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-12400 | Nine Point Circle Theorem | Let $\triangle ABC$ be a triangle.
These $9$ points:
:the feet of the altitudes of $\triangle ABC$
:the midpoints of the sides of $\triangle ABC$
:the midpoints of the lines from the vertices of $\triangle ABC$ to the orthocenter $H$ of $\triangle ABC$
all lie on the circumference of a circle.
The center $M$ lies on th... | Let the altitudes of $\triangle ABC$ be $AD$, $BE$ and $CF$.
Let $H$ be the orthocenter of $\triangle ABC$.
Let $X$, $Y$ and $Z$ bisect $AH$, $BH$ and $CH$, respectively.
Let $A_m$, $B_m$, and $C_m$ bisect $BC$, $AC$, and $AB$, respectively.
:450px
{{begin-eqn}}
{{eqn | l = \triangle AHC
| o = \sim
| r = \t... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
These $9$ points:
:the [[Definition:Foot of Altitude|feet]] of the [[Definition:Altitude of Triangle|altitudes]] of $\triangle ABC$
:the [[Definition:Midpoint|midpoints]] of the [[Definition:Side of Polygon|sides]] of $\triangle ABC$
:the [[Definiti... | Let the [[Definition:Altitude of Triangle|altitudes]] of $\triangle ABC$ be $AD$, $BE$ and $CF$.
Let $H$ be the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC$.
Let $X$, $Y$ and $Z$ [[Definition:Bisection|bisect]] $AH$, $BH$ and $CH$, respectively.
Let $A_m$, $B_m$, and $C_m$ [[Definition:Bisection|bisect]... | Nine Point Circle Theorem | https://proofwiki.org/wiki/Nine_Point_Circle_Theorem | https://proofwiki.org/wiki/Nine_Point_Circle_Theorem | [
"Nine Point Circle Theorem",
"Triangles",
"Circles",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Altitude of Triangle/Foot",
"Definition:Altitude of Triangle",
"Definition:Midpoint",
"Definition:Polygon/Side",
"Definition:Midpoint",
"Definition:Line/Segment",
"Definition:Polygon/Vertex",
"Definition:Orthocenter",
"Definition:Circle/Circumference",... | [
"Definition:Altitude of Triangle",
"Definition:Orthocenter",
"Definition:Bisection",
"Definition:Bisection",
"File:9PointCircleLabels.png",
"Triangles with One Equal Angle and Two Sides Proportional are Similar",
"Triangles with One Equal Angle and Two Sides Proportional are Similar",
"Parallelism is ... |
proofwiki-12401 | Feuerbach's Theorem | Let $\triangle ABC$ be a triangle.
The Feuerbach circle of $\triangle ABC$ is tangent to:
:the incircle of $\triangle ABC$
and:
:the $3$ excircles of $\triangle ABC$.
:800px | Recall the Third Fontené Theorem:
:the pedal circle of a point $P$ is tangent to the nine point circle
{{iff}}:
:$P$ and its isogonal conjugate $P^{-1}$ lie on a line through the circumcenter.
Let $P$ be either the incenter or an excenter of $\triangle ABC$.
By Isogonal Conjugate of Incenter or Excenter is Itself, we h... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
The [[Definition:Feuerbach Circle|Feuerbach circle]] of $\triangle ABC$ is [[Definition:Tangent Circles|tangent]] to:
:the [[Definition:Incircle of Triangle|incircle]] of $\triangle ABC$
and:
:the $3$ [[Definition:Excircle of Triangle|excircles]] of... | Recall the [[Third Fontené Theorem]]:
:the [[Definition:Pedal Circle|pedal circle]] of a point $P$ is [[Definition:Tangent Circles|tangent]] to the [[Definition:Nine Point Circle|nine point circle]]
{{iff}}:
:$P$ and its [[Definition:Isogonal Conjugate|isogonal conjugate]] $P^{-1}$ lie on a [[Definition:Straight Line|l... | Feuerbach's Theorem | https://proofwiki.org/wiki/Feuerbach's_Theorem | https://proofwiki.org/wiki/Feuerbach's_Theorem | [
"Feuerbach Circles",
"Triangles",
"Circles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Feuerbach Circle",
"Definition:Tangent Circles",
"Definition:Incircle of Triangle",
"Definition:Excircle of Triangle",
"File:9PointCircleTangentCircles.png"
] | [
"Fontené Theorems/Third",
"Definition:Pedal Circle",
"Definition:Tangent Circles",
"Definition:Feuerbach Circle",
"Definition:Isogonal Conjugate",
"Definition:Line/Straight Line",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Incircle of Triangle/Incenter",
"Definition:Excircle of ... |
proofwiki-12402 | Round Peg fits in Square Hole better than Square Peg fits in Round Hole | A round peg fits better in a square hole than a square peg fits in a round hole.
:600px | The situation is modelled by considering the ratios of the areas of:
:a square to the circle in which it is inscribed
:a square to the circle around which it has been circumscribed.
Let a square $S$ be inscribed in a circle $C$ of radius $r$.
Let $A_c$ and $A_s$ be the areas of $C$ and $S$ respectively.
From Area of Ci... | A [[Definition:Circle|round]] peg fits better in a [[Definition:Square (Geometry)|square]] hole than a [[Definition:Square (Geometry)|square]] peg fits in a [[Definition:Circle|round]] hole.
:[[File:SquareAndRoundPegsAndHoles.png|600px]] | The situation is [[Definition:Mathematical Model|modelled]] by considering the [[Definition:Ratio|ratios]] of the [[Definition:Area|areas]] of:
:a [[Definition:Square (Geometry)|square]] to the [[Definition:Circle|circle]] in which it is [[Definition:Polygon Inscribed in Circle|inscribed]]
:a [[Definition:Square (Geome... | Round Peg fits in Square Hole better than Square Peg fits in Round Hole | https://proofwiki.org/wiki/Round_Peg_fits_in_Square_Hole_better_than_Square_Peg_fits_in_Round_Hole | https://proofwiki.org/wiki/Round_Peg_fits_in_Square_Hole_better_than_Square_Peg_fits_in_Round_Hole | [
"Squares",
"Circles"
] | [
"Definition:Circle",
"Definition:Quadrilateral/Square",
"Definition:Quadrilateral/Square",
"Definition:Circle",
"File:SquareAndRoundPegsAndHoles.png"
] | [
"Definition:Mathematical Model",
"Definition:Ratio",
"Definition:Area",
"Definition:Quadrilateral/Square",
"Definition:Circle",
"Definition:Inscribe/Polygon in Circle",
"Definition:Quadrilateral/Square",
"Definition:Circle",
"Definition:Circumscribe/Circle around Polygon",
"Definition:Quadrilatera... |
proofwiki-12403 | Ratios of Sizes of Mutually Inscribed Multidimensional Cubes and Spheres | Consider:
: a cube $C_n$ of $n$ dimensions inscribed within a sphere $S_n$ of $n$ dimensions
: a sphere $S'_n$ of $n$ dimensions inscribed within a cube $C'_n$ of $n$ dimensions.
Let:
: $A_{cn}$ be the $n$ dimensional volume of $C_n$
: $A_{sn}$ be the $n$ dimensional volume of $S_n$
: $A'_{cn}$ be the $n$ dimensional ... | {{ProofWanted|Formulae for the volumes of $n$ dimensional squares and circles need to be established first.}} | Consider:
: a [[Definition:Cube (Geometry)|cube]] $C_n$ of [[Definition:Dimension (Geometry)|$n$ dimensions]] [[Definition:Inscribe|inscribed]] within a [[Definition:Sphere (Geometry)|sphere]] $S_n$ of [[Definition:Dimension (Geometry)|$n$ dimensions]]
: a [[Definition:Sphere (Geometry)|sphere]] $S'_n$ of [[Definition... | {{ProofWanted|Formulae for the volumes of $n$ dimensional squares and circles need to be established first.}} | Ratios of Sizes of Mutually Inscribed Multidimensional Cubes and Spheres | https://proofwiki.org/wiki/Ratios_of_Sizes_of_Mutually_Inscribed_Multidimensional_Cubes_and_Spheres | https://proofwiki.org/wiki/Ratios_of_Sizes_of_Mutually_Inscribed_Multidimensional_Cubes_and_Spheres | [
"Geometry",
"Squares",
"Circles"
] | [
"Definition:Cube/Geometry",
"Definition:Dimension (Geometry)",
"Definition:Inscribe",
"Definition:Sphere/Geometry",
"Definition:Dimension (Geometry)",
"Definition:Sphere/Geometry",
"Definition:Dimension (Geometry)",
"Definition:Inscribe",
"Definition:Cube/Geometry",
"Definition:Dimension (Geometry... | [] |
proofwiki-12404 | Pseudoprime Element is Prime in Arithmetic Lattice | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below arithmetic lattice.
