_of_Cubes/Proof_2 | [
"Triangular Numbers",
"Cube Numbers",
"Square of Triangular Number equals Sum of Sequence of Cubes"
] | [
"Definition:Triangular Number"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Closed Form for Triangular Numbers",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Square of Triangular Number equals Sum of Sequence of Cubes/Proof 2",
"Cube Number as Difference... |
proofwiki-12641 | Sum of Sequence of Cubes divides 3 times Sum of Sequence of Fifth Powers | :$\ds \sum_{i \mathop = 1}^n i^3 \divides 3 \sum_{i \mathop = 1}^n i^5$
where $\divides$ denotes divisibility. | {{begin-eqn}}
{{eqn | l = 3 \sum_{i \mathop = 1}^n i^5
| r = {T_n}^2 \paren {4 T_n - 1}
| c = Sum of Sequence of Fifth Powers
}}
{{eqn | r = k {T_n}^2
| c = where $k = 4 T_n - 1$
}}
{{eqn | ll= \leadsto
| l = {T_n}^2
| o = \divides
| r = 3 \sum_{i \mathop = 1}^n i^5
| c = {{Def... | :$\ds \sum_{i \mathop = 1}^n i^3 \divides 3 \sum_{i \mathop = 1}^n i^5$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | {{begin-eqn}}
{{eqn | l = 3 \sum_{i \mathop = 1}^n i^5
| r = {T_n}^2 \paren {4 T_n - 1}
| c = [[Sum of Sequence of Fifth Powers]]
}}
{{eqn | r = k {T_n}^2
| c = where $k = 4 T_n - 1$
}}
{{eqn | ll= \leadsto
| l = {T_n}^2
| o = \divides
| r = 3 \sum_{i \mathop = 1}^n i^5
| c = {... | Sum of Sequence of Cubes divides 3 times Sum of Sequence of Fifth Powers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Cubes_divides_3_times_Sum_of_Sequence_of_Fifth_Powers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Cubes_divides_3_times_Sum_of_Sequence_of_Fifth_Powers | [
"Cube Numbers",
"Fifth Powers"
] | [
"Definition:Divisor (Algebra)/Integer"
] | [
"Sum of Sequence of Fifth Powers",
"Square of Triangular Number equals Sum of Sequence of Cubes"
] |
proofwiki-12642 | Sum of Sequence of Fifth Powers | :$\ds \sum_{i \mathop = 1}^n i^5 = \dfrac { {T_n}^2 \paren {4 T_n - 1} } 3$
where $T_n$ denotes the $n$th triangular number. | <onlyinclude>
The proof proceeds by induction.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$\ds \sum_{i \mathop = 1}^n i^5 = \dfrac { {T_n}^2 \paren {4 T_n - 1} } 3$ | :$\ds \sum_{i \mathop = 1}^n i^5 = \dfrac { {T_n}^2 \paren {4 T_n - 1} } 3$
where $T_n$ denotes the $n$th [[Definition:Triangular Number|triangular number]]. | <onlyinclude>
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{i \mathop = 1}^n i^5 = \dfrac { {T_n}^2 \paren {4 T_n - 1} } 3$ | Sum of Sequence of Fifth Powers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Fifth_Powers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Fifth_Powers | [
"Fifth Powers",
"Triangular Numbers",
"Sums of Sequences"
] | [
"Definition:Triangular Number"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-12643 | Sum of Adjacent Sequences of Triangular Numbers | {{begin-eqn}}
{{eqn | l = T_1 + T_2 + T_3
| r = T_4
| c =
}}
{{eqn | l = T_5 + T_6 + T_7 + T_8
| r = T_9 + T_{10}
| c =
}}
{{eqn | l = T_{11} + T_{12} + T_{13} + T_{14} + T_{15}
| r = T_{16} + T_{17} + T_{18}
| c =
}}
{{end-eqn}}
and so on.
The $n$th line of the pattern can be wri... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = n^2 + n - 1}^{n^2 + 2 n} T_n
| r = \sum_{k \mathop = 1}^{n^2 + 2 n} T_n - \sum_{k \mathop = 1}^{n^2 + n - 2} T_n
}}
{{eqn | r = H_{n^2 + 2 n} - H_{n^2 + n - 2}
| c = {{Defof|Tetrahedral Number}}
}}
{{eqn | r = \frac {\paren {n^2 + 2 n} \paren {n^2 + 2 n + 1} \pare... | {{begin-eqn}}
{{eqn | l = T_1 + T_2 + T_3
| r = T_4
| c =
}}
{{eqn | l = T_5 + T_6 + T_7 + T_8
| r = T_9 + T_{10}
| c =
}}
{{eqn | l = T_{11} + T_{12} + T_{13} + T_{14} + T_{15}
| r = T_{16} + T_{17} + T_{18}
| c =
}}
{{end-eqn}}
and so on.
The $n$th line of the pattern can be ... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = n^2 + n - 1}^{n^2 + 2 n} T_n
| r = \sum_{k \mathop = 1}^{n^2 + 2 n} T_n - \sum_{k \mathop = 1}^{n^2 + n - 2} T_n
}}
{{eqn | r = H_{n^2 + 2 n} - H_{n^2 + n - 2}
| c = {{Defof|Tetrahedral Number}}
}}
{{eqn | r = \frac {\paren {n^2 + 2 n} \paren {n^2 + 2 n + 1} \pare... | Sum of Adjacent Sequences of Triangular Numbers | https://proofwiki.org/wiki/Sum_of_Adjacent_Sequences_of_Triangular_Numbers | https://proofwiki.org/wiki/Sum_of_Adjacent_Sequences_of_Triangular_Numbers | [
"Sums of Sequences",
"Triangular Numbers"
] | [] | [
"Closed Form for Tetrahedral Numbers",
"Closed Form for Tetrahedral Numbers"
] |
proofwiki-12644 | Fourth Power is Sum of 2 Triangular Numbers | Let $n \in \Z$ be an integer.
Then:
:$\exists a, b \in \Z_{\ge 0}: n^4 = T_a + T_b$
where $T_a$ and $T_b$ are triangular numbers.
That is, the $4$th power of an integer equals the sum of two triangular numbers. | {{begin-eqn}}
{{eqn | l = T_{n^2 - n - 1} + T_{n^2 + n - 1}
| r = \frac {\paren {n^2 - n - 1} \paren {n^2 - n} } 2 + \frac {\paren {n^2 + n - 1} \paren {n^2 + n} } 2
| c = Closed Form for Triangular Numbers
}}
{{eqn | r = \frac {n \paren {n^2 - n - 1} \paren {n - 1} } 2 + \frac {n \paren {n^2 + n - 1} \pare... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then:
:$\exists a, b \in \Z_{\ge 0}: n^4 = T_a + T_b$
where $T_a$ and $T_b$ are [[Definition:Triangular Number|triangular numbers]].
That is, the [[Definition:Integer Power|$4$th power]] of an [[Definition:Integer|integer]] equals the sum of two [[Definition:Trian... | {{begin-eqn}}
{{eqn | l = T_{n^2 - n - 1} + T_{n^2 + n - 1}
| r = \frac {\paren {n^2 - n - 1} \paren {n^2 - n} } 2 + \frac {\paren {n^2 + n - 1} \paren {n^2 + n} } 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | r = \frac {n \paren {n^2 - n - 1} \paren {n - 1} } 2 + \frac {n \paren {n^2 + n - 1} \... | Fourth Power is Sum of 2 Triangular Numbers/Proof 2 | https://proofwiki.org/wiki/Fourth_Power_is_Sum_of_2_Triangular_Numbers | https://proofwiki.org/wiki/Fourth_Power_is_Sum_of_2_Triangular_Numbers/Proof_2 | [
"Fourth Powers",
"Triangular Numbers",
"Fourth Power is Sum of 2 Triangular Numbers"
] | [
"Definition:Integer",
"Definition:Triangular Number",
"Definition:Power (Algebra)/Integer",
"Definition:Integer",
"Definition:Triangular Number"
] | [
"Closed Form for Triangular Numbers",
"Definition:Fraction/Numerator"
] |
proofwiki-12645 | Conditions for Transformation to be Canonical | Let:
:$\ds J_1 \sqbrk {\sequence {y_i}_{1 \mathop \le i \mathop \le n}, \sequence {p_i}_{1 \mathop \le i \mathop \le n} } = \int_a^b \paren {\sum_{i \mathop = 1}^n p_i y_i'-H} \rd x$
:$\ds J_2 \sqbrk {\sequence {Y_i}_{1 \mathop \le i \mathop \le n}, \sequence {P_i}_{1 \mathop \le i \mathop \le n} } = \int_a^b \paren {\... | By Conditions for Integral Functionals to have same Euler's Equations, functionals:
:$\ds \int_a^b F_1 \rd x$
and:
:$\ds \int_a^b F_2 \rd x = \int_a^b \paren {F_1 \pm \frac {\d \Phi} {\d x} } \rd x$
have same Euler's equations.
Express the first one in canonical variables $\paren {x, \sequence{y_i}_{1 \mathop \le i \... | Let:
:$\ds J_1 \sqbrk {\sequence {y_i}_{1 \mathop \le i \mathop \le n}, \sequence {p_i}_{1 \mathop \le i \mathop \le n} } = \int_a^b \paren {\sum_{i \mathop = 1}^n p_i y_i'-H} \rd x$
:$\ds J_2 \sqbrk {\sequence {Y_i}_{1 \mathop \le i \mathop \le n}, \sequence {P_i}_{1 \mathop \le i \mathop \le n} } = \int_a^b \paren ... | By [[Conditions for Integral Functionals to have same Euler's Equations]], [[Definition:Real Functional|functionals]]:
:$\ds \int_a^b F_1 \rd x$
and:
:$\ds \int_a^b F_2 \rd x = \int_a^b \paren {F_1 \pm \frac {\d \Phi} {\d x} } \rd x$
have same [[Euler's Equation for Vanishing Variation in Canonical Variables|Euler... | Conditions for Transformation to be Canonical | https://proofwiki.org/wiki/Conditions_for_Transformation_to_be_Canonical | https://proofwiki.org/wiki/Conditions_for_Transformation_to_be_Canonical | [
"Calculus of Variations"
] | [
"Definition:Functional/Real",
"Definition:Canonical Transformation"
] | [
"Conditions for Integral Functionals to have same Euler's Equations",
"Definition:Functional/Real",
"Euler's Equation for Vanishing Variation in Canonical Variables",
"Definition:Canonical Variable",
"Definition:Integration/Integrand",
"Definition:Coordinate System/Coordinate",
"Chain Rule for Real-Valu... |
proofwiki-12646 | Square of Triangular Numbers as Sum of Triangular Numbers | :${T_n}^2 = T_n + T_{n - 1} T_{n + 1}$
where $T_n$ denotes the $n$th triangular number. | {{begin-eqn}}
{{eqn | l = T_n + T_{n - 1} T_{n + 1}
| r = \frac {n \paren {n + 1} } 2 + \paren {\frac {\paren {n - 1} n} 2 \frac {\paren {n + 1} \paren {n + 2} } 2}
| c = Closed Form for Triangular Numbers
}}
{{eqn | r = \frac {2 n^2 + 2 n + \paren {n^2 - n} \paren {n^2 + 3 n + 2} } 4
| c =
}}
{{eqn ... | :${T_n}^2 = T_n + T_{n - 1} T_{n + 1}$
where $T_n$ denotes the $n$th [[Definition:Triangular Number|triangular number]]. | {{begin-eqn}}
{{eqn | l = T_n + T_{n - 1} T_{n + 1}
| r = \frac {n \paren {n + 1} } 2 + \paren {\frac {\paren {n - 1} n} 2 \frac {\paren {n + 1} \paren {n + 2} } 2}
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | r = \frac {2 n^2 + 2 n + \paren {n^2 - n} \paren {n^2 + 3 n + 2} } 4
| c =
}}
{{... | Square of Triangular Numbers as Sum of Triangular Numbers | https://proofwiki.org/wiki/Square_of_Triangular_Numbers_as_Sum_of_Triangular_Numbers | https://proofwiki.org/wiki/Square_of_Triangular_Numbers_as_Sum_of_Triangular_Numbers | [
"Triangular Numbers"
] | [
"Definition:Triangular Number"
] | [
"Closed Form for Triangular Numbers",
"Closed Form for Triangular Numbers"
] |
proofwiki-12647 | Product of Consecutive Triangular Numbers | :$2 T_n T_{n - 1} = T_{n^2 - 1}$
where $T_n$ denotes the $n$th triangular number. | {{begin-eqn}}
{{eqn | l = 2 T_n T_{n - 1}
| r = 2 \paren {\frac {n \paren {n + 1} } 2} \paren {\frac {\paren {n - 1} n} 2}
| c = Closed Form for Triangular Numbers
}}
{{eqn | r = \frac {\paren {n^2 + n} \paren {n^2 - n} } 2
| c =
}}
{{eqn | r = \frac {n^4 - n^2} 2
| c = Difference of Two Square... | :$2 T_n T_{n - 1} = T_{n^2 - 1}$
where $T_n$ denotes the $n$th [[Definition:Triangular Number|triangular number]]. | {{begin-eqn}}
{{eqn | l = 2 T_n T_{n - 1}
| r = 2 \paren {\frac {n \paren {n + 1} } 2} \paren {\frac {\paren {n - 1} n} 2}
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | r = \frac {\paren {n^2 + n} \paren {n^2 - n} } 2
| c =
}}
{{eqn | r = \frac {n^4 - n^2} 2
| c = [[Difference of Two ... | Product of Consecutive Triangular Numbers | https://proofwiki.org/wiki/Product_of_Consecutive_Triangular_Numbers | https://proofwiki.org/wiki/Product_of_Consecutive_Triangular_Numbers | [
"Triangular Numbers"
] | [
"Definition:Triangular Number"
] | [
"Closed Form for Triangular Numbers",
"Difference of Two Squares",
"Closed Form for Triangular Numbers"
] |
proofwiki-12648 | Sum of Sequence of Reciprocals of Triangular Numbers | :$\ds \sum_{k \mathop \ge 1} \dfrac 1 {T_k} = 2$
where $T_k$ denotes the $k$th triangular number. | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {T_k}
| r = \sum_{k \mathop \ge 1} \dfrac 2 {k \paren {k + 1} }
| c = Closed Form for Triangular Numbers
}}
{{eqn | r = 2 \sum_{k \mathop \ge 1} \dfrac 1 {k \paren {k + 1} }
| c =
}}
{{eqn | r = 2 \times 1
| c = {{Corollary|Sum from 1 to... | :$\ds \sum_{k \mathop \ge 1} \dfrac 1 {T_k} = 2$
where $T_k$ denotes the $k$th [[Definition:Triangular Number|triangular number]]. | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {T_k}
| r = \sum_{k \mathop \ge 1} \dfrac 2 {k \paren {k + 1} }
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | r = 2 \sum_{k \mathop \ge 1} \dfrac 1 {k \paren {k + 1} }
| c =
}}
{{eqn | r = 2 \times 1
| c = {{Corollary|Sum from ... | Sum of Sequence of Reciprocals of Triangular Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Reciprocals_of_Triangular_Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Reciprocals_of_Triangular_Numbers | [
"Triangular Numbers",
"Sums of Sequences"
] | [
"Definition:Triangular Number"
] | [
"Closed Form for Triangular Numbers"
] |
proofwiki-12649 | Triangular Number whose Square is Triangular | The only triangular number with less than $660$ digits, whose square is also triangular, is $6$. | We have that:
::${T_3}^2 = 6^2 = 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8$
To establish that it is the only one yet known can be established by an exhaustive search.