Let $p \in S$.
Then if $p$ is pseudoprime element, then $p$ is prime element. | By Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice:
:$\ll$ is a multiplicative relation.
Thus by Way Below Relation is Multiplicative implies Pseudoprime Element is Prime:
:the result holds.
{{qed}} | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Arithmetic Ordered Set|arithmetic]] [[Definition:Lattice (Order Theory)|lattice]].
Let $p \in S$.
Then if $p$ is [[Definition:Pseudoprime (Order Theory)|pseudoprime element]], then $p$ is [[Definition:Prime ... | By [[Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice]]:
:$\ll$ is a [[Definition:Multiplicative Relation|multiplicative relation]].
Thus by [[Way Below Relation is Multiplicative implies Pseudoprime Element is Prime]]:
:the result holds.
{{qed}} | Pseudoprime Element is Prime in Arithmetic Lattice | https://proofwiki.org/wiki/Pseudoprime_Element_is_Prime_in_Arithmetic_Lattice | https://proofwiki.org/wiki/Pseudoprime_Element_is_Prime_in_Arithmetic_Lattice | [
"Prime Elements",
"Continuous Lattices"
] | [
"Definition:Bounded Below Set",
"Definition:Arithmetic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Pseudoprime (Order Theory)",
"Definition:Prime Element (Order Theory)"
] | [
"Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice",
"Definition:Multiplicative Relation",
"Way Below Relation is Multiplicative implies Pseudoprime Element is Prime"
] |
proofwiki-12405 | Every Pseudoprime Element is Prime implies Lattice is Arithmetic | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic distributive lattice.
Assume that
:for every element $p$ of $S$ if $p$ is pseudoprime element, then $p$ is prime element.
Then $L$ is arithmetic. | By If Every Element Pseudoprime is Prime then Way Below Relation is Multiplicative:
:$\ll$ is a multiplicative relation.
where $\ll$ denotes the way below relation.
Thus by Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice:
:the result holds.
{{qed}} | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Distributive Lattice|distributive lattice]].
Assume that
:for every [[Definition:Element|element]] $p$ of $S$ if $p$ is [[Definition:Pseudoprime (Order Theor... | By [[If Every Element Pseudoprime is Prime then Way Below Relation is Multiplicative]]:
:$\ll$ is a [[Definition:Multiplicative Relation|multiplicative relation]].
where $\ll$ denotes the [[Definition:Element is Way Below|way below relation]].
Thus by [[Arithmetic iff Way Below Relation is Multiplicative in Algebraic ... | Every Pseudoprime Element is Prime implies Lattice is Arithmetic | https://proofwiki.org/wiki/Every_Pseudoprime_Element_is_Prime_implies_Lattice_is_Arithmetic | https://proofwiki.org/wiki/Every_Pseudoprime_Element_is_Prime_implies_Lattice_is_Arithmetic | [
"Prime Elements",
"Continuous Lattices"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Distributive Lattice",
"Definition:Element",
"Definition:Pseudoprime (Order Theory)",
"Definition:Prime Element (Order Theory)",
"Definition:Arithmetic Ordered Set"
] | [
"If Every Element Pseudoprime is Prime then Way Below Relation is Multiplicative",
"Definition:Multiplicative Relation",
"Definition:Element is Way Below",
"Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice"
] |
proofwiki-12406 | General Periodicity Property/Corollary | Let $f: \R \to \R$ be a real function.
Then $L$ is a periodic element of $f$ {{iff}}:
:$\forall x \in \R: \map f {x \bmod L} = \map f x$
where $x \bmod L$ is the modulo operation. | === Necessary Condition ===
Let $f: \R \to \R$ be a real function with a periodic element $L$.
Then:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f {n L + r} \quad 0 < r < \size L
| c = {{Defof|Quotient (Integer Division)/Real|Quotient}}
}}
{{eqn | r = \map f r
| c = General Periodicity Property
}}
{... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Then $L$ is a [[Definition:Periodic Element|periodic element]] of $f$ {{iff}}:
:$\forall x \in \R: \map f {x \bmod L} = \map f x$
where $x \bmod L$ is the [[Definition:Modulo Operation|modulo operation]]. | === Necessary Condition ===
Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] with a [[Definition:Periodic Element|periodic element]] $L$.
Then:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f {n L + r} \quad 0 < r < \size L
| c = {{Defof|Quotient (Integer Division)/Real|Quotient}}
}}
{{... | General Periodicity Property/Corollary | https://proofwiki.org/wiki/General_Periodicity_Property/Corollary | https://proofwiki.org/wiki/General_Periodicity_Property/Corollary | [
"Periodic Functions"
] | [
"Definition:Real Function",
"Definition:Periodic Function/Periodic Element",
"Definition:Modulo Operation"
] | [
"Definition:Real Function",
"Definition:Periodic Function/Periodic Element",
"General Periodicity Property",
"Definition:Real Function"
] |
proofwiki-12407 | Characterization of Euler's Number by Inequality | Let $a$ be a (strictly) positive real number.
Then:
:$a = e \iff \forall x \in \R: a^x \ge x + 1$
where $e$ denotes Euler's number. | === Forward Implication ===
Proved in Exponential Function Inequality.
{{qed|lemma}} | Let $a$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Then:
:$a = e \iff \forall x \in \R: a^x \ge x + 1$
where $e$ denotes [[Definition:Euler's Number|Euler's number]]. | === Forward Implication ===
Proved in [[Exponential Function Inequality]].
{{qed|lemma}} | Characterization of Euler's Number by Inequality | https://proofwiki.org/wiki/Characterization_of_Euler's_Number_by_Inequality | https://proofwiki.org/wiki/Characterization_of_Euler's_Number_by_Inequality | [
"Euler's Number"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Euler's Number"
] | [
"Exponential of x not less than 1+x",
"Exponential of x not less than 1+x"
] |
proofwiki-12408 | 3 Configurations of 9 Lines with 3 Intersection Points on each Line | There exist exactly $3$ essentially different configurations of $9$ straight lines each of which has exactly $3$ points of intersection.
This is one: there are two others.
:400px | {{ProofWanted|The precise meaning of the term "essentially different" needs to be established, for a start}} | There exist exactly $3$ essentially different configurations of $9$ [[Definition:Straight Line|straight lines]] each of which has exactly $3$ [[Definition:Point|points]] of [[Definition:Intersection (Geometry)|intersection]].
This is one: there are two others.
:[[File:PappusHexagonTheorem.png|400px]] | {{ProofWanted|The precise meaning of the term "essentially different" needs to be established, for a start}} | 3 Configurations of 9 Lines with 3 Intersection Points on each Line | https://proofwiki.org/wiki/3_Configurations_of_9_Lines_with_3_Intersection_Points_on_each_Line | https://proofwiki.org/wiki/3_Configurations_of_9_Lines_with_3_Intersection_Points_on_each_Line | [
"Projective Geometry"
] | [
"Definition:Line/Straight Line",
"Definition:Point",
"Definition:Intersection (Geometry)",
"File:PappusHexagonTheorem.png"
] | [] |
proofwiki-12409 | Power of Positive Real Number is Positive/Real Number | Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $r \in \R$ be a real number.
Then:
:$x^r > 0$
where $x^r$ denotes the $x$ to the power of $r$. | From the definition of $x$ to the power of $r$:
:$x^r = \map \exp {r \ln x}$
The result follows from Exponential of Real Number is Strictly Positive.
{{qed}}
Category:Power of Positive Real Number is Positive
h0m8t9tu4wm2hmo6bal3kzgkd208761 | Let $x \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $r \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$x^r > 0$
where $x^r$ denotes the [[Definition:Power to Real Number|$x$ to the power of $r$]]. | From the definition of [[Definition:Power to Real Number|$x$ to the power of $r$]]:
:$x^r = \map \exp {r \ln x}$
The result follows from [[Exponential of Real Number is Strictly Positive]].
{{qed}}
[[Category:Power of Positive Real Number is Positive]]
h0m8t9tu4wm2hmo6bal3kzgkd208761 | Power of Positive Real Number is Positive/Real Number | https://proofwiki.org/wiki/Power_of_Positive_Real_Number_is_Positive/Real_Number | https://proofwiki.org/wiki/Power_of_Positive_Real_Number_is_Positive/Real_Number | [
"Power of Positive Real Number is Positive"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Real Number",
"Definition:Power (Algebra)/Real Number"
] | [
"Definition:Power (Algebra)/Real Number",
"Exponential of Real Number is Strictly Positive",
"Category:Power of Positive Real Number is Positive"
] |
proofwiki-12410 | Element is Finite iff Element is Compact in Lattice of Power Set | Let $X$ be a set.
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be a lattice of power set.
Let $x \in \powerset X$.
Then $x$ is a finite set {{iff}} $x$ is a compact element. | === Sufficient Condition when Empty ===
The case when $x = \O$
By Empty Set is Bottom of Lattice of Power Set:
:$x = \bot$
where $\bot$ denotes the bottom of $L$.
By Bottom is Way Below Any Element:
:$x \ll x$
where $\ll$ denotes the way below relation.
Thus by definition
:$x$ is a finite set {{iff}} $x$ is a compact e... | Let $X$ be a [[Definition:Set|set]].
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be a [[Definition:Lattice (Order Theory)|lattice]] of [[Definition:Power Set|power set]].
Let $x \in \powerset X$.
Then $x$ is a [[Definition:Finite Set|finite set]] {{iff}} $x$ is a [[Definition:Compact Element|compact elem... | === Sufficient Condition when Empty ===
The case when $x = \O$
By [[Empty Set is Bottom of Lattice of Power Set]]:
:$x = \bot$
where $\bot$ denotes the [[Definition:Bottom of Lattice|bottom]] of $L$.
By [[Bottom is Way Below Any Element]]:
:$x \ll x$
where $\ll$ denotes the [[Definition:Element is Way Below|way belo... | Element is Finite iff Element is Compact in Lattice of Power Set | https://proofwiki.org/wiki/Element_is_Finite_iff_Element_is_Compact_in_Lattice_of_Power_Set | https://proofwiki.org/wiki/Element_is_Finite_iff_Element_is_Compact_in_Lattice_of_Power_Set | [
"Way Below Relation",
"Power Set"
] | [
"Definition:Set",
"Definition:Lattice (Order Theory)",
"Definition:Power Set",
"Definition:Finite Set",
"Definition:Compact Element"
] | [
"Empty Set is Bottom of Lattice of Power Set",
"Definition:Bottom of Lattice",
"Bottom is Way Below Any Element",
"Definition:Element is Way Below",
"Definition:Finite Set",
"Definition:Compact Element",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Compact Element",
"Definition:Co... |
proofwiki-12411 | Function with Limit at Infinity of Exponential Order Zero | Let $f: \hointr 0 \to \to \R$ be a real function.
Let $f$ be continuous everywhere on their domains, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$.
{{explain|Establish whether it is "finite subinterval" that is needed here, or what we have alr... | Denote $\ds L = \lim_{t \mathop \to +\infty} \map f t$.
Define the constant mapping:
:$\map C t = - L$
Further define:
:$\map g t = \map f t + \map C t$
From:
:Constant Function is of Exponential Order Zero,
:Sum of Functions of Exponential Order,
it is sufficient to prove that $g$ is of exponential order $0$.
Fix $\ep... | Let $f: \hointr 0 \to \to \R$ be a [[Definition:Real Function|real function]].
Let $f$ be [[Definition:Continuity|continuous]] everywhere on their [[Definition:Domain of Mapping|domains]], except possibly for some [[Definition:Finite Set|finite number]] of [[Definition:Discontinuity of the First Kind|discontinuities o... | Denote $\ds L = \lim_{t \mathop \to +\infty} \map f t$.
Define the [[Definition:Constant Mapping|constant mapping]]:
:$\map C t = - L$
Further define:
:$\map g t = \map f t + \map C t$
From:
:[[Constant Function is of Exponential Order Zero]],
:[[Sum of Functions of Exponential Order]],
it is sufficient to prov... | Function with Limit at Infinity of Exponential Order Zero | https://proofwiki.org/wiki/Function_with_Limit_at_Infinity_of_Exponential_Order_Zero | https://proofwiki.org/wiki/Function_with_Limit_at_Infinity_of_Exponential_Order_Zero | [
"Exponential Order"
] | [
"Definition:Real Function",
"Definition:Continuous",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Finite Set",
"Definition:Discontinuity (Real Analysis)/First Kind",
"Definition:Finite Subinterval",
"Definition:Subdivision of Interval/Finite",
"Definition:Limit of Real Function/Limit at Infin... | [
"Definition:Constant Mapping",
"Constant Function is of Exponential Order Zero",
"Sum of Functions of Exponential Order",
"Definition:Exponential Order/Real Index",
"Definition:Limit of Real Function/Limit at Infinity/Positive",
"Exponential of Zero",
"Definition:Exponential Order/Real Index",
"Catego... |
proofwiki-12412 | Bounded Function is of Exponential Order Zero | Let $f: \hointr 0 \to \to \mathbb F$ be a function, where $\mathbb F \in \set {\R, \C}$.
Let $f$ be continuous everywhere on its domain, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$.
{{explain|Establish whether it is "finite subinterval" that... | Let $U$ be an upper bound of $f$.
Let $L$ be a lower bound of $f$.
Let $K > \max \set {\size U, \size L}$.
Then:
{{begin-eqn}}
{{eqn | q = \forall t \ge 1
| l = \size {\map f t}
| o = <
| r = K
| c = {{Defof|Bounded Mapping}}
}}
{{eqn | r = K e^{0 t}
| c = Exponential of Zero
}}
{{end-eqn... | Let $f: \hointr 0 \to \to \mathbb F$ be a [[Definition:Function|function]], where $\mathbb F \in \set {\R, \C}$.
Let $f$ be [[Definition:Continuity|continuous]] everywhere on its [[Definition:Domain of Mapping|domain]], except possibly for some [[Definition:Finite Set|finite number]] of [[Definition:Discontinuity of t... | Let $U$ be an [[Definition:Upper Bound of Mapping|upper bound of $f$]].
Let $L$ be a [[Definition:Lower Bound of Mapping|lower bound of $f$]].
Let $K > \max \set {\size U, \size L}$.
Then:
{{begin-eqn}}
{{eqn | q = \forall t \ge 1
| l = \size {\map f t}
| o = <
| r = K
| c = {{Defof|Bounded... | Bounded Function is of Exponential Order Zero | https://proofwiki.org/wiki/Bounded_Function_is_of_Exponential_Order_Zero | https://proofwiki.org/wiki/Bounded_Function_is_of_Exponential_Order_Zero | [
"Exponential Order"
] | [
"Definition:Function",
"Definition:Continuous",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Finite Set",
"Definition:Discontinuity (Real Analysis)/First Kind",
"Definition:Finite Subinterval",
"Definition:Subdivision of Interval/Finite",
"Definition:Bounded Mapping",
"Definition:Exponentia... | [
"Definition:Upper Bound of Mapping",
"Definition:Lower Bound of Mapping",
"Exponential of Zero",
"Definition:Exponential Order/Real Index"
] |
proofwiki-12413 | Arctangent is of Exponential Order Zero | Let $\arctan: \R \to \openint {-\dfrac \pi 2} {\dfrac \pi 2}$ be the real arctangent.
Then $\arctan$ is of exponential order $0$. | Follows from Function with Limit at Infinity of Exponential Order Zero.
{{qed}}
Category:Arctangent Function
Category:Exponential Order
Category:Limits of Real Functions
fn6s7w3ymg5h2cf712ycea4qix5bgia | Let $\arctan: \R \to \openint {-\dfrac \pi 2} {\dfrac \pi 2}$ be the [[Definition:Real Arctangent|real arctangent]].
Then $\arctan$ is of [[Definition:Exponential Order to Real Index|exponential order $0$]]. | Follows from [[Function with Limit at Infinity of Exponential Order Zero]].
{{qed}}
[[Category:Arctangent Function]]
[[Category:Exponential Order]]
[[Category:Limits of Real Functions]]
fn6s7w3ymg5h2cf712ycea4qix5bgia | Arctangent is of Exponential Order Zero | https://proofwiki.org/wiki/Arctangent_is_of_Exponential_Order_Zero | https://proofwiki.org/wiki/Arctangent_is_of_Exponential_Order_Zero | [
"Arctangent Function",
"Exponential Order",
"Limits of Real Functions"
] | [
"Definition:Inverse Tangent/Real/Arctangent",
"Definition:Exponential Order/Real Index"
] | [
"Function with Limit at Infinity of Exponential Order Zero",
"Category:Arctangent Function",
"Category:Exponential Order",
"Category:Limits of Real Functions"
] |
proofwiki-12414 | Arccotangent is of Exponential Order Zero | Let $\arccot: \R \to \openint 0 \pi$ be the real arccotangent.
Then $\arccot$ is of exponential order $0$. | Follows from Function with Limit at Infinity of Exponential Order Zero.
{{qed}}
{{MissingLinks|limit of $\arccot$}}
Category:Exponential Order
ri29ldq9urkxmn6a26kq9sxapo9m553 | Let $\arccot: \R \to \openint 0 \pi$ be the [[Definition:Real Arccotangent|real arccotangent]].