{{qed}} | The only [[Definition:Triangular Number|triangular number]] with less than $660$ digits, whose [[Definition:Square (Algebra)|square]] is also [[Definition:Triangular Number|triangular]], is $6$. | We have that:
::${T_3}^2 = 6^2 = 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8$
To establish that it is the only one yet known can be established by an exhaustive search.
{{qed}} | Triangular Number whose Square is Triangular | https://proofwiki.org/wiki/Triangular_Number_whose_Square_is_Triangular | https://proofwiki.org/wiki/Triangular_Number_whose_Square_is_Triangular | [
"Triangular Numbers",
"6"
] | [
"Definition:Triangular Number",
"Definition:Square/Function",
"Definition:Triangular Number"
] | [] |
proofwiki-12650 | Factors of Integer Congruent to 5 modulo 6 | Let $m$ be an positive integer.
Let $m \equiv 5 \pmod 6$.
Then $m$ has two divisors whose sum is divisible by $6$. | {{questionable|This is so trivial it's pointless. Both $m$ and $1$ are factors, totalling $6 n$, trivially a multiple of $6$. I expect this should be "prime factors".}} | Let $m$ be an [[Definition:Positive Integer|positive integer]].
Let $m \equiv 5 \pmod 6$.
Then $m$ has two [[Definition:Divisor of Integer|divisors]] whose [[Definition:Integer Addition|sum]] is [[Definition:Divisor of Integer|divisible]] by $6$. | {{questionable|This is so trivial it's pointless. Both $m$ and $1$ are factors, totalling $6 n$, trivially a multiple of $6$. I expect this should be "prime factors".}} | Factors of Integer Congruent to 5 modulo 6 | https://proofwiki.org/wiki/Factors_of_Integer_Congruent_to_5_modulo_6 | https://proofwiki.org/wiki/Factors_of_Integer_Congruent_to_5_modulo_6 | [
"6"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-12651 | Top in Ordered Set of Topology | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $P = \left({\tau, \subseteq}\right)$ be an inclusion ordered set of $\tau$.
Then $P$ is bounded above and $\top_P = S$
where $\top_P$ denotes the greatest element in $P$. | By definition of topological space:
:$S \in \tau$
By definition of subset:
:$\forall A \in \tau: A \subseteq S$
Hence $P$ is bounded above.
Thus by definition of the greatest element:
:$\top_P = S$
{{qed}} | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $P = \left({\tau, \subseteq}\right)$ be an [[Definition:Subset|inclusion]] [[Definition:Ordered Set|ordered set]] of $\tau$.
Then $P$ is [[Definition:Bounded Above Set|bounded above]] and $\top_P = S$
where $\top_P$ denote... | By definition of [[Definition:Topological Space|topological space]]:
:$S \in \tau$
By definition of [[Definition:Subset|subset]]:
:$\forall A \in \tau: A \subseteq S$
Hence $P$ is [[Definition:Bounded Above Set|bounded above]].
Thus by definition of the [[Definition:Greatest Element|greatest element]]:
:$\top_P = S$... | Top in Ordered Set of Topology | https://proofwiki.org/wiki/Top_in_Ordered_Set_of_Topology | https://proofwiki.org/wiki/Top_in_Ordered_Set_of_Topology | [
"Topology",
"Order Theory"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Ordered Set",
"Definition:Bounded Above Set",
"Definition:Greatest Element"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Bounded Above Set",
"Definition:Greatest Element"
] |
proofwiki-12652 | Triangular Numbers which are also Square | Let $A_n$ be the $n$th non-negative integer whose square is also a triangular number.
Then:
:$A_n = \begin {cases} 0 & : n = 0 \\ 1 & : n = 1 \\ 6 A_{n - 1} - A_{n - 2} & : n > 1 \end {cases}$ | Let $n \in \Z_{>0}$ be such that $n^2$ is a triangular number.
Then we have:
{{begin-eqn}}
{{eqn | q = \exists m \in \Z_{>0}
| l = n^2
| r = \dfrac {m \paren {m + 1} } 2
| c = Closed Form for Triangular Numbers
}}
{{eqn | ll= \leadstoandfrom
| l = 2 n^2
| r = m^2 + m
| c =
}}
{{eqn ... | Let $A_n$ be the $n$th [[Definition:Positive Integer|non-negative integer]] whose [[Definition:Square (Algebra)|square]] is also a [[Definition:Triangular Number|triangular number]].
Then:
:$A_n = \begin {cases} 0 & : n = 0 \\ 1 & : n = 1 \\ 6 A_{n - 1} - A_{n - 2} & : n > 1 \end {cases}$ | Let $n \in \Z_{>0}$ be such that $n^2$ is a [[Definition:Triangular Number|triangular number]].
Then we have:
{{begin-eqn}}
{{eqn | q = \exists m \in \Z_{>0}
| l = n^2
| r = \dfrac {m \paren {m + 1} } 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | ll= \leadstoandfrom
| l = 2 n^2
... | Triangular Numbers which are also Square | https://proofwiki.org/wiki/Triangular_Numbers_which_are_also_Square | https://proofwiki.org/wiki/Triangular_Numbers_which_are_also_Square | [
"Triangular Numbers",
"Square Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Square/Function",
"Definition:Triangular Number"
] | [
"Definition:Triangular Number",
"Closed Form for Triangular Numbers",
"Completing the Square",
"Definition:Pell's Equation",
"Pell's Equation/Examples/8",
"Definition:Positive/Integer",
"Definition:Sequence",
"Definition:Triangular Number",
"Definition:Sequence",
"Definition:Fraction/Numerator",
... |
proofwiki-12653 | Index of Square Triangular Number from Preceding | Let $T_n$ be the $n$th triangular number.
Let $T_n$ be square.
Then $T_{4 n \paren {n + 1} }$ is also square. | {{begin-eqn}}
{{eqn | l = T_{4 n \paren {n + 1} }
| r = \frac {\paren {4 n \paren {n + 1} } \paren {4 n \paren {n + 1} + 1} } 2
| c = Closed Form for Triangular Numbers
}}
{{eqn | ll= \leadsto
| l = T_{4 n \paren {n + 1} }
| r = \frac {n \paren {n + 1} } 2 \times 4 \paren {4 n \paren {n + 1} + 1... | Let $T_n$ be the $n$th [[Definition:Triangular Number|triangular number]].
Let $T_n$ be [[Definition:Square Number|square]].
Then $T_{4 n \paren {n + 1} }$ is also [[Definition:Square Number|square]]. | {{begin-eqn}}
{{eqn | l = T_{4 n \paren {n + 1} }
| r = \frac {\paren {4 n \paren {n + 1} } \paren {4 n \paren {n + 1} + 1} } 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | ll= \leadsto
| l = T_{4 n \paren {n + 1} }
| r = \frac {n \paren {n + 1} } 2 \times 4 \paren {4 n \paren {n + 1}... | Index of Square Triangular Number from Preceding | https://proofwiki.org/wiki/Index_of_Square_Triangular_Number_from_Preceding | https://proofwiki.org/wiki/Index_of_Square_Triangular_Number_from_Preceding | [
"Triangular Numbers",
"Square Numbers"
] | [
"Definition:Triangular Number",
"Definition:Square Number",
"Definition:Square Number"
] | [
"Closed Form for Triangular Numbers",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number"
] |
proofwiki-12654 | Triangular Number cannot be Cube | Let $T_n$ be the $n$th triangular number such that $n > 1$.
Then $T_n$ cannot be a cube. | Suppose $T_n = x^3$ for some $x \in \Z$.
Then by Odd Square is Eight Triangles Plus One:
:$\exists y \in \Z: 8 T_n + 1 = \paren {2 x}^3 + 1 = y^2$
By Cube which is One Less than a Square:
:$2 x = 2$, $y = 3$
giving the unique solution:
:$T_n = 1^3 = 1$
{{qed}} | Let $T_n$ be the $n$th [[Definition:Triangular Number|triangular number]] such that $n > 1$.
Then $T_n$ cannot be a [[Definition:Cube Number|cube]]. | Suppose $T_n = x^3$ for some $x \in \Z$.
Then by [[Odd Square is Eight Triangles Plus One]]:
:$\exists y \in \Z: 8 T_n + 1 = \paren {2 x}^3 + 1 = y^2$
By [[Cube which is One Less than a Square]]:
:$2 x = 2$, $y = 3$
giving the unique solution:
:$T_n = 1^3 = 1$
{{qed}} | Triangular Number cannot be Cube | https://proofwiki.org/wiki/Triangular_Number_cannot_be_Cube | https://proofwiki.org/wiki/Triangular_Number_cannot_be_Cube | [
"Triangular Numbers",
"Cube Numbers"
] | [
"Definition:Triangular Number",
"Definition:Cube Number"
] | [
"Odd Square is Eight Triangles Plus One",
"Cube which is One Less than a Square"
] |
proofwiki-12655 | Numbers of form 31 x 16^n are sum of 16 4th Powers | Let $m \in \Z$ be an integer of the form $31 \times 16^n$ for $n \in \Z_{\ge 0}$.
Then in order to express $m$ as the sum of fourth powers, you need $16$ of them. | We have:
:$31 \times 16^n = \paren {2^{n + 1} }^4 + 15 \times \paren {2^n}^4$
so every integer of the form $31 \times 16^n$ for $n \in \Z_{\ge 0}$ can be expressed as the sum of 16 fourth powers.
Now we show that we cannot use less than $16$ fourth powers.
Observe that for an even number $2 k$:
{{begin-eqn}}
{{eqn | l ... | Let $m \in \Z$ be an [[Definition:Integer|integer]] of the form $31 \times 16^n$ for $n \in \Z_{\ge 0}$.
Then in order to express $m$ as the [[Definition:Integer Addition|sum]] of [[Definition:Biquadrate|fourth powers]], you need $16$ of them. | We have:
:$31 \times 16^n = \paren {2^{n + 1} }^4 + 15 \times \paren {2^n}^4$
so every [[Definition:Integer|integer]] of the form $31 \times 16^n$ for $n \in \Z_{\ge 0}$ can be expressed as the [[Definition:Integer Addition|sum]] of 16 [[Definition:Biquadrate|fourth powers]].
Now we show that we cannot use less than ... | Numbers of form 31 x 16^n are sum of 16 4th Powers | https://proofwiki.org/wiki/Numbers_of_form_31_x_16^n_are_sum_of_16_4th_Powers | https://proofwiki.org/wiki/Numbers_of_form_31_x_16^n_are_sum_of_16_4th_Powers | [
"Hilbert-Waring Theorem",
"16"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Biquadrate"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Biquadrate",
"Definition:Biquadrate",
"Definition:Even Integer",
"Definition:Odd Integer",
"Binomial Theorem",
"Definition:Even Integer",
"Definition:Biquadrate",
"Definition:Biquadrate",
"Definition:Odd Integer",
"Definition:Di... |
proofwiki-12656 | Integers with Prime Values of Divisor Sum | The sequence of integers whose divisor sum is prime begins:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} 2
| r = 3
}}
{{eqn | l = \map {\sigma_1} 4
| r = 7
}}
{{eqn | l = \map {\sigma_1} 6
| r = 13
}}
{{eqn | l = \map {\sigma_1} {16}
| r = 31
}}
{{eqn | l = \map {\sigma_1} {25}
| r = 31
}}
{{... | Apart from $2$, all primes are odd.
From Divisor Sum is Odd iff Argument is Square or Twice Square, for $\map {\sigma_1} n$ to be odd it needs to be of the form $m^2$ or $2 m^2$.