Then $\arccot$ is of [[Definition:Exponential Order to Real Index|exponential order $0$]]. | Follows from [[Function with Limit at Infinity of Exponential Order Zero]].
{{qed}}
{{MissingLinks|limit of $\arccot$}}
[[Category:Exponential Order]]
ri29ldq9urkxmn6a26kq9sxapo9m553 | Arccotangent is of Exponential Order Zero | https://proofwiki.org/wiki/Arccotangent_is_of_Exponential_Order_Zero | https://proofwiki.org/wiki/Arccotangent_is_of_Exponential_Order_Zero | [
"Exponential Order"
] | [
"Definition:Inverse Cotangent/Real/Arccotangent",
"Definition:Exponential Order/Real Index"
] | [
"Function with Limit at Infinity of Exponential Order Zero",
"Category:Exponential Order"
] |
proofwiki-12415 | Limit at Infinity of Sine Integral Function | Let $\Si: \R \to \R$ denote the sine integral function.
Then $\Si$ has a (finite) limit at infinity:
:$\ds \lim_{x \mathop \to +\infty} \map \Si x = \frac \pi 2$ | The limit:
:$\ds \lim_{x \mathop \to +\infty} \map \Si x = \lim_{x \mathop \to +\infty} \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t$
is the Dirichlet Integral.
{{qed}} | Let $\Si: \R \to \R$ denote the [[Definition:Sine Integral Function|sine integral function]].
Then $\Si$ has a [[Definition:Limit at Infinity|(finite) limit at infinity]]:
:$\ds \lim_{x \mathop \to +\infty} \map \Si x = \frac \pi 2$ | The limit:
:$\ds \lim_{x \mathop \to +\infty} \map \Si x = \lim_{x \mathop \to +\infty} \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t$
is the [[Dirichlet Integral]].
{{qed}} | Limit at Infinity of Sine Integral Function | https://proofwiki.org/wiki/Limit_at_Infinity_of_Sine_Integral_Function | https://proofwiki.org/wiki/Limit_at_Infinity_of_Sine_Integral_Function | [
"Sine Integral Function"
] | [
"Definition:Sine Integral Function",
"Definition:Limit of Real Function/Limit at Infinity/Positive"
] | [
"Dirichlet Integral"
] |
proofwiki-12416 | Natural Number Power is of Exponential Order Epsilon | Let $n \in \N$ be a natural number.
Then:
:$t \mapsto t^n$
is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude. | The proof proceeds by induction on $n$, where $n$ is the degree of the polynomial. | Let $n \in \N$ be a [[Definition:Natural Numbers|natural number]].
Then:
:$t \mapsto t^n$
is of [[Definition:Exponential Order to Real Index|exponential order $\epsilon$]] for any $\epsilon > 0$ arbitrarily small in magnitude. | The proof proceeds by [[Principle of Mathematical Induction|induction]] on $n$, where $n$ is the [[Definition:Degree of Polynomial|degree of the polynomial]]. | Natural Number Power is of Exponential Order Epsilon | https://proofwiki.org/wiki/Natural_Number_Power_is_of_Exponential_Order_Epsilon | https://proofwiki.org/wiki/Natural_Number_Power_is_of_Exponential_Order_Epsilon | [
"Exponential Order"
] | [
"Definition:Natural Numbers",
"Definition:Exponential Order/Real Index"
] | [
"Principle of Mathematical Induction",
"Definition:Degree of Polynomial",
"Principle of Mathematical Induction"
] |
proofwiki-12417 | Raising Exponential Order | Let $\map f t: \R \to \mathbb F$ a function, where $\mathbb F \in \set {\R, \C}$.
Let $f$ be continuous on the real interval $\hointr 0 \to$, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$.
{{explain|Establish whether it is "finite subinterval"... | From the definition of exponential order, there exist strictly positive real numbers $M$ and $K$ such that:
:$\forall t \ge M: \size {\map f t} < K e^{a t}$
From Exponential is Strictly Increasing, we have:
:$K e^{a t} < K e^{b t}$
Therefore:
:$\forall t \ge M: \size {\map f t} < K e^{b t}$
The result follows from the ... | Let $\map f t: \R \to \mathbb F$ a [[Definition:Function|function]], where $\mathbb F \in \set {\R, \C}$.
Let $f$ be [[Definition:Continuous Mapping|continuous]] on the [[Definition:Unbounded Closed Real Interval|real interval]] $\hointr 0 \to$, except possibly for some [[Definition:Finite Set|finite number]] of [[Def... | From the definition of [[Definition:Exponential Order to Real Index|exponential order]], there exist [[Definition:Strictly Positive Real Number|strictly positive real numbers]] $M$ and $K$ such that:
:$\forall t \ge M: \size {\map f t} < K e^{a t}$
From [[Exponential is Strictly Increasing]], we have:
:$K e^{a t} < K ... | Raising Exponential Order | https://proofwiki.org/wiki/Raising_Exponential_Order | https://proofwiki.org/wiki/Raising_Exponential_Order | [
"Exponential Order"
] | [
"Definition:Function",
"Definition:Continuous Mapping",
"Definition:Real Interval/Unbounded Closed",
"Definition:Finite Set",
"Definition:Discontinuity (Real Analysis)/First Kind",
"Definition:Finite Subinterval",
"Definition:Subdivision of Interval/Finite",
"Definition:Exponential Order/Real Index",
... | [
"Definition:Exponential Order/Real Index",
"Definition:Strictly Positive/Real Number",
"Exponential is Strictly Increasing",
"Definition:Exponential Order/Real Index",
"Category:Exponential Order"
] |
proofwiki-12418 | Factorial is not of Exponential Order | Let $\Gamma$ denote the gamma function.
Let $\map f t = \map \Gamma {t + 1} = t!$.
Then:
:$f$ is not of exponential order.
That is, it grows faster than any exponential. | From Gamma Function is Continuous on Positive Reals, $f$ is continuous for $t \ge 0$.
Set $t > 0$.
From Stirling's Formula:
:$t! \sim \sqrt {2 \pi t} \paren {\dfrac t e}^t$
where $\sim$ denotes asymptotic equality.
That is,
{{begin-eqn}}
{{eqn | l = t!
| o = \sim
| r = \sqrt {2 \pi t} \paren {\frac t e}^t
... | Let $\Gamma$ denote the [[Definition:Gamma Function|gamma function]].
Let $\map f t = \map \Gamma {t + 1} = t!$.
Then:
:$f$ is not of [[Definition:Exponential Order|exponential order]].
That is, it grows faster than any exponential. | From [[Gamma Function is Continuous on Positive Reals]], $f$ is continuous for $t \ge 0$.
Set $t > 0$.
From [[Stirling's Formula]]:
:$t! \sim \sqrt {2 \pi t} \paren {\dfrac t e}^t$
where $\sim$ denotes [[Definition:Asymptotic Equality|asymptotic equality]].
That is,
{{begin-eqn}}
{{eqn | l = t!
| o = \sim
... | Factorial is not of Exponential Order | https://proofwiki.org/wiki/Factorial_is_not_of_Exponential_Order | https://proofwiki.org/wiki/Factorial_is_not_of_Exponential_Order | [
"Gamma Function",
"Exponential Order"
] | [
"Definition:Gamma Function",
"Definition:Exponential Order"
] | [
"Gamma Function is Continuous on Positive Reals",
"Stirling's Formula",
"Definition:Asymptotic Equality",
"Definition:Exponential Order",
"Definition:Sufficiently Large",
"X to the x is not of Exponential Order",
"Category:Gamma Function",
"Category:Exponential Order"
] |
proofwiki-12419 | X to the x is not of Exponential Order | Let $f: \R_{>0} \to \R$ be definedas:
:$\forall x \in \R_{>0}: \map f x = x^x$.
Then:
:$f$ is not of exponential order.
That is, it grows faster than any exponential. | === Lemma ===
{{:X to the x is not of Exponential Order/Lemma}}{{qed|lemma}}
By the definition of power:
:$\map f t = \map \exp {t \ln t}$
The theorem is equivalent to that there do not exist strictly positive real constants $M$, $K$, $a$ such that:
:$\forall t \ge M: \size {\map f t} < K e^{a t}$
{{AimForCont}} such ... | Let $f: \R_{>0} \to \R$ be definedas:
:$\forall x \in \R_{>0}: \map f x = x^x$.
Then:
:$f$ is not of [[Definition:Exponential Order|exponential order]].