Suppose $n$ has two coprime divisors $p$ and $q$, each to power $k_p$ and $k_q$ respectively.
Then $\map {\sigma_1} n$ will have $\map {\sigm... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] whose [[Definition:Divisor Sum Function|divisor sum]] is [[Definition:Prime Number|prime]] begins:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} 2
| r = 3
}}
{{eqn | l = \map {\sigma_1} 4
| r = 7
}}
{{eqn | l = \map {\sigma_1} 6
... | Apart from $2$, all [[Definition:Prime Number|primes]] are [[Definition:Odd Integer|odd]].
From [[Divisor Sum is Odd iff Argument is Square or Twice Square]], for $\map {\sigma_1} n$ to be [[Definition:Odd Integer|odd]] it needs to be of the form $m^2$ or $2 m^2$.
Suppose $n$ has two [[Definition:Coprime Integers|cop... | Integers with Prime Values of Divisor Sum | https://proofwiki.org/wiki/Integers_with_Prime_Values_of_Divisor_Sum | https://proofwiki.org/wiki/Integers_with_Prime_Values_of_Divisor_Sum | [
"Divisor Sum Function",
"Prime Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Divisor Sum Function",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Odd Integer",
"Divisor Sum is Odd iff Argument is Square or Twice Square",
"Definition:Odd Integer",
"Definition:Coprime/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition... |
proofwiki-12657 | Equality of Integers to the Power of Each Other | $2$ and $4$ are the only pair of positive integers $m, n$ such that $m \ne n$ such that:
:$m^n = n^m$
Thus:
$2^4 = 4^2 = 16$ | {{WLOG}} suppose $m > n$.
Write $m = n + x$, where $x$ is an integer.
Then:
{{begin-eqn}}
{{eqn | l = m^n
| r = n^m
}}
{{eqn | l = \paren {n + x}^n
| r = n^{n + x}
}}
{{eqn | l = \paren {1 + \frac x n}^n
| r = n^x
| c = dividing both sides by $n^n$
}}
{{end-eqn}}
From Real Sequence $\paren {1 + ... | $2$ and $4$ are the only pair of [[Definition:Positive Integer|positive integers]] $m, n$ such that $m \ne n$ such that:
:$m^n = n^m$
Thus:
$2^4 = 4^2 = 16$ | {{WLOG}} suppose $m > n$.
Write $m = n + x$, where $x$ is an [[Definition:Integer|integer]].
Then:
{{begin-eqn}}
{{eqn | l = m^n
| r = n^m
}}
{{eqn | l = \paren {n + x}^n
| r = n^{n + x}
}}
{{eqn | l = \paren {1 + \frac x n}^n
| r = n^x
| c = dividing both sides by $n^n$
}}
{{end-eqn}}
From ... | Equality of Integers to the Power of Each Other/Proof 2 | https://proofwiki.org/wiki/Equality_of_Integers_to_the_Power_of_Each_Other | https://proofwiki.org/wiki/Equality_of_Integers_to_the_Power_of_Each_Other/Proof_2 | [
"16",
"Number Theory",
"Equality of Integers to the Power of Each Other"
] | [
"Definition:Positive/Integer"
] | [
"Definition:Integer",
"Real Sequence (1 + x over n)^n is Convergent",
"Definition:Increasing/Real Function",
"Definition:Limit of Sequence/Real Numbers",
"Bernoulli's Inequality",
"Bernoulli's Inequality"
] |
proofwiki-12658 | Element of Ordered Set of Topology is Dense iff is Everywhere Dense | Let $T = \struct {S, \tau}$ be a topological space.
Let $P = \struct {\tau, \preceq}$ be an ordered set
where $\mathord \preceq = \mathord \subseteq \cap \paren {\tau \times \tau}$
Let $A \in \tau$.
Then $A$ is a dense element in $P$ {{iff}} $A$ is everywhere dense. | === Sufficient Condition ===
Assume that
:$A$ is a dense element in $P$.
By Bottom in Ordered Set of Topology:
:$\bot_P = \O$
We will prove that
:for every open subset $U$ of $S$: $U \ne \O \implies U \cap A \ne \O$
Let $U$ be an open subset of $S$ such that
:$U \ne \O$
By definition of open set:
:$U \in \tau$
By defin... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $P = \struct {\tau, \preceq}$ be an [[Definition:Ordered Set|ordered set]]
where $\mathord \preceq = \mathord \subseteq \cap \paren {\tau \times \tau}$
Let $A \in \tau$.
Then $A$ is a [[Definition:Dense (Lattice Theory)/Element... | === Sufficient Condition ===
Assume that
:$A$ is a [[Definition:Dense (Lattice Theory)/Element|dense element]] in $P$.
By [[Bottom in Ordered Set of Topology]]:
:$\bot_P = \O$
We will prove that
:for every [[Definition:Open Set (Topology)|open]] [[Definition:Subset|subset]] $U$ of $S$: $U \ne \O \implies U \cap A \n... | Element of Ordered Set of Topology is Dense iff is Everywhere Dense | https://proofwiki.org/wiki/Element_of_Ordered_Set_of_Topology_is_Dense_iff_is_Everywhere_Dense | https://proofwiki.org/wiki/Element_of_Ordered_Set_of_Topology_is_Dense_iff_is_Everywhere_Dense | [
"Denseness"
] | [
"Definition:Topological Space",
"Definition:Ordered Set",
"Definition:Dense (Lattice Theory)/Element",
"Definition:Everywhere Dense"
] | [
"Definition:Dense (Lattice Theory)/Element",
"Bottom in Ordered Set of Topology",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Topological Space",
"Meet in Inclusion Ordered Set",
"Definition:D... |
proofwiki-12659 | Square whose Perimeter equals its Area | The $4 \times 4$ square is the only square whose area in square units equals its perimeter in units.
The area and perimeter of this square are $16$. | Let $S$ be a square whose area equals its perimeter.
Let $A$ be the area of $S$.
Let $P$ be the perimeter of $S$.
Let $b$ be the length of one side of $S$.
From Area of Square:
:$A = b^2$
From Perimeter of Rectangle:
:$P = 2 b + 2 b = 4 b$
Setting $A = P$
:$b^2 = 4 b$
and so:
:$b = 4$
and so:
:$A = 16 = P$
{{qed}} | The $4 \times 4$ [[Definition:Square (Geometry)|square]] is the only [[Definition:Square (Geometry)|square]] whose [[Definition:Area|area]] in square units equals its [[Definition:Perimeter|perimeter]] in units.
The [[Definition:Area|area]] and [[Definition:Perimeter|perimeter]] of this [[Definition:Square (Geometry)|... | Let $S$ be a [[Definition:Square (Geometry)|square]] whose [[Definition:Area|area]] equals its [[Definition:Perimeter|perimeter]].
Let $A$ be the [[Definition:Area|area]] of $S$.
Let $P$ be the [[Definition:Perimeter|perimeter]] of $S$.
Let $b$ be the [[Definition:Length (Linear Measure)|length]] of one [[Definition... | Square whose Perimeter equals its Area | https://proofwiki.org/wiki/Square_whose_Perimeter_equals_its_Area | https://proofwiki.org/wiki/Square_whose_Perimeter_equals_its_Area | [
"Squares"
] | [
"Definition:Quadrilateral/Square",
"Definition:Quadrilateral/Square",
"Definition:Area",
"Definition:Perimeter",
"Definition:Area",
"Definition:Perimeter",
"Definition:Quadrilateral/Square"
] | [
"Definition:Quadrilateral/Square",
"Definition:Area",
"Definition:Perimeter",
"Definition:Area",
"Definition:Perimeter",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Area of Square",
"Perimeter of Rectangle"
] |
proofwiki-12660 | Noether's Theorem (Hamiltonian Mechanics) | Let there be an infinitesimal transformation of generalised coordinates such that:
:$q_i \to \tilde q_i = q_i + q_i^\alpha \tuple {q, \dot q, t} \varepsilon_\alpha + \hbox {terms vanishing on shell}$
where $\varepsilon$ is not time-dependent.
Under this transformation, let the variation of the Lagrangian be:
:$L \tuple... | {{proof wanted}}
{{Namedfor|Emmy Noether|cat = Noether E}}
Category:Partial Differential Equations
Category:Physics
Category:Hamiltonian Mechanics
Category:Calculus of Variations
ftjzbexnj494jsdgbxk0nuhx7nf6e3h | Let there be an [[Definition:Infinitesimal|infinitesimal]] transformation of generalised coordinates such that:
:$q_i \to \tilde q_i = q_i + q_i^\alpha \tuple {q, \dot q, t} \varepsilon_\alpha + \hbox {terms vanishing on shell}$
where $\varepsilon$ is not time-dependent.
Under this transformation, let the variation... | {{proof wanted}}
{{Namedfor|Emmy Noether|cat = Noether E}}
[[Category:Partial Differential Equations]]
[[Category:Physics]]
[[Category:Hamiltonian Mechanics]]
[[Category:Calculus of Variations]]
ftjzbexnj494jsdgbxk0nuhx7nf6e3h | Noether's Theorem (Hamiltonian Mechanics) | https://proofwiki.org/wiki/Noether's_Theorem_(Hamiltonian_Mechanics) | https://proofwiki.org/wiki/Noether's_Theorem_(Hamiltonian_Mechanics) | [
"Partial Differential Equations",
"Physics",
"Hamiltonian Mechanics",
"Calculus of Variations"
] | [
"Definition:Infinitesimal"
] | [
"Category:Partial Differential Equations",
"Category:Physics",
"Category:Hamiltonian Mechanics",
"Category:Calculus of Variations"
] |
proofwiki-12661 | Magic Constant of Order 4 Magic Square | The magic constant of the order $4$ magic square is $34$. | Let $M_4$ denote an order $4$ magic square
By Sum of Terms of Magic Square, the total of all the entries in $M_4$ is given by:
:$T_4 = \dfrac {4^2 \left({4^2 + 1}\right)} 2 = \dfrac {16 \times 17} 2 = 136$
As there are $4$ rows of $M_4$, the magic constant of $M_4$ is given by:
:$S_4 = \dfrac {136} 4 = 34$
{{qed}} | The [[Definition:Magic Constant|magic constant]] of the [[Magic Square/Examples/Order 4|order $4$ magic square]] is $34$. | Let $M_4$ denote an [[Magic Square/Examples/Order 4|order $4$ magic square]]
By [[Sum of Terms of Magic Square]], the total of all the entries in $M_4$ is given by:
:$T_4 = \dfrac {4^2 \left({4^2 + 1}\right)} 2 = \dfrac {16 \times 17} 2 = 136$
As there are $4$ [[Definition:Row of Matrix|rows]] of $M_4$, the [[Definit... | Magic Constant of Order 4 Magic Square/Proof 1 | https://proofwiki.org/wiki/Magic_Constant_of_Order_4_Magic_Square | https://proofwiki.org/wiki/Magic_Constant_of_Order_4_Magic_Square/Proof_1 | [
"Magic Squares",
"34",
"Magic Constant of Order 4 Magic Square"
] | [
"Definition:Magic Square/Magic Constant",
"Magic Square/Examples/Order 4"
] | [
"Magic Square/Examples/Order 4",
"Sum of Terms of Magic Square",
"Definition:Matrix/Row",
"Definition:Magic Square/Magic Constant"
] |
proofwiki-12662 | Magic Constant of Order 4 Magic Square | The magic constant of the order $4$ magic square is $34$. | Let $M_n$ denote the magic square of order $n$.
By Magic Constant of Magic Square, the magic constant of $M_n$ is given by:
:$S_n = \dfrac {n \left({n^2 + 1}\right)} 2$
Setting $n = 4$:
:$S_4 = \dfrac {4 \times 17} 2 = 34$
{{qed}} | The [[Definition:Magic Constant|magic constant]] of the [[Magic Square/Examples/Order 4|order $4$ magic square]] is $34$. | Let $M_n$ denote the [[Definition:Magic Square|magic square]] of [[Definition:Order of Magic Square|order $n$]].