That is, it grows faster than any [[Definition:Exponential Function|exponential]]. | === [[X to the x is not of Exponential Order/Lemma|Lemma]] ===
{{:X to the x is not of Exponential Order/Lemma}}{{qed|lemma}}
By the definition of [[Definition:Power to Real Number|power]]:
:$\map f t = \map \exp {t \ln t}$
The theorem is equivalent to that there do not exist [[Definition:Strictly Positive Real Nu... | X to the x is not of Exponential Order | https://proofwiki.org/wiki/X_to_the_x_is_not_of_Exponential_Order | https://proofwiki.org/wiki/X_to_the_x_is_not_of_Exponential_Order | [
"Exponential Order"
] | [
"Definition:Exponential Order",
"Definition:Exponential Function"
] | [
"X to the x is not of Exponential Order/Lemma",
"Definition:Power (Algebra)/Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Real Number",
"Definition:Constant",
"X to the x is not of Exponential Order/Lemma",
"Definition:Constant",
"Definition:Contradiction",
"Proof by Contradi... |
proofwiki-12420 | Empty Set is Bottom of Lattice of Power Set | Let $X$ be a set.
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be the lattice of the power set of $X$.
Then:
:$\O = \bot$
where $\bot$ denotes the bottom of $L$. | By Empty Set is Subset of All Sets:
:$\forall S \in \powerset X: \O \subseteq S$
By Empty Set is Element of Power Set:
:$\O \in \powerset X$
Thus by definition of the smallest element:
:$\O = \bot$
{{qed}} | Let $X$ be a [[Definition:Set|set]].
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be the [[Definition:Lattice (Order Theory)|lattice]] of the [[Definition:Power Set|power set]] of $X$.
Then:
:$\O = \bot$
where $\bot$ denotes the [[Definition:Bottom of Lattice|bottom]] of $L$. | By [[Empty Set is Subset of All Sets]]:
:$\forall S \in \powerset X: \O \subseteq S$
By [[Empty Set is Element of Power Set]]:
:$\O \in \powerset X$
Thus by definition of the [[Definition:Smallest Element|smallest element]]:
:$\O = \bot$
{{qed}} | Empty Set is Bottom of Lattice of Power Set | https://proofwiki.org/wiki/Empty_Set_is_Bottom_of_Lattice_of_Power_Set | https://proofwiki.org/wiki/Empty_Set_is_Bottom_of_Lattice_of_Power_Set | [
"Lattice Theory",
"Power Set"
] | [
"Definition:Set",
"Definition:Lattice (Order Theory)",
"Definition:Power Set",
"Definition:Bottom of Lattice"
] | [
"Empty Set is Subset of All Sets",
"Empty Set is Element of Power Set",
"Definition:Smallest Element"
] |
proofwiki-12421 | Limit of x to the x | Let $f: \R_{>0} \to \R$ be defined as:
:$\forall x \in \R_{>0}: \map f x = x^x$
Then:
:$\ds \lim_{x \mathop \to 0^+} x^x = 1$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0^+} x^x
| r = \lim_{x \mathop \to 0^+} \map \exp {x \ln x}
| c = {{Defof|Power (Algebra)|Power|subdef = Real Number|index = 1}}
}}
{{eqn | r = \map \exp {\lim_{x \mathop \to 0^+} x \ln x}
| c = Exponential Function is Continuous
}}
{{eqn | r = \map \exp {... | Let $f: \R_{>0} \to \R$ be defined as:
:$\forall x \in \R_{>0}: \map f x = x^x$
Then:
:$\ds \lim_{x \mathop \to 0^+} x^x = 1$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0^+} x^x
| r = \lim_{x \mathop \to 0^+} \map \exp {x \ln x}
| c = {{Defof|Power (Algebra)|Power|subdef = Real Number|index = 1}}
}}
{{eqn | r = \map \exp {\lim_{x \mathop \to 0^+} x \ln x}
| c = [[Exponential Function is Continuous/Real Numbers|Exponential... | Limit of x to the x/Proof 1 | https://proofwiki.org/wiki/Limit_of_x_to_the_x | https://proofwiki.org/wiki/Limit_of_x_to_the_x/Proof_1 | [
"Examples of Limits of Real Functions",
"Limit of x to the x"
] | [] | [
"Exponential Function is Continuous/Real Numbers",
"L'Hôpital's Rule",
"Exponential of Zero"
] |
proofwiki-12422 | Limit of x to the x | Let $f: \R_{>0} \to \R$ be defined as:
:$\forall x \in \R_{>0}: \map f x = x^x$
Then:
:$\ds \lim_{x \mathop \to 0^+} x^x = 1$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0^+} x^x
| r = \lim_{x \mathop \to 0^+} \map \exp {x \ln x}
| c = {{Defof|Power (Algebra)|Power|subdef = Real Number|index = 1|disp}}
}}
{{eqn | r = \map \exp {\lim_{x \mathop \to 0^+} x \ln x}
| c = Exponential Function is Continuous
}}
{{eqn | r = \exp 0... | Let $f: \R_{>0} \to \R$ be defined as:
:$\forall x \in \R_{>0}: \map f x = x^x$
Then:
:$\ds \lim_{x \mathop \to 0^+} x^x = 1$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0^+} x^x
| r = \lim_{x \mathop \to 0^+} \map \exp {x \ln x}
| c = {{Defof|Power (Algebra)|Power|subdef = Real Number|index = 1|disp}}
}}
{{eqn | r = \map \exp {\lim_{x \mathop \to 0^+} x \ln x}
| c = [[Exponential Function is Continuous/Real Numbers|Expone... | Limit of x to the x/Proof 2 | https://proofwiki.org/wiki/Limit_of_x_to_the_x | https://proofwiki.org/wiki/Limit_of_x_to_the_x/Proof_2 | [
"Examples of Limits of Real Functions",
"Limit of x to the x"
] | [] | [
"Exponential Function is Continuous/Real Numbers",
"Exponential of Zero"
] |
proofwiki-12423 | Hermite-Lindemann-Weierstrass Theorem/Weaker | Let $a$ be a non-zero algebraic number (possibly complex).
Then:
:$e^a$ is transcendental
where $e$ is Euler's number. | This follows trivially from Hermite-Lindemann-Weierstrass Theorem by taking $n = 1$.
{{qed}}
{{Namedfor|Charles Hermite|name2 = Carl Louis Ferdinand von Lindemann|name3 = Karl Theodor Wilhelm Weierstrass}}
Category:Hermite-Lindemann-Weierstrass Theorem
dc6lltdyk5rumubw9fjpblyf7pt2x9p | Let $a$ be a non-zero [[Definition:Algebraic Number|algebraic number]] (possibly [[Definition:Complex Number|complex]]).
Then:
:$e^a$ is [[Definition:Transcendental Number|transcendental]]
where $e$ is [[Definition:Euler's Number|Euler's number]]. | This follows trivially from [[Hermite-Lindemann-Weierstrass Theorem]] by taking $n = 1$.
{{qed}}
{{Namedfor|Charles Hermite|name2 = Carl Louis Ferdinand von Lindemann|name3 = Karl Theodor Wilhelm Weierstrass}}
[[Category:Hermite-Lindemann-Weierstrass Theorem]]
dc6lltdyk5rumubw9fjpblyf7pt2x9p | Hermite-Lindemann-Weierstrass Theorem/Weaker | https://proofwiki.org/wiki/Hermite-Lindemann-Weierstrass_Theorem/Weaker | https://proofwiki.org/wiki/Hermite-Lindemann-Weierstrass_Theorem/Weaker | [
"Hermite-Lindemann-Weierstrass Theorem"
] | [
"Definition:Algebraic Number",
"Definition:Complex Number",
"Definition:Transcendental Number",
"Definition:Euler's Number"
] | [
"Hermite-Lindemann-Weierstrass Theorem",
"Category:Hermite-Lindemann-Weierstrass Theorem"
] |
proofwiki-12424 | Schanuel's Conjecture Implies Transcendence of Log Pi | Let Schanuel's Conjecture be true.
Then the logarithm of $\pi$ (pi):
:$\ln \pi$
is transcendental. | Assume the truth of Schanuel's Conjecture.
From Schanuel's Conjecture Implies Algebraic Independence of Pi and Log of Pi over the Rationals, $\ln \pi$ and $\pi$ are algebraically independent over the rational numbers $\Q$.
Therefore, if Schanuel's Conjecture holds, $\ln \pi$ must be transcendental.
{{qed}}
Category:Tra... | Let [[Schanuel's Conjecture]] be true.
Then the [[Definition:Logarithm|logarithm]] of [[Definition:Pi|$\pi$ (pi)]]:
:$\ln \pi$
is [[Definition:Transcendental Number|transcendental]]. | Assume the truth of [[Schanuel's Conjecture]].
From [[Schanuel's Conjecture Implies Algebraic Independence of Pi and Log of Pi over the Rationals]], $\ln \pi$ and $\pi$ are [[Definition:Algebraically Independent|algebraically independent]] over the [[Definition:Rational Number|rational numbers $\Q$]].