By [[Magic Constant of Magic Square]], the [[Definition:Magic Constant|magic constant]] of $M_n$ is given by:
:$S_n = \dfrac {n \left({n^2 + 1}\right)} 2$
Setting $n = 4$:
:$S_4 = \dfrac {4 \times 17} 2 = ... | Magic Constant of Order 4 Magic Square/Proof 2 | https://proofwiki.org/wiki/Magic_Constant_of_Order_4_Magic_Square | https://proofwiki.org/wiki/Magic_Constant_of_Order_4_Magic_Square/Proof_2 | [
"Magic Squares",
"34",
"Magic Constant of Order 4 Magic Square"
] | [
"Definition:Magic Square/Magic Constant",
"Magic Square/Examples/Order 4"
] | [
"Definition:Magic Square",
"Definition:Magic Square/Order",
"Magic Constant of Magic Square",
"Definition:Magic Square/Magic Constant"
] |
proofwiki-12663 | Sum of Cubes on Diagonals of Moessner's Order 4 Magic Square | The sums of the cubes of the entries on the diagonals of Moessner's order $4$ magic square are equal. | Recall Moessner's order $4$ magic square:
{{:Definition:Moessner's Order 4 Magic Square}}
{{begin-eqn}}
{{eqn | l = 12^3 + 3^3 + 14^3 + 5^3
| r = 1728 + 27 + 2744 + 125
| c =
}}
{{eqn | r = 4624
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 9^3 + 2^3 + 15^3 + 8^3
| r = 729 + 8 + 3375 + 512
... | The [[Definition:Integer Addition|sums]] of the [[Definition:Cube (Algebra)|cubes]] of the entries on the diagonals of [[Definition:Moessner's Order 4 Magic Square|Moessner's order $4$ magic square]] are equal. | Recall [[Definition:Moessner's Order 4 Magic Square|Moessner's order $4$ magic square]]:
{{:Definition:Moessner's Order 4 Magic Square}}
{{begin-eqn}}
{{eqn | l = 12^3 + 3^3 + 14^3 + 5^3
| r = 1728 + 27 + 2744 + 125
| c =
}}
{{eqn | r = 4624
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 9^3 + 2^... | Sum of Cubes on Diagonals of Moessner's Order 4 Magic Square | https://proofwiki.org/wiki/Sum_of_Cubes_on_Diagonals_of_Moessner's_Order_4_Magic_Square | https://proofwiki.org/wiki/Sum_of_Cubes_on_Diagonals_of_Moessner's_Order_4_Magic_Square | [
"Magic Squares"
] | [
"Definition:Addition/Integers",
"Definition:Cube/Algebra",
"Magic Square/Examples/Order 4/Alfred Moessner"
] | [
"Magic Square/Examples/Order 4/Alfred Moessner"
] |
proofwiki-12664 | Sum of Squares on Pairs of Rows and Columns of Moessner's Order 4 Magic Square | The sums of the squares of the entries are equal on the following pairs of rows and columns of Moessner's order $4$ magic square:
:Rows $1$ and $4$
:Rows $2$ and $3$
:Columns $1$ and $4$
:Columns $2$ and $3$. | Recall Moessner's order $4$ magic square:
{{:Definition:Moessner's Order 4 Magic Square}} | The [[Definition:Integer Addition|sums]] of the [[Definition:Square (Algebra)|squares]] of the entries are equal on the following pairs of rows and columns of [[Definition:Moessner's Order 4 Magic Square|Moessner's order $4$ magic square]]:
:Rows $1$ and $4$
:Rows $2$ and $3$
:Columns $1$ and $4$
:Columns $2$ and $3$. | Recall [[Definition:Moessner's Order 4 Magic Square|Moessner's order $4$ magic square]]:
{{:Definition:Moessner's Order 4 Magic Square}} | Sum of Squares on Pairs of Rows and Columns of Moessner's Order 4 Magic Square | https://proofwiki.org/wiki/Sum_of_Squares_on_Pairs_of_Rows_and_Columns_of_Moessner's_Order_4_Magic_Square | https://proofwiki.org/wiki/Sum_of_Squares_on_Pairs_of_Rows_and_Columns_of_Moessner's_Order_4_Magic_Square | [
"Magic Squares"
] | [
"Definition:Addition/Integers",
"Definition:Square/Function",
"Magic Square/Examples/Order 4/Alfred Moessner"
] | [
"Magic Square/Examples/Order 4/Alfred Moessner"
] |
proofwiki-12665 | Equivalence of Definitions of Quasiperfect Number | The following definitions of a quasiperfect number are equivalent: | By definition of abundance:
:$\map A n = \map {\sigma_1} n - 2 n$
By definition of divisor sum function:
:$\map {\sigma_1} n$ is the sum of all the divisors of $n$.
Thus $\map {\sigma_1} n - n$ is the sum of the aliquot parts of $n$.
Hence the result.
{{qed}}
Category:Quasiperfect Numbers
doe8kd8xsd3rd8z2o3wt2ibh0ik4sq... | The following definitions of a [[Definition:Quasiperfect Number|quasiperfect number]] are [[Definition:Logical Equivalence|equivalent]]: | By definition of [[Definition:Abundance|abundance]]:
:$\map A n = \map {\sigma_1} n - 2 n$
By definition of [[Definition:Divisor Sum Function|divisor sum function]]:
:$\map {\sigma_1} n$ is the [[Definition:Integer Addition|sum]] of all the [[Definition:Divisor of Integer|divisors]] of $n$.
Thus $\map {\sigma_1} n -... | Equivalence of Definitions of Quasiperfect Number | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Quasiperfect_Number | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Quasiperfect_Number | [
"Quasiperfect Numbers"
] | [
"Definition:Quasiperfect Number",
"Definition:Logical Equivalence"
] | [
"Definition:Abundance",
"Definition:Divisor Sum Function",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer/Aliquot Part",
"Category:Quasiperfect Numbers"
] |
proofwiki-12666 | Quasiperfect Number is Square of Odd Integer | Let $n$ be a quasiperfect number.
Then:
:$n = \paren {2 k + 1}^2$
for some $k \in \Z_{>0}$.
That is, a quasiperfect number is the square of an odd integer. | By definition of quasiperfect number:
:$\map {\sigma_1} n = 2 n + 1$
where $\map {\sigma_1} n$ denotes the divisor sum of $n$.
That is, $\map {\sigma_1} n$ is odd.
Then from Divisor Sum is Odd iff Argument is Square or Twice Square:
$n$ is either square or twice a square.
Suppose $n = 2^k m^2$ is a quasiperfect number,... | Let $n$ be a [[Definition:Quasiperfect Number|quasiperfect number]].
Then:
:$n = \paren {2 k + 1}^2$
for some $k \in \Z_{>0}$.
That is, a [[Definition:Quasiperfect Number|quasiperfect number]] is the [[Definition:Square Number|square]] of an [[Definition:Odd Integer|odd integer]]. | By definition of [[Definition:Quasiperfect Number|quasiperfect number]]:
:$\map {\sigma_1} n = 2 n + 1$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum]] of $n$.
That is, $\map {\sigma_1} n$ is [[Definition:Odd Integer|odd]].
Then from [[Divisor Sum is Odd iff Argument is Square or... | Quasiperfect Number is Square of Odd Integer | https://proofwiki.org/wiki/Quasiperfect_Number_is_Square_of_Odd_Integer | https://proofwiki.org/wiki/Quasiperfect_Number_is_Square_of_Odd_Integer | [
"Quasiperfect Numbers"
] | [
"Definition:Quasiperfect Number",
"Definition:Quasiperfect Number",
"Definition:Square Number",
"Definition:Odd Integer"
] | [
"Definition:Quasiperfect Number",
"Definition:Divisor Sum Function",
"Definition:Odd Integer",
"Divisor Sum is Odd iff Argument is Square or Twice Square",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Quasiperfect Number",
"Definition:Odd Integer",
"Divisor Sum Function is Mul... |
proofwiki-12667 | Bottom in Ordered Set of Topology | Let $T = \struct {S, \tau}$ be a topological space.
Let $P = \struct {\tau, \subseteq}$ be an inclusion ordered set of $\tau$.
Then $P$ is bounded below and $\bot_P = \O$ | By Empty Set is Element of Topology:
:$\O \in \tau$
By Empty Set is Subset of All Sets:
:$\forall A \in \tau: \O \subseteq A$
Hence $P$ is bounded below.
Thus by definition of the smallest element:
:$\bot_P = \O$
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $P = \struct {\tau, \subseteq}$ be an [[Definition:Subset|inclusion]] [[Definition:Ordered Set|ordered set]] of $\tau$.
Then $P$ is [[Definition:Bounded Below Set|bounded below]] and $\bot_P = \O$ | By [[Empty Set is Element of Topology]]:
:$\O \in \tau$
By [[Empty Set is Subset of All Sets]]:
:$\forall A \in \tau: \O \subseteq A$
Hence $P$ is [[Definition:Bounded Below Set|bounded below]].
Thus by definition of the [[Definition:Smallest Element|smallest element]]:
:$\bot_P = \O$
{{qed}} | Bottom in Ordered Set of Topology | https://proofwiki.org/wiki/Bottom_in_Ordered_Set_of_Topology | https://proofwiki.org/wiki/Bottom_in_Ordered_Set_of_Topology | [
"Topology",
"Order Theory"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Ordered Set",
"Definition:Bounded Below Set"
] | [
"Empty Set is Element of Topology",
"Empty Set is Subset of All Sets",
"Definition:Bounded Below Set",
"Definition:Smallest Element"
] |
proofwiki-12668 | Long Period Prime/Examples/17 | The prime number $17$ is a long period prime:
:$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$ | From Reciprocal of $17$:
{{:Reciprocal of 17}}
Counting the digits, it is seen that this has a period of recurrence of $16$.
Hence the result.
{{qed}} | The [[Definition:Prime Number|prime number]] $17$ is a [[Definition:Long Period Prime|long period prime]]:
:$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$ | From [[Reciprocal of 17|Reciprocal of $17$]]:
{{:Reciprocal of 17}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $16$.
Hence the result.
{{qed}} | Long Period Prime/Examples/17 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/17 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/17 | [
"17",
"Examples of Long Period Primes"
] | [
"Definition:Prime Number",
"Definition:Long Period Prime"
] | [
"Reciprocal of 17",
"Definition:Basis Expansion/Recurrence/Period"
] |
proofwiki-12669 | Prime Dudeney Number | The only prime Dudeney number is $17$: | We have that:
{{begin-eqn}}
{{eqn | l = 17^3
| r = 4913
| c =
}}
{{eqn | l = 17
| r = 4 + 9 + 1 + 3
| c =
}}
{{end-eqn}}
From Sequence of Dudeney Numbers, the complete set of positive integers with this property are:
:$0, 1, 8, 17, 18, 26, 27$
Of these, only $17$ is prime.
{{qed}} | The only [[Definition:Prime Number|prime]] [[Definition:Dudeney Number|Dudeney number]] is $17$: | We have that:
{{begin-eqn}}
{{eqn | l = 17^3
| r = 4913
| c =
}}
{{eqn | l = 17
| r = 4 + 9 + 1 + 3
| c =
}}
{{end-eqn}}
From [[Sequence of Dudeney Numbers]], the complete [[Definition:Set|set]] of [[Definition:Positive Integer|positive integers]] with this property are:
:$0, 1, 8, 17, 18, ... | Prime Dudeney Number | https://proofwiki.org/wiki/Prime_Dudeney_Number | https://proofwiki.org/wiki/Prime_Dudeney_Number | [
"17",
"Prime Numbers",
"Dudeney Numbers"
] | [
"Definition:Prime Number",
"Definition:Dudeney Number"
] | [
"Sequence of Dudeney Numbers",
"Definition:Set",
"Definition:Positive/Integer",
"Definition:Prime Number"
] |
proofwiki-12670 | Sequence of Dudeney Numbers | The only Dudeney numbers are:
:$0, 1, 8, 17, 18, 26, 27$
two of which are themselves cubes, and one of which is prime. | We have trivially that:
{{begin-eqn}}
{{eqn | l = 0^3
| r = 0
| c =
}}
{{eqn | l = 1^3
| r = 1
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = 8^3
| r = 512
| c =
}}
{{eqn | l = 8
| r = 5 + 1 + 2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 17^3
| r = 491... | The only [[Definition:Dudeney Number|Dudeney numbers]] are:
:$0, 1, 8, 17, 18, 26, 27$
two of which are themselves [[Definition:Cube Number|cubes]], and one of which is [[Definition:Prime Number|prime]]. | We have trivially that:
{{begin-eqn}}
{{eqn | l = 0^3
| r = 0
| c =
}}
{{eqn | l = 1^3
| r = 1
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = 8^3
| r = 512
| c =
}}
{{eqn | l = 8
| r = 5 + 1 + 2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 17^3
| r... | Sequence of Dudeney Numbers | https://proofwiki.org/wiki/Sequence_of_Dudeney_Numbers | https://proofwiki.org/wiki/Sequence_of_Dudeney_Numbers | [
"Dudeney Numbers",
"8",
"17",
"18",
"26",
"27"
] | [
"Definition:Dudeney Number",
"Definition:Cube Number",
"Definition:Prime Number"
] | [
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Cube Number",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Digit",
"Bernoulli's Inequality"
] |
proofwiki-12671 | Derivation of Hamilton-Jacobi Equation | Let $\map S {x_0, x_1, \mathbf y} = \map S {x, \mathbf y}$ be the geodetic distance, where $x_0$ is fixed and $x_1=x$.
Let $H$ be Hamiltonian.
Then the following equation holds:
:$\dfrac {\partial S} {\partial x} + \map H {x, \mathbf y, \nabla_{\mathbf y} S} = 0$
and is known as the Hamilton-Jacobi Equation. | Consider the increment $\Delta S$:
:$\Delta S = \map S {x + \Delta x, \mathbf y + \Delta \mathbf y} - \map S {x, \mathbf y}$
Note that the change of function $\mathbf y$ denoted by $\Delta \mathbf y$ is dependent on the manner $\Delta x$ is chosen through the definition of geodetic distance.
For sufficiently smooth $S$... | Let $\map S {x_0, x_1, \mathbf y} = \map S {x, \mathbf y}$ be [[Definition:Geodetic Distance|the geodetic distance]], where $x_0$ is fixed and $x_1=x$.
Let $H$ be [[Definition:Hamiltonian|Hamiltonian]].