Therefore, if [... | Schanuel's Conjecture Implies Transcendence of Log Pi | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_Log_Pi | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_Log_Pi | [
"Transcendental Numbers",
"Pi",
"Logarithms",
"Schanuel's Conjecture"
] | [
"Schanuel's Conjecture",
"Definition:Logarithm",
"Definition:Pi",
"Definition:Transcendental Number"
] | [
"Schanuel's Conjecture",
"Schanuel's Conjecture Implies Algebraic Independence of Pi and Log of Pi over the Rationals",
"Definition:Algebraically Independent",
"Definition:Rational Number",
"Schanuel's Conjecture",
"Definition:Transcendental Number",
"Category:Transcendental Numbers",
"Category:Pi",
... |
proofwiki-12425 | Schanuel's Conjecture Implies Transcendence of Pi by Euler's Number | Let Schanuel's Conjecture be true.
Then $\pi \times e$ is transcendental. | Assume the truth of Schanuel's Conjecture.
By Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals, $\pi$ and $e$ are algebraically independent over the rational numbers $\Q$.
That is, no non-trivial polynomials $\map f {x, y}$ with rational coefficients satisfy:
:$\map f {\p... | Let [[Schanuel's Conjecture]] be true.
Then $\pi \times e$ is [[Definition:Transcendental Number|transcendental]]. | Assume the truth of [[Schanuel's Conjecture]].
By [[Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals]], $\pi$ and $e$ are [[Definition:Algebraically Independent|algebraically independent]] over the [[Definition:Rational Number|rational numbers $\Q$]].
That is, no non-tr... | Schanuel's Conjecture Implies Transcendence of Pi by Euler's Number | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_Pi_by_Euler's_Number | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_Pi_by_Euler's_Number | [
"Transcendental Numbers",
"Pi",
"Euler's Number",
"Schanuel's Conjecture"
] | [
"Schanuel's Conjecture",
"Definition:Transcendental Number"
] | [
"Schanuel's Conjecture",
"Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals",
"Definition:Algebraically Independent",
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Algebraic Number",
"Definition:Rational Number",
"Definition:Rati... |
proofwiki-12426 | Schanuel's Conjecture Implies Transcendence of Pi plus Euler's Number | Let Schanuel's Conjecture be true.
Then $\pi + e$ is transcendental. | Assume the truth of Schanuel's Conjecture.
By Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals, $\pi$ and $e$ are algebraically independent over the rational numbers $\Q$.
That is, no non-trivial polynomials $\map f {x, y}$ with rational coefficients satisfy:
:$\map f {\p... | Let [[Schanuel's Conjecture]] be true.
Then $\pi + e$ is [[Definition:Transcendental Number|transcendental]]. | Assume the truth of [[Schanuel's Conjecture]].
By [[Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals]], $\pi$ and $e$ are [[Definition:Algebraically Independent|algebraically independent]] over the [[Definition:Rational Number|rational numbers $\Q$]].
That is, no non-tr... | Schanuel's Conjecture Implies Transcendence of Pi plus Euler's Number | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_Pi_plus_Euler's_Number | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_Pi_plus_Euler's_Number | [
"Transcendental Numbers",
"Pi",
"Euler's Number",
"Schanuel's Conjecture"
] | [
"Schanuel's Conjecture",
"Definition:Transcendental Number"
] | [
"Schanuel's Conjecture",
"Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals",
"Definition:Algebraically Independent",
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Algebraic Number",
"Definition:Rational Number",
"Definition:Rati... |
proofwiki-12427 | Transcendence of Sum or Product of Transcendentals | Let $a$ and $b$ be two transcendental numbers.
Then at least one of $a + b$ and $a \times b$ is transcendental. | Proof by Contradiction:
{{AimForCont}} $a + b$ and $a \times b$ are both not transcendental.
Hence by definition, they are both algebraic.
Hence, $\left({z - a}\right) \left({z - b}\right)$ is a polynomial with algebraic coefficients.
Therefore, $a$ and $b$ must both be algebraic.
However, this contradicts with the ass... | Let $a$ and $b$ be two [[Definition:Transcendental Number|transcendental numbers]].
Then at least one of $a + b$ and $a \times b$ is [[Definition:Transcendental Number|transcendental]]. | [[Proof by Contradiction]]:
{{AimForCont}} $a + b$ and $a \times b$ are both not [[Definition:Transcendental Number|transcendental]].
Hence by definition, they are both [[Definition:Algebraic Number|algebraic]].
Hence, $\left({z - a}\right) \left({z - b}\right)$ is a polynomial with [[Definition:Algebraic Number|alg... | Transcendence of Sum or Product of Transcendentals | https://proofwiki.org/wiki/Transcendence_of_Sum_or_Product_of_Transcendentals | https://proofwiki.org/wiki/Transcendence_of_Sum_or_Product_of_Transcendentals | [
"Transcendental Numbers"
] | [
"Definition:Transcendental Number",
"Definition:Transcendental Number"
] | [
"Proof by Contradiction",
"Definition:Transcendental Number",
"Definition:Algebraic Number",
"Definition:Algebraic Number",
"Definition:Algebraic Number",
"Definition:Transcendental Number",
"Proof by Contradiction",
"Definition:Transcendental Number",
"Category:Transcendental Numbers"
] |
proofwiki-12428 | Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals | Let Schanuel's Conjecture be true.
Then $\pi$ (pi) and $e$ (Euler's number) are algebraically independent over the rational numbers $\Q$. | Assume the truth of Schanuel's Conjecture.
Let $z_1 = 1$ and $z_2 = i \pi$.
Note that $z_1$ is wholly real and $z_2$ is wholly imaginary.
Hence, by Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals, they are linearly independent over $\Q$.
By Schanuel's Conjecture, the extension... | Let [[Schanuel's Conjecture]] be true.
Then [[Definition:Pi|$\pi$ (pi)]] and [[Definition:Euler's Number|$e$ (Euler's number)]] are [[Definition:Algebraically Independent|algebraically independent]] over the [[Definition:Rational Number|rational numbers $\Q$]]. | Assume the truth of [[Schanuel's Conjecture]].
Let $z_1 = 1$ and $z_2 = i \pi$.
Note that $z_1$ is [[Definition:Wholly Real|wholly real]] and $z_2$ is [[Definition:Wholly Imaginary|wholly imaginary]].
Hence, by [[Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals]], they are [... | Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Algebraic_Independence_of_Pi_and_Euler's_Number_over_the_Rationals | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Algebraic_Independence_of_Pi_and_Euler's_Number_over_the_Rationals | [
"Transcendental Numbers",
"Pi",
"Euler's Number",
"Schanuel's Conjecture"
] | [
"Schanuel's Conjecture",
"Definition:Pi",
"Definition:Euler's Number",
"Definition:Algebraically Independent",
"Definition:Rational Number"
] | [
"Schanuel's Conjecture",
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number/Wholly Imaginary",
"Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals",
"Definition:Linearly Independent/Set",
"Schanuel's Conjecture",
"Definition:Field Extension/Comple... |
proofwiki-12429 | Image of Mapping from Finite Set is Finite | Let $X, Y$ be sets.
Let $f: X \to Y$ be a mapping.
Let $X$ be a finite set.
Then $f \sqbrk X$ is a finite set. | By definition of surjection:
:$f: X \to f \sqbrk X$ is a surjection.
The case when $X \ne \O$:
By Surjection iff Cardinal Inequality:
:$\card {f \sqbrk X} \le \card X$
Thus by Set of Cardinality not Greater than Cardinality of Finite Set is Finite:
:$f \sqbrk X$ is finite.
{{qed|lemma}}
The case when $X = \O$:
By {{Cor... | Let $X, Y$ be [[Definition:Set|sets]].
Let $f: X \to Y$ be a [[Definition:Mapping|mapping]].
Let $X$ be a [[Definition:Finite Set|finite set]].
Then $f \sqbrk X$ is a [[Definition:Finite Set|finite set]]. | By definition of [[Definition:Surjection|surjection]]:
:$f: X \to f \sqbrk X$ is a [[Definition:Surjection|surjection]].