Then the following [[Definition:Differential Equation|equation]] holds:
:$\dfrac {\partial S} {\partial x} + \map... | Consider the increment $\Delta S$:
:$\Delta S = \map S {x + \Delta x, \mathbf y + \Delta \mathbf y} - \map S {x, \mathbf y}$
Note that the change of [[Definition:Function|function]] $\mathbf y$ denoted by $\Delta \mathbf y$ is dependent on the manner $\Delta x$ is chosen through [[Definition:Geodetic Distance|the def... | Derivation of Hamilton-Jacobi Equation | https://proofwiki.org/wiki/Derivation_of_Hamilton-Jacobi_Equation | https://proofwiki.org/wiki/Derivation_of_Hamilton-Jacobi_Equation | [
"Calculus of Variations"
] | [
"Definition:Geodetic Distance",
"Definition:Hamiltonian",
"Definition:Differential Equation",
"Definition:Hamilton-Jacobi Equation"
] | [
"Definition:Function",
"Definition:Geodetic Distance",
"Definition:Smooth Real Function",
"Definition:Differential of Mapping",
"Definition:Geodetic Distance",
"Definition:Extremum/Functional",
"Definition:Line/Curve",
"Definition:Point",
"Definition:Point",
"Definition:Increment/Functional",
"D... |
proofwiki-12672 | Partial Derivatives of Solution of Hamilton-Jacobi Equation are First Integrals of Euler's Equations | Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$, $\bsalpha = \sequence {\alpha_i}_{1 \mathop \le i \mathop \le m}$ be vectors, where $m \le n$.
Let $S = \map S {x, \mathbf y, \bsalpha}$ be a solution of the Hamilton-Jacobi equation, where $\bsalpha$ are parameters.
Then each partial derivative:
:$\dfr... | Consider the total derivative of $\dfrac {\partial S} {\partial \alpha_i}$ {{WRT|Differentiation}} $x$:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \frac {\partial S} {\partial \alpha_i}
| r = \frac {\partial^2 S} {\partial x \partial\alpha_i} + \sum_{j \mathop = 1}^n \frac {\partial^2 S} {\partial y_j \partial \al... | Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$, $\bsalpha = \sequence {\alpha_i}_{1 \mathop \le i \mathop \le m}$ be [[Definition:Vector (Linear Algebra)|vectors]], where $m \le n$.
Let $S = \map S {x, \mathbf y, \bsalpha}$ be [[Definition:Solution to Differential Equation|a solution]] of the [[Defi... | Consider the [[Definition:Total Derivative|total derivative]] of $\dfrac {\partial S} {\partial \alpha_i}$ {{WRT|Differentiation}} $x$:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \frac {\partial S} {\partial \alpha_i}
| r = \frac {\partial^2 S} {\partial x \partial\alpha_i} + \sum_{j \mathop = 1}^n \frac {\partia... | Partial Derivatives of Solution of Hamilton-Jacobi Equation are First Integrals of Euler's Equations | https://proofwiki.org/wiki/Partial_Derivatives_of_Solution_of_Hamilton-Jacobi_Equation_are_First_Integrals_of_Euler's_Equations | https://proofwiki.org/wiki/Partial_Derivatives_of_Solution_of_Hamilton-Jacobi_Equation_are_First_Integrals_of_Euler's_Equations | [
"Calculus of Variations"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Differential Equation/Solution",
"Definition:Hamilton-Jacobi Equation",
"Definition:Parameter of Differential Equation",
"Definition:Partial Derivative",
"Definition:First Integral of System of Differential Equations",
"Euler's Equation for Vanishing Varia... | [
"Definition:Total Derivative",
"Definition:Total Derivative",
"Definition:Parameter of Differential Equation",
"Definition:Differential Equation/Solution",
"Definition:Hamilton-Jacobi Equation",
"Definition:Partial Derivative",
"Derivative of Composite Function",
"Definition:Hamilton-Jacobi Equation",... |
proofwiki-12673 | Jacobi's Theorem | Let $\mathbf y = \sequence {y_i}_{1 \le i \le n}$, $\boldsymbol \alpha = \sequence {\alpha_i}_{1 \le i \le n}$, $\boldsymbol \beta = \sequence {\beta_i}_{1 \le i \le n}$ be vectors, where $\alpha_i$ and $ \beta_i$ are parameters.
Let $S = \map S {x, \mathbf y, \boldsymbol \alpha}$ be a a complete solution of the Hamilt... | Consider canonical Euler's equations:
:$\dfrac {\d y_i} {\d x} = \dfrac {\partial H} {\partial p_i}$
:$\dfrac {\d p_i} {\d x} = -\dfrac {\partial H} {\partial y_i}$
Apply a canonical transformation:
:$\tuple {x, \mathbf y, \mathbf p, H} \to \tuple {x, \boldsymbol \alpha, \boldsymbol \beta, H^*}$
where $\Phi = S$.
By Co... | Let $\mathbf y = \sequence {y_i}_{1 \le i \le n}$, $\boldsymbol \alpha = \sequence {\alpha_i}_{1 \le i \le n}$, $\boldsymbol \beta = \sequence {\beta_i}_{1 \le i \le n}$ be [[Definition:Vector (Linear Algebra)|vectors]], where $\alpha_i$ and $ \beta_i$ are [[Definition:Parameter of Differential Equation|parameters]].
... | Consider [[Euler's Equation for Vanishing Variation in Canonical Variables|canonical Euler's equations]]:
:$\dfrac {\d y_i} {\d x} = \dfrac {\partial H} {\partial p_i}$
:$\dfrac {\d p_i} {\d x} = -\dfrac {\partial H} {\partial y_i}$
Apply a [[Definition:Canonical Transformation|canonical transformation]]:
:$\tuple {x... | Jacobi's Theorem/Proof 2 | https://proofwiki.org/wiki/Jacobi's_Theorem | https://proofwiki.org/wiki/Jacobi's_Theorem/Proof_2 | [
"Calculus of Variations",
"Jacobi's Theorem"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Parameter of Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Hamilton-Jacobi Equation",
"Definition:Determinant",
"Definition:Differential Equation/Solution/General Solution",
"Euler's Equation for Vanishin... | [
"Euler's Equation for Vanishing Variation in Canonical Variables",
"Definition:Canonical Transformation",
"Conditions for Transformation to be Canonical",
"Definition:Differential Equation/Solution",
"Definition:Hamilton-Jacobi Equation",
"Definition:Coordinate",
"Euler's Equation for Vanishing Variatio... |
proofwiki-12674 | Equivalence of Definitions of Principal Ideal of Preordered Set | Let $\struct {S, \preceq}$ be a preordered set.
Let $I$ be an ideal in $S$.
{{TFAE|def = Principal Ideal of Preordered Set}} | === Definition $1$ implies Definition $2$ ===
Assume that
:$\exists x \in I: x$ is upper bound for $I$
We will prove that
:$I \subseteq x^\preceq$
Let $y \in I$.
By definition of upper bound:
:$y \preceq x$
Thus by definition of lower closure of element:
:$y \in x^\preceq$
{{qed|lemma}}
We will prove that
:$x^\preceq \... | Let $\struct {S, \preceq}$ be a [[Definition:Preordered Set|preordered set]].
Let $I$ be an [[Definition:Ideal (Order Theory)|ideal]] in $S$.
{{TFAE|def = Principal Ideal of Preordered Set}} | === Definition $1$ implies Definition $2$ ===
Assume that
:$\exists x \in I: x$ is [[Definition:Upper Bound of Set|upper bound]] for $I$
We will prove that
:$I \subseteq x^\preceq$
Let $y \in I$.
By definition of [[Definition:Upper Bound of Set|upper bound]]:
:$y \preceq x$
Thus by definition of [[Definition:Lower... | Equivalence of Definitions of Principal Ideal of Preordered Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Principal_Ideal_of_Preordered_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Principal_Ideal_of_Preordered_Set | [
"Principal Ideals of Preordered Sets"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Ideal (Order Theory)"
] | [
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Lower Closure/Element",
"Definition:Lower Closure/Element",
"Definition:Lower Section",
"Definition:Set Equality",
"Definition:Lower Closure/Element",
"Definition:Lower Closure/Element"
] |
proofwiki-12675 | Seventeen Different Wallpaper Patterns | There are $17$ different symmetry groups for a wallpaper pattern. | {{ProofWanted|Lots of background work needed.}} | There are $17$ different [[Definition:Symmetry Group|symmetry groups]] for a [[Definition:Wallpaper Pattern|wallpaper pattern]]. | {{ProofWanted|Lots of background work needed.}} | Seventeen Different Wallpaper Patterns | https://proofwiki.org/wiki/Seventeen_Different_Wallpaper_Patterns | https://proofwiki.org/wiki/Seventeen_Different_Wallpaper_Patterns | [
"Wallpaper Patterns",
"Symmetry Groups",
"17"
] | [
"Definition:Symmetry Group",
"Definition:Wallpaper Pattern"
] | [] |
proofwiki-12676 | Smallest Odd Number not of form 2 a squared plus p | $17$ is the smallest odd number $n$ greater than $3$ which cannot be expressed in the form:
:$n = 2 a^2 + p$
where:
:$p$ is prime
:$a \in \Z_{>0}$ is a (strictly) positive integer. | First note that $3$ is not so expressible:
{{begin-eqn}}
{{eqn | l = 3 - 2 \times 1^2
| r = 1
| c = which is not prime
}}
{{end-eqn}}
and so $3$ cannot be written in such a form.
Then we have:
{{begin-eqn}}
{{eqn | l = 5
| r = 2 \times 1^2 + 3
| c =
}}
{{eqn | l = 7
| r = 2 \times 1^2 + 5... | $17$ is the smallest [[Definition:Odd Number|odd number]] $n$ greater than $3$ which cannot be expressed in the form:
:$n = 2 a^2 + p$
where:
:$p$ is [[Definition:Prime Number|prime]]
:$a \in \Z_{>0}$ is a [[Definition:Strictly Positive Integer|(strictly) positive integer]]. | First note that $3$ is not so expressible:
{{begin-eqn}}
{{eqn | l = 3 - 2 \times 1^2
| r = 1
| c = which is not [[Definition:Prime Number|prime]]
}}
{{end-eqn}}
and so $3$ cannot be written in such a form.
Then we have:
{{begin-eqn}}
{{eqn | l = 5
| r = 2 \times 1^2 + 3
| c =
}}
{{eqn | l... | Smallest Odd Number not of form 2 a squared plus p | https://proofwiki.org/wiki/Smallest_Odd_Number_not_of_form_2_a_squared_plus_p | https://proofwiki.org/wiki/Smallest_Odd_Number_not_of_form_2_a_squared_plus_p | [
"17",
"Prime Numbers",
"Polynomial Expressions for Primes",
"Goldbach's Lesser Conjecture"
] | [
"Definition:Odd Integer",
"Definition:Prime Number",
"Definition:Strictly Positive/Integer"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-12677 | Compact Element iff Principal Ideal | Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $P = \struct {\map {\operatorname {Ids} } L, \precsim}$ be an inclusion ordered set
where
:$\map {\operatorname {Ids} } L$ denotes the set of all ideals in $L$,
:$\mathord\precsim = \mathord\subseteq \cap \paren {\map {\operatorname {Ids} } L... | By Ideals are Continuous Lattice Subframe of Power Set:
:$P$ is continuous lattice subframe of $\struct {\powerset S, \subseteq'}$
where
:$\powerset S$ denotes the power set of $S$,
:$\mathord\subseteq' = \mathord\subseteq \cap \struct {\powerset S \times \powerset S}$ | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Let $P = \struct {\map {\operatorname {Ids} } L, \precsim}$ be an [[Definition:Subset|inclusion]] [[Definition:Ordered Set|ordered set]]
where
:$\map {\operatorname {Ids} } L$ den... | By [[Ideals are Continuous Lattice Subframe of Power Set]]:
:$P$ is [[Definition:Continuous Lattice Subframe|continuous lattice subframe]] of $\struct {\powerset S, \subseteq'}$
where
:$\powerset S$ denotes the [[Definition:Power Set|power set]] of $S$,
:$\mathord\subseteq' = \mathord\subseteq \cap \struct {\powerset ... | Compact Element iff Principal Ideal | https://proofwiki.org/wiki/Compact_Element_iff_Principal_Ideal | https://proofwiki.org/wiki/Compact_Element_iff_Principal_Ideal | [
"Join and Meet Semilattices",
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Subset",
"Definition:Ordered Set",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)",
"Definition:Compact Element",
"Definition:Principal Ideal of Preordered Set"
] | [
"Ideals are Continuous Lattice Subframe of Power Set",
"Definition:Continuous Lattice Subframe",
"Definition:Power Set"
] |
proofwiki-12678 | Integers as Sum of Three Pairwise Coprime Integers | Let $n$ be an integer greater than $17$.
Then $n$ is the sum of $3$ integers greater than $1$ which are pairwise coprime. | === Case $1$: $n$ is even ===
There is some integer $k > 2$ such that one of the following holds:
:$n = 6 k = 2 + 3 + \paren {6 k - 5}$
:$n = 6 k + 2 = 4 + 3 + \paren {6 k - 7}$
:$n = 6 k + 4 = 2 + 3 + \paren {6 k - 5}$
All terms are greater than $1$.
The first two terms are powers of $2$ and $3$, so they are coprime.
... | Let $n$ be an [[Definition:Integer|integer]] greater than $17$.
Then $n$ is the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Integer|integers]] greater than $1$ which are [[Definition:Pairwise Coprime Integers|pairwise coprime]]. | === Case $1$: $n$ is even ===
There is some [[Definition:Integer|integer]] $k > 2$ such that one of the following holds:
:$n = 6 k = 2 + 3 + \paren {6 k - 5}$
:$n = 6 k + 2 = 4 + 3 + \paren {6 k - 7}$
:$n = 6 k + 4 = 2 + 3 + \paren {6 k - 5}$
All terms are greater than $1$.