The case when $X \ne \O$:
By [[Surjection iff Cardinal Inequality]]:
:$\card {f \sqbrk X} \le \card X$
Thus by [[Set of Cardinality not Greater than Cardinality of Finite Set is Finite]]:
:$f \sqb... | Image of Mapping from Finite Set is Finite | https://proofwiki.org/wiki/Image_of_Mapping_from_Finite_Set_is_Finite | https://proofwiki.org/wiki/Image_of_Mapping_from_Finite_Set_is_Finite | [
"Images",
"Finite Sets"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Finite Set",
"Definition:Finite Set"
] | [
"Definition:Surjection",
"Definition:Surjection",
"Surjection iff Cardinal Inequality",
"Set of Cardinality not Greater than Cardinality of Finite Set is Finite",
"Definition:Finite Set"
] |
proofwiki-12430 | Magic Constant of Order 3 Magic Square | The magic constant of the order $3$ magic square is $15$. | Let $M_3$ denote the order $3$ magic square
By Sum of Terms of Magic Square, the total of all the entries in $M_3$ is given by:
:$T_3 = \dfrac {3^2 \left({3^2 + 1}\right)} 2 = \dfrac {9 \times 10} 2 = 45$
As there are $3$ rows of $M_3$, the magic constant of $M_3$ is given by:
:$S_3 = \dfrac {45} 3 = 15$
{{qed}} | The [[Definition:Magic Constant|magic constant]] of the [[Magic Square/Examples/Order 3|order $3$ magic square]] is $15$. | Let $M_3$ denote the [[Magic Square/Examples/Order 3|order $3$ magic square]]
By [[Sum of Terms of Magic Square]], the total of all the entries in $M_3$ is given by:
:$T_3 = \dfrac {3^2 \left({3^2 + 1}\right)} 2 = \dfrac {9 \times 10} 2 = 45$
As there are $3$ [[Definition:Row of Matrix|rows]] of $M_3$, the [[Definiti... | Magic Constant of Order 3 Magic Square/Proof 1 | https://proofwiki.org/wiki/Magic_Constant_of_Order_3_Magic_Square | https://proofwiki.org/wiki/Magic_Constant_of_Order_3_Magic_Square/Proof_1 | [
"Magic Squares",
"15",
"Magic Constant of Order 3 Magic Square"
] | [
"Definition:Magic Square/Magic Constant",
"Magic Square/Examples/Order 3"
] | [
"Magic Square/Examples/Order 3",
"Sum of Terms of Magic Square",
"Definition:Matrix/Row",
"Definition:Magic Square/Magic Constant"
] |
proofwiki-12431 | Magic Constant of Order 3 Magic Square | The magic constant of the order $3$ magic square is $15$. | 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 $n = 3$:
:$S_3 = \dfrac {3 \times 10} 2 = 15$
{{qed}} | The [[Definition:Magic Constant|magic constant]] of the [[Magic Square/Examples/Order 3|order $3$ magic square]] is $15$. | 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 $n = 3$:
:$S_3 = \dfrac {3 \times 10} 2 = ... | Magic Constant of Order 3 Magic Square/Proof 2 | https://proofwiki.org/wiki/Magic_Constant_of_Order_3_Magic_Square | https://proofwiki.org/wiki/Magic_Constant_of_Order_3_Magic_Square/Proof_2 | [
"Magic Squares",
"15",
"Magic Constant of Order 3 Magic Square"
] | [
"Definition:Magic Square/Magic Constant",
"Magic Square/Examples/Order 3"
] | [
"Definition:Magic Square",
"Definition:Magic Square/Order",
"Magic Constant of Magic Square",
"Definition:Magic Square/Magic Constant"
] |
proofwiki-12432 | Sum of Terms of Magic Square | The total of all the entries in a magic square of order $n$ is given by:
:$T_n = \dfrac {n^2 \paren {n^2 + 1} } 2$ | Let $M_n$ denote a magic square of order $n$.
$M_n$ is by definition a square matrix of order $n$ containing the positive integers from $1$ upwards.
Thus there are $n^2$ entries in $M_n$, going from $1$ to $n^2$.
Thus:
{{begin-eqn}}
{{eqn | l = T_n
| r = \sum_{k \mathop = 1}^{n^2} k
| c =
}}
{{eqn | r = \f... | The total of all the entries in a [[Definition:Magic Square|magic square]] of [[Definition:Order of Magic Square|order $n$]] is given by:
:$T_n = \dfrac {n^2 \paren {n^2 + 1} } 2$ | Let $M_n$ denote a [[Definition:Magic Square|magic square]] of [[Definition:Order of Magic Square|order $n$]].
$M_n$ is by definition a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]] containing the [[Definition:Positive Integer|positive integers]] from $1$ upwards.
Thus ... | Sum of Terms of Magic Square | https://proofwiki.org/wiki/Sum_of_Terms_of_Magic_Square | https://proofwiki.org/wiki/Sum_of_Terms_of_Magic_Square | [
"Magic Squares"
] | [
"Definition:Magic Square",
"Definition:Magic Square/Order"
] | [
"Definition:Magic Square",
"Definition:Magic Square/Order",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Positive/Integer",
"Closed Form for Triangular Numbers"
] |
proofwiki-12433 | Schanuel's Conjecture Implies Transcendence of 2 to the power of Euler's Number | Let Schanuel's Conjecture be true.
Then $2$ to the power of Euler's number $e$:
:$2^e$
is transcendental, where $e$ is Euler's number. | Assume the truth of Schanuel's Conjecture.
Let $z_1 = \ln \ln 2$, $z_2 = 1 + \ln \ln 2$, $z_3 = \ln 2$, and $z_4 = e \ln 2$.
By Lemma, they are linearly independent over the rational numbers $\Q$.
Observe that $z_3 = e^{z_1}$ and $z_4 = e^{z_2}$.
By Schanuel's Conjecture, the extension field $\map \Q {z_1, z_2, z_3, z_... | Let [[Schanuel's Conjecture]] be true.
Then $2$ to the [[Definition:Power to Real Number|power]] of [[Definition:Euler's Number|Euler's number $e$]]:
:$2^e$
is [[Definition:Transcendental Number|transcendental]], where $e$ is [[Definition:Euler's Number|Euler's number]]. | Assume the truth of [[Schanuel's Conjecture]].
Let $z_1 = \ln \ln 2$, $z_2 = 1 + \ln \ln 2$, $z_3 = \ln 2$, and $z_4 = e \ln 2$.
By [[Schanuel's Conjecture Implies Transcendence of 2 to the power of Euler's Number/Lemma|Lemma]], they are [[Definition:Linearly Independent Set|linearly independent]] over the [[Definiti... | Schanuel's Conjecture Implies Transcendence of 2 to the power of Euler's Number | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_2_to_the_power_of_Euler's_Number | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_2_to_the_power_of_Euler's_Number | [
"Transcendental Numbers",
"2",
"Euler's Number",
"Schanuel's Conjecture"
] | [
"Schanuel's Conjecture",
"Definition:Power (Algebra)/Real Number",
"Definition:Euler's Number",
"Definition:Transcendental Number",
"Definition:Euler's Number"
] | [
"Schanuel's Conjecture",
"Schanuel's Conjecture Implies Transcendence of 2 to the power of Euler's Number/Lemma",
"Definition:Linearly Independent/Set",
"Definition:Rational Number",
"Schanuel's Conjecture",
"Definition:Field Extension",
"Definition:Transcendence Degree",
"Definition:Rational Number",... |
proofwiki-12434 | Schanuel's Conjecture Implies Transcendence of Pi to the power of Euler's Number | Let Schanuel's Conjecture be true.
Then $\pi$(pi) to the power of Euler's number $e$:
:$\pi^e$
is transcendental. | Assume the truth of Schanuel's Conjecture.
Let $z_1 = \ln \ln \pi$, $z_2 = 1 + \ln \ln \pi$, $z_3 = \ln \pi$, $z_4 = e \ln \pi$, and $z_5 = i \pi$.
By Lemma, they are linearly independent over the rational numbers $\Q$.
Observe that $z_3 = e^{z_1}$ and $z_4 = e^{z_2}$.
By Schanuel's Conjecture, the extension field $\Q ... | Let [[Schanuel's Conjecture]] be true.
Then [[Definition:Pi|$\pi$(pi)]] to the [[Definition:Power to Real Number|power]] of [[Definition:Euler's Number|Euler's number $e$]]:
:$\pi^e$
is [[Definition:Transcendental Number|transcendental]]. | Assume the truth of [[Schanuel's Conjecture]].
Let $z_1 = \ln \ln \pi$, $z_2 = 1 + \ln \ln \pi$, $z_3 = \ln \pi$, $z_4 = e \ln \pi$, and $z_5 = i \pi$.
By [[Schanuel's Conjecture Implies Transcendence of Pi to the power of Euler's Number/Lemma|Lemma]], they are [[Definition:Linearly Independent Set|linearly independe... | Schanuel's Conjecture Implies Transcendence of Pi to the power of Euler's Number | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_Pi_to_the_power_of_Euler's_Number | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_Pi_to_the_power_of_Euler's_Number | [
"Transcendental Numbers",
"Pi",
"Euler's Number",
"Schanuel's Conjecture"
] | [
"Schanuel's Conjecture",
"Definition:Pi",
"Definition:Power (Algebra)/Real Number",
"Definition:Euler's Number",
"Definition:Transcendental Number"
] | [
"Schanuel's Conjecture",
"Schanuel's Conjecture Implies Transcendence of Pi to the power of Euler's Number/Lemma",
"Definition:Linearly Independent/Set",
"Definition:Rational Number",
"Schanuel's Conjecture",
"Definition:Field Extension",
"Definition:Transcendence Degree",
"Definition:Rational Number"... |
proofwiki-12435 | Schanuel's Conjecture Implies Transcendence of Euler's Number to the power of Euler's Number | Let Schanuel's Conjecture be true.