The first two terms are [[Definition:I... | Integers as Sum of Three Pairwise Coprime Integers | https://proofwiki.org/wiki/Integers_as_Sum_of_Three_Pairwise_Coprime_Integers | https://proofwiki.org/wiki/Integers_as_Sum_of_Three_Pairwise_Coprime_Integers | [
"Number Theory",
"17"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Integer",
"Definition:Pairwise Coprime/Integers"
] | [
"Definition:Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Coprime/Integers",
"Definition:Divisor",
"Definition:Pairwise Coprime/Integers",
"Definition:Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Divisor",
"Definition:Coprime/Integers",
"Definition:Coprime/Integers",
... |
proofwiki-12679 | Reversal formed by Repdigits of Base minus 1 by Addition and Multiplication | Let $b \in \Z_{>1}$ be an integer greater than $1$.
Let $n = b^k - 1$ for some integer $k$ such that $k \ge 1$.
Then:
: $n + n$ is the reversal of $\paren {b - 1} n$
when both are expressed in base $b$ representation. | By Power of Base minus 1 is Repdigit Base minus 1, $n$ is a repdigit number consisting of $k$ occurrences of $b - 1$.
Let $a = b - 1$.
Thus $n$ can be expressed in base $b$ as:
:$n = {\overbrace {\sqbrk {aaa \cdots a} }^k}_b$
We have that:
{{begin-eqn}}
{{eqn | l = n + n
| r = 2 n
| c =
}}
{{eqn | r = \sqb... | Let $b \in \Z_{>1}$ be an [[Definition:Integer|integer]] greater than $1$.
Let $n = b^k - 1$ for some [[Definition:Integer|integer]] $k$ such that $k \ge 1$.
Then:
: $n + n$ is the [[Definition:Reversal|reversal]] of $\paren {b - 1} n$
when both are expressed in [[Definition:Number Base|base $b$]] representation. | By [[Power of Base minus 1 is Repdigit Base minus 1]], $n$ is a [[Definition:Repdigit Number|repdigit number]] consisting of $k$ occurrences of $b - 1$.
Let $a = b - 1$.
Thus $n$ can be expressed in [[Definition:Number Base|base $b$]] as:
:$n = {\overbrace {\sqbrk {aaa \cdots a} }^k}_b$
We have that:
{{begin-eqn}}... | Reversal formed by Repdigits of Base minus 1 by Addition and Multiplication | https://proofwiki.org/wiki/Reversal_formed_by_Repdigits_of_Base_minus_1_by_Addition_and_Multiplication | https://proofwiki.org/wiki/Reversal_formed_by_Repdigits_of_Base_minus_1_by_Addition_and_Multiplication | [
"Number Bases",
"Reversal formed by Repdigits of Base minus 1 by Addition and Multiplication",
"Repdigit Numbers",
"Reversals"
] | [
"Definition:Integer",
"Definition:Integer",
"Definition:Reversal",
"Definition:Number Base"
] | [
"Power of Base minus 1 is Repdigit Base minus 1",
"Definition:Repdigit Number",
"Definition:Number Base",
"Multiple of Repdigit Base minus 1",
"Multiple of Repdigit Base minus 1"
] |
proofwiki-12680 | Power of Base minus 1 is Repdigit Base minus 1 | Let $b \in \Z_{>1}$ be an integer greater than $1$.
Let $n = b^k - 1$ for some integer $k$ such that $k \ge 1$.
Let $n$ be expressed in base $b$ representation.
Then $n$ is a repdigit number consisting of $k$ instances of digit $b - 1$. | {{begin-eqn}}
{{eqn | l = \dfrac {b^k - 1} {b - 1}
| r = \sum_{j \mathop = 0}^{k - 1} b^j
| c = Sum of Geometric Sequence
}}
{{eqn | ll= \leadsto
| l = n = b^k - 1
| r = \paren {b - 1} \sum_{j \mathop = 0}^{k - 1} b^j
| c =
}}
{{eqn | r = \sum_{j \mathop = 0}^{k - 1} \paren {b - 1} b^j
... | Let $b \in \Z_{>1}$ be an [[Definition:Integer|integer]] greater than $1$.
Let $n = b^k - 1$ for some [[Definition:Integer|integer]] $k$ such that $k \ge 1$.
Let $n$ be expressed in [[Definition:Number Base|base $b$]] representation.
Then $n$ is a [[Definition:Repdigit Number|repdigit number]] consisting of $k$ ins... | {{begin-eqn}}
{{eqn | l = \dfrac {b^k - 1} {b - 1}
| r = \sum_{j \mathop = 0}^{k - 1} b^j
| c = [[Sum of Geometric Sequence]]
}}
{{eqn | ll= \leadsto
| l = n = b^k - 1
| r = \paren {b - 1} \sum_{j \mathop = 0}^{k - 1} b^j
| c =
}}
{{eqn | r = \sum_{j \mathop = 0}^{k - 1} \paren {b - 1} b^... | Power of Base minus 1 is Repdigit Base minus 1 | https://proofwiki.org/wiki/Power_of_Base_minus_1_is_Repdigit_Base_minus_1 | https://proofwiki.org/wiki/Power_of_Base_minus_1_is_Repdigit_Base_minus_1 | [
"Repdigit Numbers",
"Number Bases"
] | [
"Definition:Integer",
"Definition:Integer",
"Definition:Number Base",
"Definition:Repdigit Number",
"Definition:Digit"
] | [
"Sum of Geometric Sequence",
"Definition:Number Base",
"Definition:Digit",
"Category:Repdigit Numbers",
"Category:Number Bases"
] |
proofwiki-12681 | Multiple of Repdigit Base minus 1 | Let $b \in \Z_{>1}$ be an integer greater than $1$.
Let $n$ be a repdigit number of $k$ instances of the digit $b - 1$ for some integer $k$ such that $k \ge 1$.
Let $m \in \Z_{>1}$ be an integer such that $1 < m < b$.
Then $m \times n$, when expressed in base $b$, is of the form:
:$m n = \sqbrk {r d d \cdots d s}_b$
wh... | {{begin-eqn}}
{{eqn | l = n
| r = \sum_{j \mathop = 0}^{k - 1} \paren {b - 1} b^j
| c = Basis Representation Theorem
}}
{{eqn | r = b^k - 1
| c = Sum of Geometric Sequence
}}
{{eqn | ll= \leadsto
| l = m n
| r = m \paren {b^k - 1}
| c =
}}
{{eqn | r = \paren {m - 1} b^k + b^k - 1 + ... | Let $b \in \Z_{>1}$ be an [[Definition:Integer|integer]] greater than $1$.
Let $n$ be a [[Definition:Repdigit Number|repdigit number]] of $k$ instances of the [[Definition:Digit|digit]] $b - 1$ for some [[Definition:Integer|integer]] $k$ such that $k \ge 1$.
Let $m \in \Z_{>1}$ be an [[Definition:Integer|integer]] su... | {{begin-eqn}}
{{eqn | l = n
| r = \sum_{j \mathop = 0}^{k - 1} \paren {b - 1} b^j
| c = [[Basis Representation Theorem]]
}}
{{eqn | r = b^k - 1
| c = [[Sum of Geometric Sequence]]
}}
{{eqn | ll= \leadsto
| l = m n
| r = m \paren {b^k - 1}
| c =
}}
{{eqn | r = \paren {m - 1} b^k + b^... | Multiple of Repdigit Base minus 1 | https://proofwiki.org/wiki/Multiple_of_Repdigit_Base_minus_1 | https://proofwiki.org/wiki/Multiple_of_Repdigit_Base_minus_1 | [
"Repdigit Numbers",
"Multiple of Repdigit Base minus 1"
] | [
"Definition:Integer",
"Definition:Repdigit Number",
"Definition:Digit",
"Definition:Integer",
"Definition:Integer",
"Definition:Number Base"
] | [
"Basis Representation Theorem",
"Sum of Geometric Sequence",
"Sum of Geometric Sequence"
] |
proofwiki-12682 | Only Number Twice Sum of Digits is 18 | There exists only one (strictly) positive integer that is exactly twice the sum of its digits. | Let $n$ be equal to twice the sum of its digits.
Suppose $n$ has $1$ digit.
Then $n$ equals the sum of its digits.
Thus $n$ does not have only $1$ digit.
Suppose $n$ has $3$ digits.
Then $n > 99$.
But the highest number that can be formed by the sum of $3$ digits is $9 + 9 + 9 = 27$.
Similarly for if $n$ has more than ... | There exists only one [[Definition:Strictly Positive Integer|(strictly) positive integer]] that is exactly twice the [[Definition:Integer Addition|sum]] of its [[Definition:Digit|digits]]. | Let $n$ be equal to twice the [[Definition:Integer Addition|sum]] of its [[Definition:Digit|digits]].
Suppose $n$ has $1$ [[Definition:Digit|digit]].
Then $n$ equals the [[Definition:Integer Addition|sum]] of its [[Definition:Digit|digits]].
Thus $n$ does not have only $1$ [[Definition:Digit|digit]].
Suppose $n$ h... | Only Number Twice Sum of Digits is 18 | https://proofwiki.org/wiki/Only_Number_Twice_Sum_of_Digits_is_18 | https://proofwiki.org/wiki/Only_Number_Twice_Sum_of_Digits_is_18 | [
"18"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Addition/Integers",
"Definition:Digit"
] | [
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit"
] |
proofwiki-12683 | Long Period Prime/Examples/19 | The prime number $19$ is a long period prime:
:$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$ | From Reciprocal of $19$:
{{:Reciprocal of 19}}
Counting the digits, it is seen that this has a period of recurrence of $18$.
Hence the result.
{{qed}} | The [[Definition:Prime Number|prime number]] $19$ is a [[Definition:Long Period Prime|long period prime]]:
:$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$ | From [[Reciprocal of 19|Reciprocal of $19$]]:
{{:Reciprocal of 19}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $18$.
Hence the result.
{{qed}} | Long Period Prime/Examples/19 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/19 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/19 | [
"19",
"Examples of Long Period Primes"
] | [
"Definition:Prime Number",
"Definition:Long Period Prime"
] | [
"Reciprocal of 19",
"Definition:Basis Expansion/Recurrence/Period"
] |
proofwiki-12684 | Divisibility by 19 | Let $n$ be an integer expressed in the form:
:$n = 100 a + b$
Then $n$ is divisible by $19$ {{iff}} $a + 4 b$ is divisible by $19$. | Let $a, b, c \in \Z$.
{{begin-eqn}}
{{eqn | l = 100 a + b
| r = 19 c
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = 400 a + 4 b
| r = 19 \paren {4 c}
| c = Multiply by $4$
}}
{{eqn | ll= \leadstoandfrom
| l = 399 a + a + 4 b
| r = 19 \paren {4 c}
| c = Separate the $a$ value... | Let $n$ be an [[Definition:Integer|integer]] expressed in the form:
:$n = 100 a + b$
Then $n$ is [[Definition:Divisor of Integer|divisible]] by $19$ {{iff}} $a + 4 b$ is [[Definition:Divisor of Integer|divisible]] by $19$. | Let $a, b, c \in \Z$.
{{begin-eqn}}
{{eqn | l = 100 a + b
| r = 19 c
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = 400 a + 4 b
| r = 19 \paren {4 c}
| c = Multiply by $4$
}}
{{eqn | ll= \leadstoandfrom
| l = 399 a + a + 4 b
| r = 19 \paren {4 c}
| c = Separate the $a$ valu... | Divisibility by 19 | https://proofwiki.org/wiki/Divisibility_by_19 | https://proofwiki.org/wiki/Divisibility_by_19 | [
"19",
"Divisibility Tests"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-12685 | Diameter of Closure of Subset is Diameter of Subset | Let $M = \struct {A, d}$ be a metric space.
Let $S \subseteq A$ be bounded in $M$.
Then:
:$\map \diam S = \map \diam {S^-}$
where $\map \diam S$ denotes the diameter of $S$, and $S^-$ denotes the closure of $S$ in $M$. | {{AimForCont}} that $\map \diam S \ne \map \diam {S^-}$.
$S \subseteq S^-$ by Subset of Metric Space is Subset of its Closure, so it then follows that:
:$\map \diam S < \map \diam {S^-}$
Then there exists $x, y \in S^-$ such that $\map d {x, y} > \map \diam S$.
By Point in Closure of Subset of Metric Space iff Limit of... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $S \subseteq A$ be [[Definition:Bounded Metric Space|bounded]] in $M$.
Then:
:$\map \diam S = \map \diam {S^-}$
where $\map \diam S$ denotes the [[Definition:Diameter of Subset of Metric Space|diameter]] of $S$, and $S^-$ denotes the [[Defi... | {{AimForCont}} that $\map \diam S \ne \map \diam {S^-}$.
$S \subseteq S^-$ by [[Subset of Metric Space is Subset of its Closure]], so it then follows that:
:$\map \diam S < \map \diam {S^-}$
Then there exists $x, y \in S^-$ such that $\map d {x, y} > \map \diam S$.
By [[Point in Closure of Subset of Metric Space i... | Diameter of Closure of Subset is Diameter of Subset | https://proofwiki.org/wiki/Diameter_of_Closure_of_Subset_is_Diameter_of_Subset | https://proofwiki.org/wiki/Diameter_of_Closure_of_Subset_is_Diameter_of_Subset | [
"Metric Spaces",
"Set Closures"
] | [
"Definition:Metric Space",
"Definition:Bounded Metric Space",
"Definition:Diameter of Subset of Metric Space",
"Definition:Closure (Topology)/Metric Space"
] | [
"Subset of Metric Space is Subset of its Closure",
"Point in Closure of Subset of Metric Space iff Limit of Sequence",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Metric Space/Metric",
"Definition:Convergent Sequence/Metric Space",
"Distance Function of Metric Space... |
proofwiki-12686 | Magic Hexagon of Order 3 is Unique | Apart from the trivial order $1$ magic hexagon, there exists only one magic hexagon: the order $3$ magic hexagon:
{{:Order 3 Magic Hexagon}} | We first prove that only order $3$ magic hexagons exist apart from the order $1$ magic hexagon, by calculating the magic constant for each order.
If we ignore the center tile, the rest of the hexagon can be divided into $3$ parallelogram-like parts:
For example, the above can be divided into $\set {3, 17, 18, 19, 7, 1}... | Apart from the trivial [[Order 1 Magic Hexagon|order $1$ magic hexagon]], there exists only one [[Definition:Magic Hexagon|magic hexagon]]: the [[Order 3 Magic Hexagon|order $3$ magic hexagon]]:
{{:Order 3 Magic Hexagon}} | We first prove that only [[Order 3 Magic Hexagon|order $3$ magic hexagons]] exist apart from the [[Order 1 Magic Hexagon|order $1$ magic hexagon]], by calculating the magic constant for each order.
If we ignore the center tile, the rest of the hexagon can be divided into $3$ parallelogram-like parts:
For example, th... | Magic Hexagon of Order 3 is Unique | https://proofwiki.org/wiki/Magic_Hexagon_of_Order_3_is_Unique | https://proofwiki.org/wiki/Magic_Hexagon_of_Order_3_is_Unique | [
"Magic Hexagons"
] | [
"Magic Hexagon/Examples/Order 1",
"Definition:Magic Hexagon",
"Magic Hexagon/Examples/Order 3"
] | [
"Magic Hexagon/Examples/Order 3",
"Magic Hexagon/Examples/Order 1",
"Definition:Addition/Integers",
"Definition:Addition/Integers",
"Sum of Arithmetic Sequence",
"Definition:Addition/Integers",
"Definition:Addition/Integers",
"Definition:Integer",
"Definition:Integer",
"Definition:Integer",
"Def... |
proofwiki-12687 | Magic Constant of Order 3 Magic Hexagon | The magic constant of the order 3 magic hexagon is $38$.
It is noted that the central cell is $5$, the same as that of the order 3 magic square. | {{ProofWanted|Plenty background work needed}} | The [[Definition:Magic Constant of Magic Hexagon|magic constant]] of the [[Order 3 Magic Hexagon|order 3 magic hexagon]] is $38$.
It is noted that the central cell is $5$, the same as that of the [[Definition:Order 3 Magic Square|order 3 magic square]]. | {{ProofWanted|Plenty background work needed}} | Magic Constant of Order 3 Magic Hexagon | https://proofwiki.org/wiki/Magic_Constant_of_Order_3_Magic_Hexagon | https://proofwiki.org/wiki/Magic_Constant_of_Order_3_Magic_Hexagon | [
"Magic Hexagons",
"38"
] | [
"Definition:Magic Hexagon/Magic Constant",
"Magic Hexagon/Examples/Order 3",
"Magic Square/Examples/Order 3"
] | [] |
proofwiki-12688 | Sum of Sequence of Alternating Positive and Negative Factorials being Prime | Let $n \in \Z_{\ge 0}$ be a positive integer.
Let:
{{begin-eqn}}
{{eqn | l = m
| r = \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!
| c =
}}
{{eqn | r = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1
| c =
}}
{{end-eqn}}
The sequence of $n$ such that $m$ is prime... | Let $\map f n$ be defined as:
:$\map f n := \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$
First we observe that for $n > 1$:
:$\map f n := n! - \map f {n - 1}$
We have:
{{begin-eqn}}
{{eqn | l = \map f 1
| r = 1!
| c =
}}
{{eqn | r = 1
| c = which is not prime
}}
{{end-eqn}}
{{begin... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Let:
{{begin-eqn}}
{{eqn | l = m
| r = \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!
| c =
}}
{{eqn | r = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1
| c =
}}
{{end-eqn}}
The [[D... | Let $\map f n$ be defined as:
:$\map f n := \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$
First we observe that for $n > 1$:
:$\map f n := n! - \map f {n - 1}$
We have:
{{begin-eqn}}
{{eqn | l = \map f 1
| r = 1!
| c =
}}
{{eqn | r = 1
| c = which is not [[Definition:Prime Numbe... | Sum of Sequence of Alternating Positive and Negative Factorials being Prime | https://proofwiki.org/wiki/Sum_of_Sequence_of_Alternating_Positive_and_Negative_Factorials_being_Prime | https://proofwiki.org/wiki/Sum_of_Sequence_of_Alternating_Positive_and_Negative_Factorials_being_Prime | [
"Factorials",
"Prime Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Integer Sequence",
"Definition:Prime Number",
"Definition:Integer Sequence"
] | [
"Definition:Prime Number",
"Factorial/Examples",
"Definition:Prime Number",
"Factorial/Examples",
"Definition:Prime Number",
"Factorial/Examples",
"Definition:Prime Number",
"Factorial/Examples",
"Definition:Prime Number",
"Factorial/Examples",
"Definition:Prime Number",
"Factorial/Examples",
... |
proofwiki-12689 | Prime to Own Power minus 1 over Prime minus 1 being Prime | Let $n \in \Z_{>1}$ be an integer greater than $1$.
Then $\dfrac {n^n - 1} {n - 1}$ is a prime for $n$ equal to:
:$2, 3, 19, 31$
{{OEIS|A088790}} | {{ProofWanted|Can easily be proved that $n$ must itself be prime for the expression to be prime. Then it's a matter of checking them all.}}
Note that if $4 p + 1$ is prime for prime $p$, then $\dfrac {p^p - 1} {p - 1}$ is divisible by $4 p + 1$:
Let $q = 4 p + 1$ be prime.
By First Supplement to Law of Quadratic Recipr... | Let $n \in \Z_{>1}$ be an [[Definition:Integer|integer]] greater than $1$.
Then $\dfrac {n^n - 1} {n - 1}$ is a [[Definition:Prime Number|prime]] for $n$ equal to:
:$2, 3, 19, 31$
{{OEIS|A088790}} | {{ProofWanted|Can easily be proved that $n$ must itself be prime for the expression to be prime. Then it's a matter of checking them all.}}
Note that if $4 p + 1$ is prime for prime $p$, then $\dfrac {p^p - 1} {p - 1}$ is divisible by $4 p + 1$:
Let $q = 4 p + 1$ be prime.
By [[First Supplement to Law of Quadratic ... | Prime to Own Power minus 1 over Prime minus 1 being Prime | https://proofwiki.org/wiki/Prime_to_Own_Power_minus_1_over_Prime_minus_1_being_Prime | https://proofwiki.org/wiki/Prime_to_Own_Power_minus_1_over_Prime_minus_1_being_Prime | [
"Prime Numbers"
] | [
"Definition:Integer",
"Definition:Prime Number"
] | [
"First Supplement to Law of Quadratic Reciprocity",
"Square of Sum",
"Congruence of Powers",
"Fermat's Little Theorem"
] |
proofwiki-12690 | Squares Ending in 5 Occurrences of 2-Digit Pattern | Let $n$ be a square number whose decimal representation ends in the pattern $\mathtt {xyxyxyxyxy}$.
Then $\mathtt {xy}$ is one of:
:$21, 29, 61, 69, 84$
The smallest examples of such numbers are:
{{begin-eqn}}
{{eqn | l = 508 \, 853 \, 989^2
| r = \phantom {0 \,} 258 \, 932 \, 38 \mathbf {2 \, 121 \, 212 \, 121}
... | === Case $1$: $\mathtt {xy}$ is odd ===
Consider the last $3$ digits of $n$: $\mathtt {yxy}$.
By Odd Square Modulo 8:
:$n \equiv \mathtt {yxy} \equiv 1 \pmod 8$
By Square Modulo 5:
:$n \equiv \mathtt {yxy} \equiv 0, 1, 4 \pmod 5$
We have:
{{begin-eqn}}
{{eqn | l = \mathtt {yxy}
| o = \equiv
| r = 25, 1, 9
... | Let $n$ be a [[Definition:Square Number|square number]] whose [[Definition:Decimal Expansion|decimal representation]] ends in the pattern $\mathtt {xyxyxyxyxy}$.
Then $\mathtt {xy}$ is one of:
:$21, 29, 61, 69, 84$
The smallest examples of such numbers are:
{{begin-eqn}}
{{eqn | l = 508 \, 853 \, 989^2
| r = \p... | === Case $1$: $\mathtt {xy}$ is odd ===
Consider the last $3$ [[Definition:Digit|digits]] of $n$: $\mathtt {yxy}$.
By [[Odd Square Modulo 8]]:
:$n \equiv \mathtt {yxy} \equiv 1 \pmod 8$
By [[Square Modulo 5]]:
:$n \equiv \mathtt {yxy} \equiv 0, 1, 4 \pmod 5$
We have:
{{begin-eqn}}
{{eqn | l = \mathtt {yxy}
| ... | Squares Ending in 5 Occurrences of 2-Digit Pattern | https://proofwiki.org/wiki/Squares_Ending_in_5_Occurrences_of_2-Digit_Pattern | https://proofwiki.org/wiki/Squares_Ending_in_5_Occurrences_of_2-Digit_Pattern | [
"Number Theory",
"Recreational Mathematics"
] | [
"Definition:Square Number",
"Definition:Decimal Expansion"
] | [
"Definition:Digit",
"Odd Square Modulo 8",
"Square Modulo 5",
"Chinese Remainder Theorem",
"Definition:Odd Integer",
"Definition:Decimal Expansion",
"Divisibility by 5",
"Definition:Prime Number",
"Odd Square Modulo 8",
"Chinese Remainder Theorem",
"Definition:Digit",
"Square Modulo 5",
"Chi... |
proofwiki-12691 | Ideals are Continuous Lattice Subframe of Power Set | Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $I = \paren {\map {\operatorname{Ids} } L, \precsim}$ be an inclusion ordered set
where
:$\map {\operatorname{Ids} } L$ denotes the set of all ideals in $L$,
:$\mathord \precsim = \mathord \subseteq \cap \paren {\map {\operatorname{Ids} } L \... | By definition of subset:
:$\map {\operatorname{Ids} } L \subseteq \powerset S$
Then
:$\mathord \precsim = \mathord \precsim' \cap \paren {\map {\operatorname{Ids} } L \times \map {\operatorname{Ids} } L}$
Hence $I$ is ordered subset of $P$. | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Let $I = \paren {\map {\operatorname{Ids} } L, \precsim}$ be an [[Definition:Subset|inclusion]] [[Definition:Ordered Set|ordered set]]
where
:$\map {\operatorname{Ids} } L$ denote... | By definition of [[Definition:Subset|subset]]:
:$\map {\operatorname{Ids} } L \subseteq \powerset S$
Then
:$\mathord \precsim = \mathord \precsim' \cap \paren {\map {\operatorname{Ids} } L \times \map {\operatorname{Ids} } L}$
Hence $I$ is [[Definition:Ordered Subset|ordered subset]] of $P$. | Ideals are Continuous Lattice Subframe of Power Set | https://proofwiki.org/wiki/Ideals_are_Continuous_Lattice_Subframe_of_Power_Set | https://proofwiki.org/wiki/Ideals_are_Continuous_Lattice_Subframe_of_Power_Set | [
"Join and Meet Semilattices"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Subset",
"Definition:Ordered Set",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)",
"Definition:Subset",
"Definition:Ordered Set",
"Definition:Power Set",
"Definition:Continuous Lattice Subframe"
] | [
"Definition:Subset",
"Definition:Ordered Subset",
"Definition:Subset",
"Definition:Subset"
] |
proofwiki-12692 | Perfect Square Dissection of Lowest Order | The perfect square dissection of lowest order is of an integer square of side $112$ into $21$ parts. | :700px
{{ProofWanted|That this is the smallest still needs to be proved.}} | The [[Definition:Perfect Square Dissection|perfect square dissection]] of lowest [[Definition:Order of Perfect Square Dissection|order]] is of an [[Definition:Integer Square|integer square]] of [[Definition:Side of Polygon|side]] $112$ into $21$ parts. | :[[File:SmallestPerfectSquareDissection.png|700px]]
{{ProofWanted|That this is the smallest still needs to be proved.}} | Perfect Square Dissection of Lowest Order | https://proofwiki.org/wiki/Perfect_Square_Dissection_of_Lowest_Order | https://proofwiki.org/wiki/Perfect_Square_Dissection_of_Lowest_Order | [
"Perfect Square Dissections",
"21",
"112",
"Recreational Mathematics"
] | [
"Definition:Perfect Square Dissection",
"Definition:Perfect Square Dissection/Order",
"Definition:Integer Square",
"Definition:Polygon/Side"
] | [
"File:SmallestPerfectSquareDissection.png"
] |
proofwiki-12693 | Smallest Number Expressible as Sum of at most Three Triangular Numbers in 4 ways | $21$ is the smallest number which can be expressed as the sum of at most $3$ triangular numbers in $4$ ways. | By inspection:
{{begin-eqn}}
{{eqn | l = 1
| r = T_1
| c = $1$ way
}}
{{eqn | l = 2
| r = T_1 + T_1
| c = $1$ way
}}
{{eqn | l = 3
| r = T_1 + T_1 + T_1
| c =
}}
{{eqn | r = T_2
| c = $2$ ways
}}
{{eqn | l = 4
| r = T_2 + T_1
| c = $1$ way
}}
{{eqn | l = 5
| ... | $21$ is the smallest number which can be expressed as the [[Definition:Integer Addition|sum]] of at most $3$ [[Definition:Triangular Number|triangular numbers]] in $4$ ways. | By inspection:
{{begin-eqn}}
{{eqn | l = 1
| r = T_1
| c = $1$ way
}}
{{eqn | l = 2
| r = T_1 + T_1
| c = $1$ way
}}
{{eqn | l = 3
| r = T_1 + T_1 + T_1
| c =
}}
{{eqn | r = T_2
| c = $2$ ways
}}
{{eqn | l = 4
| r = T_2 + T_1
| c = $1$ way
}}
{{eqn | l = 5
|... | Smallest Number Expressible as Sum of at most Three Triangular Numbers in 4 ways | https://proofwiki.org/wiki/Smallest_Number_Expressible_as_Sum_of_at_most_Three_Triangular_Numbers_in_4_ways | https://proofwiki.org/wiki/Smallest_Number_Expressible_as_Sum_of_at_most_Three_Triangular_Numbers_in_4_ways | [
"Triangular Numbers",
"21"
] | [
"Definition:Addition/Integers",
"Definition:Triangular Number"
] | [] |
proofwiki-12694 | Necessary Condition for Twice Differentiable Functional to have Minimum | Let $J \sqbrk y$ be a twice differentiable functional.
Let $\delta J \sqbrk {\hat y; h} = 0$.
Suppose, for $y = \hat y$ and all admissible $h$:
:$\delta^2 J \sqbrk {y; h} \ge 0$
Then $J$ has a minimum for $y=\hat y$ if
{{explain|if what?}} | By definition, $ \Delta J \sqbrk y$ can be expressed as:
:$\Delta J \sqbrk {y; h} = \delta J \sqbrk {y; h} + \delta^2 J \sqbrk {y; h} + \epsilon \size h^2$
By assumption:
:$\delta J \sqbrk {\hat y; h} = 0$
Hence:
:$\Delta J \sqbrk {\hat y; h} = \delta^2 J \sqbrk {\hat y; h} + \epsilon \size h^2$
Therefore, for sufficie... | Let $J \sqbrk y$ be a [[Definition:Twice Differentiable Functional|twice differentiable functional]].
Let $\delta J \sqbrk {\hat y; h} = 0$.
Suppose, for $y = \hat y$ and all admissible $h$:
:$\delta^2 J \sqbrk {y; h} \ge 0$
Then $J$ has a [[Definition:Minimum Value of Functional|minimum]] for $y=\hat y$ if
{{e... | By definition, $ \Delta J \sqbrk y$ can be expressed as:
:$\Delta J \sqbrk {y; h} = \delta J \sqbrk {y; h} + \delta^2 J \sqbrk {y; h} + \epsilon \size h^2$
By [[Condition for Differentiable Functional to have Extremum|assumption]]:
:$\delta J \sqbrk {\hat y; h} = 0$
Hence:
:$\Delta J \sqbrk {\hat y; h} = \delta^2 ... | Necessary Condition for Twice Differentiable Functional to have Minimum | https://proofwiki.org/wiki/Necessary_Condition_for_Twice_Differentiable_Functional_to_have_Minimum | https://proofwiki.org/wiki/Necessary_Condition_for_Twice_Differentiable_Functional_to_have_Minimum | [
"Calculus of Variations"
] | [
"Definition:Twice Differentiable/Functional",
"Definition:Minimum Value of Functional"
] | [
"Condition for Differentiable Functional to have Extremum",
"Definition:Sign of Number",
"Definition:Minimum Value of Functional",
"Definition:Contradiction",
"Definition:Function"
] |
proofwiki-12695 | Numbers Equal to Number of Digits in Factorial | For $n \in \set {1, 22, 23, 24}$ the number of digits in the decimal representation of $n!$ is equal to $n$. | First we note that:
:$1! = 1$
which has $1$ digit.
Then from Examples of Factorials:
:$2! = 2$
which has $1$ digit.
Multiplying $n$ by a $1$-digit number increases the number of digits of $n$ by no more than $1$.
Thus from $n = 3$ to $n = 9$, $n!$ has no more than $n - 1$ digits.
From $21$ Factorial:
:$21! = 51 \, 090 ... | For $n \in \set {1, 22, 23, 24}$ the number of [[Definition:Digit|digits]] in the [[Definition:Decimal Expansion|decimal representation]] of $n!$ is equal to $n$. | First we note that:
:$1! = 1$
which has $1$ digit.
Then from [[Factorial/Examples|Examples of Factorials]]:
:$2! = 2$
which has $1$ digit.
Multiplying $n$ by a $1$-[[Definition:Digit|digit]] number increases the number of [[Definition:Digit|digits]] of $n$ by no more than $1$.
Thus from $n = 3$ to $n = 9$, $n!$ ha... | Numbers Equal to Number of Digits in Factorial | https://proofwiki.org/wiki/Numbers_Equal_to_Number_of_Digits_in_Factorial | https://proofwiki.org/wiki/Numbers_Equal_to_Number_of_Digits_in_Factorial | [
"Factorials"
] | [
"Definition:Digit",
"Definition:Decimal Expansion"
] | [
"Factorial/Examples",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Factorial/Examples/21",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Factorial/Examples/22",
"Definition:Digit",
"Factorial/Examples/23",
"Definition:Digit",
"Factorial/Ex... |
proofwiki-12696 | Square of Small-Digit Palindromic Number is Palindromic | Let $n$ be an integer such that the sum of the squares of the digits of $n$ in decimal representation is less than $10$.
Let $n$ be palindromic.
Then $n^2$ is also palindromic.
The sequence of such numbers begins:
:$0, 1, 2, 3, 11, 22, 101, 111, 121, 202, 212, 1001, 1111, \dots$
{{OEIS|A057135}} | Let $\ds n = \sum_{k \mathop = 0}^m a_k 10^k$ be a number satisfying the conditions above.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = \sum_{k \mathop = 0}^m a_k^2
| o = <
| r = 10
}}
{{eqn | n = 2
| l = a_k
| r = a_{m - k}
| rr = \forall k: 0 \le k \le m
}}
{{end-eqn}}
Consider $\ds n^2... | Let $n$ be an [[Definition:Integer|integer]] such that the [[Definition:Integer Addition|sum]] of the [[Definition:Square Function|squares]] of the [[Definition:Digit|digits]] of $n$ in [[Definition:Decimal Expansion|decimal representation]] is less than $10$.
Let $n$ be [[Definition:Palindromic Number|palindromic]].... | Let $\ds n = \sum_{k \mathop = 0}^m a_k 10^k$ be a number satisfying the conditions above.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = \sum_{k \mathop = 0}^m a_k^2
| o = <
| r = 10
}}
{{eqn | n = 2
| l = a_k
| r = a_{m - k}
| rr = \forall k: 0 \le k \le m
}}
{{end-eqn}}
Consider $\ds ... | Square of Small-Digit Palindromic Number is Palindromic | https://proofwiki.org/wiki/Square_of_Small-Digit_Palindromic_Number_is_Palindromic | https://proofwiki.org/wiki/Square_of_Small-Digit_Palindromic_Number_is_Palindromic | [
"Square of Small-Digit Palindromic Number is Palindromic",
"Palindromic Numbers",
"Square Numbers",
"Recreational Mathematics"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Square/Function",
"Definition:Digit",
"Definition:Decimal Expansion",
"Definition:Palindromic Number",
"Definition:Palindromic Number",
"Definition:Integer Sequence"
] | [
"Definition:Multiplication of Polynomials",
"Cauchy's Inequality",
"Basis Representation Theorem",
"Definition:Palindromic Number"
] |
proofwiki-12697 | Sufficient Condition for Twice Differentiable Functional to have Minimum | Let $J$ be a twice differentiable functional.
Let $J$ have an extremum for $y=\hat y$.
Let the second variation $\delta^2 J \sqbrk {\hat y; h}$ be strongly positive {{WRT}} $h$.
Then $J$ acquires the minimum for $y = \hat y$ . | By assumption, $J$ has an extremum for $y = \hat y$:
:$\delta J \sqbrk {\hat y; h} = 0$
The increment is expressible then as:
:$\Delta J \sqbrk {\hat y; h} = \delta^2 J \sqbrk {\hat y; h} + \epsilon \size h^2$
where $\epsilon \to 0$ as $\size h \to 0$.
By assumption, the second variation is strongly positive:
:$\delta^... | Let $J$ be a [[Definition:Twice Differentiable Functional|twice differentiable functional]].
Let $J$ have an [[Definition:Extremum of Functional|extremum]] for $y=\hat y$.
Let the [[Definition:Second Variation of Functional|second variation]] $\delta^2 J \sqbrk {\hat y; h}$ be [[Definition:Strongly Positive Quadratic... | By assumption, $J$ has an [[Definition:Extremum of Functional|extremum]] for $y = \hat y$:
:$\delta J \sqbrk {\hat y; h} = 0$
The [[Definition:Increment of Functional|increment]] is expressible then as:
:$\Delta J \sqbrk {\hat y; h} = \delta^2 J \sqbrk {\hat y; h} + \epsilon \size h^2$
where $\epsilon \to 0$ as $\s... | Sufficient Condition for Twice Differentiable Functional to have Minimum | https://proofwiki.org/wiki/Sufficient_Condition_for_Twice_Differentiable_Functional_to_have_Minimum | https://proofwiki.org/wiki/Sufficient_Condition_for_Twice_Differentiable_Functional_to_have_Minimum | [
"Calculus of Variations"
] | [
"Definition:Twice Differentiable/Functional",
"Definition:Extremum/Functional",
"Definition:Twice Differentiable/Functional",
"Definition:Strongly Positive Quadratic Functional",
"Definition:Minimum Value of Functional"
] | [
"Definition:Extremum/Functional",
"Definition:Increment/Functional",
"Definition:Twice Differentiable/Functional",
"Definition:Strongly Positive Quadratic Functional",
"Membership is Left Compatible with Ordinal Addition",
"Membership is Left Compatible with Ordinal Multiplication",
"Definition:Incremen... |
proofwiki-12698 | Three times Number whose Divisor Sum is Square | Let $n \in \Z_{>0}$ be a positive integer.
Let the divisor sum $\map {\sigma_1} n$ of $n$ be square.
Let $3$ not be a divisor of $n$.
Then the divisor sum of $3 n$ is square. | Let $\map {\sigma_1} n = k^2$.
We have from {{DSFLink|3}}:
:{{:Numbers whose Divisor Sum is Square/Examples/3}}
As $3$ is not a divisor of $n$, it follows that $3$ and $n$ are coprime.
Thus:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {3 n}
| r = \map {\sigma_1} {3 n} \map {\sigma_1} {3 n}
| c = Divisor Sum F... | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
Let the [[Definition:Divisor Sum Function|divisor sum]] $\map {\sigma_1} n$ of $n$ be [[Definition:Square Number|square]].
Let $3$ not be a [[Definition:Divisor of Integer|divisor]] of $n$.
Then the [[Definition:Divisor Sum Function|divisor ... | Let $\map {\sigma_1} n = k^2$.
We have from {{DSFLink|3}}:
:{{:Numbers whose Divisor Sum is Square/Examples/3}}
As $3$ is not a [[Definition:Divisor of Integer|divisor]] of $n$, it follows that $3$ and $n$ are [[Definition:Coprime Integers|coprime]].
Thus:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {3 n}
| r = ... | Three times Number whose Divisor Sum is Square/Proof 1 | https://proofwiki.org/wiki/Three_times_Number_whose_Divisor_Sum_is_Square | https://proofwiki.org/wiki/Three_times_Number_whose_Divisor_Sum_is_Square/Proof_1 | [
"Numbers whose Divisor Sum is Square",
"Three times Number whose Divisor Sum is Square"
] | [
"Definition:Positive/Integer",
"Definition:Divisor Sum Function",
"Definition:Square Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Sum Function",
"Definition:Square Number"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Coprime/Integers",
"Divisor Sum Function is Multiplicative"
] |
proofwiki-12699 | Three times Number whose Divisor Sum is Square | Let $n \in \Z_{>0}$ be a positive integer.
Let the divisor sum $\map {\sigma_1} n$ of $n$ be square.
Let $3$ not be a divisor of $n$.
Then the divisor sum of $3 n$ is square. | From Numbers whose Divisor Sum is Square:
:{{:Numbers whose Divisor Sum is Square/Examples/3}}
The result follows as a specific instance of Product of Coprime Numbers whose Divisor Sum is Square has Square Divisor Sum.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
Let the [[Definition:Divisor Sum Function|divisor sum]] $\map {\sigma_1} n$ of $n$ be [[Definition:Square Number|square]].
Let $3$ not be a [[Definition:Divisor of Integer|divisor]] of $n$.
Then the [[Definition:Divisor Sum Function|divisor ... | From [[Numbers whose Divisor Sum is Square/Examples/3|Numbers whose Divisor Sum is Square]]:
:{{:Numbers whose Divisor Sum is Square/Examples/3}}
The result follows as a specific instance of [[Product of Coprime Numbers whose Divisor Sum is Square has Square Divisor Sum]].
{{qed}} | Three times Number whose Divisor Sum is Square/Proof 2 | https://proofwiki.org/wiki/Three_times_Number_whose_Divisor_Sum_is_Square | https://proofwiki.org/wiki/Three_times_Number_whose_Divisor_Sum_is_Square/Proof_2 | [
"Numbers whose Divisor Sum is Square",
"Three times Number whose Divisor Sum is Square"
] | [
"Definition:Positive/Integer",
"Definition:Divisor Sum Function",
"Definition:Square Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Sum Function",
"Definition:Square Number"
] | [
"Numbers whose Divisor Sum is Square/Examples/3",
"Product of Coprime Numbers whose Divisor Sum is Square has Square Divisor Sum"
] |
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