Then Euler's number $e$ to the power of itself:
:$e^e$
is transcendental. | Assume the truth of Schanuel's Conjecture.
Let $z_1 = 1$, $z_2 = e$.
By Euler's Number is Irrational, $z_1$ and $z_2$ are linearly independent over $\Q$.
By Schanuel's Conjecture, the extension field $\Q \left({z_1, z_2, e^{z_1}, e^{z_2}}\right)$ has transcendence degree at least $2$ over $\Q$.
That is, the extension f... | Let [[Schanuel's Conjecture]] be true.
Then [[Definition:Euler's Number|Euler's number $e$]] to the [[Definition:Power to Real Number|power]] of itself:
:$e^e$
is [[Definition:Transcendental Number|transcendental]]. | Assume the truth of [[Schanuel's Conjecture]].
Let $z_1 = 1$, $z_2 = e$.
By [[Euler's Number is Irrational]], $z_1$ and $z_2$ are [[Definition:Linearly Independent|linearly independent]] over $\Q$.
By [[Schanuel's Conjecture]], the [[Definition:Field Extension/Complex|extension field]] $\Q \left({z_1, z_2, e^{z_1}, ... | Schanuel's Conjecture Implies Transcendence of Euler's Number to the power of Euler's Number | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_Euler's_Number_to_the_power_of_Euler's_Number | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Transcendence_of_Euler's_Number_to_the_power_of_Euler's_Number | [
"Transcendental Numbers",
"Euler's Number",
"Schanuel's Conjecture"
] | [
"Schanuel's Conjecture",
"Definition:Euler's Number",
"Definition:Power (Algebra)/Real Number",
"Definition:Transcendental Number"
] | [
"Schanuel's Conjecture",
"Euler's Number is Irrational",
"Definition:Linearly Independent",
"Schanuel's Conjecture",
"Definition:Field Extension/Complex",
"Definition:Transcendence Degree",
"Definition:Field Extension/Complex",
"Definition:Transcendence Degree",
"Definition:Algebraic Number",
"Sch... |
proofwiki-12436 | Magic Constant of Magic Square | The magic constant of a magic square of order $n$ is given by:
:$S_n = \dfrac {n \paren {n^2 + 1} } 2$ | Let $M_n$ denote a magic square of order $n$.
By Sum of Terms of Magic Square, the total of all the entries in a magic square of order $n$ is given by:
:$T_n = \dfrac {n^2 \paren {n^2 + 1} } 2$
There are $n$ rows in $M_n$, each one with the same magic constant.
Thus the magic constant $S_n$ of the magic square $M_n$ is... | The [[Definition:Magic Constant|magic constant]] of a [[Definition:Magic Square|magic square]] of [[Definition:Order of Magic Square|order $n$]] is given by:
:$S_n = \dfrac {n \paren {n^2 + 1} } 2$ | Let $M_n$ denote a [[Definition:Magic Square|magic square]] of [[Definition:Order of Magic Square|order $n$]].
By [[Sum of Terms of Magic Square]], the total of all the entries in a [[Definition:Magic Square|magic square]] of [[Definition:Order of Magic Square|order $n$]] is given by:
:$T_n = \dfrac {n^2 \paren {n^2 ... | Magic Constant of Magic Square | https://proofwiki.org/wiki/Magic_Constant_of_Magic_Square | https://proofwiki.org/wiki/Magic_Constant_of_Magic_Square | [
"Magic Squares"
] | [
"Definition:Magic Square/Magic Constant",
"Definition:Magic Square",
"Definition:Magic Square/Order"
] | [
"Definition:Magic Square",
"Definition:Magic Square/Order",
"Sum of Terms of Magic Square",
"Definition:Magic Square",
"Definition:Magic Square/Order",
"Definition:Matrix/Row",
"Definition:Magic Square/Magic Constant",
"Definition:Magic Square/Magic Constant",
"Definition:Magic Square",
"Sum of Te... |
proofwiki-12437 | Magic Square of Order 3 is Unique | Up to rotations and reflections, the magic square of order $3$ is unique:
{{:Magic Square/Examples/Order 3}} | Let $M_3$ denote the magic square of order $3$.
Each row, column and diagonal of $M_3$ must be a different set of $3$ elements of $\N_9$, where $\N_9$ denotes the set $\set {1, 2, 3, 4, 5, 6, 7, 8, 9}$.
The sets of $3$ elements of $\N_9$ adding to $15$ can be stated:
:$\set {1, 5, 9}, \set {1, 6, 8}$
:$\set {2, 4, 9}, ... | Up to rotations and reflections, the [[Definition:Order 3 Magic Square|magic square of order $3$]] is [[Definition:Unique|unique]]:
{{:Magic Square/Examples/Order 3}} | Let $M_3$ denote the [[Definition:Order 3 Magic Square|magic square of order $3$]].
Each [[Definition:Row of Matrix|row]], [[Definition:Column of Matrix|column]] and [[Definition:Diagonal of Array|diagonal]] of $M_3$ must be a different [[Definition:Set|set]] of $3$ [[Definition:Element|elements]] of $\N_9$, where $\N... | Magic Square of Order 3 is Unique | https://proofwiki.org/wiki/Magic_Square_of_Order_3_is_Unique | https://proofwiki.org/wiki/Magic_Square_of_Order_3_is_Unique | [
"Magic Squares"
] | [
"Magic Square/Examples/Order 3",
"Definition:Unique"
] | [
"Magic Square/Examples/Order 3",
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Definition:Array/Diagonal",
"Definition:Set",
"Definition:Element",
"Definition:Set",
"Definition:Set",
"Definition:Element",
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Definition:Array/Diagonal",... |
proofwiki-12438 | Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals | Let $z_1$ be a non-zero wholly real number.
Let $z_2$ be a non-zero wholly imaginary number.
Then, $z_1$ and $z_2$ are linearly independent over the rational numbers $\Q$, where the group is the complex numbers $\C$. | From Rational Numbers form Subfield of Complex Numbers, the unitary module $\struct {\C, +, \times}_\Q$ over $\Q$ satisfies the unitary module axioms:
* Ring Action: $\C$ is closed under multiplication, so $\Q \times \C \subset \C$.
* Distributive: $\times$ distributes over $+$.
* Associativity: $\times$ is associative... | Let $z_1$ be a non-zero [[Definition:Wholly Real|wholly real]] number.
Let $z_2$ be a non-zero [[Definition:Wholly Imaginary|wholly imaginary]] number.
Then, $z_1$ and $z_2$ are [[Definition:Linearly Independent Set|linearly independent]] over the [[Definition:Rational Number|rational numbers $\Q$]], where the [[Def... | From [[Rational Numbers form Subfield of Complex Numbers]], the [[Definition:Unitary Module|unitary module]] $\struct {\C, +, \times}_\Q$ over $\Q$ satisfies the [[Axiom:Unitary Module Axioms|unitary module axioms]]:
* [[Definition:Left Linear Ring Action|Ring Action]]: $\C$ is closed under multiplication, so $\Q \tim... | Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals | https://proofwiki.org/wiki/Wholly_Real_Number_and_Wholly_Imaginary_Number_are_Linearly_Independent_over_the_Rationals | https://proofwiki.org/wiki/Wholly_Real_Number_and_Wholly_Imaginary_Number_are_Linearly_Independent_over_the_Rationals | [
"Complex Analysis"
] | [
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Linearly Independent/Set",
"Definition:Rational Number",
"Definition:Group",
"Definition:Complex Number"
] | [
"Rational Numbers form Subfield of Complex Numbers",
"Definition:Unitary Module over Ring",
"Axiom:Unitary Left Module Axioms",
"Definition:Linear Ring Action/Left",
"Definition:Distributive Operation",
"Definition:Associative Operation",
"Definition:Multiplicative Identity",
"Definition:Complex Numbe... |
proofwiki-12439 | Schanuel's Conjecture Implies Algebraic Independence of Pi and Log of Pi over the Rationals | Let Schanuel's Conjecture be true.
Then $\pi$ (pi) and the logarithm of $\pi$ (pi):
:$\ln \pi$
are algebraically independent over the rational numbers $\Q$. | Assume the truth of Schanuel's Conjecture.
Let $z_1 = \ln \pi$, $z_2 = i \pi$.
Note that $z_1$ is wholly real and $z_2$ is wholly imaginary.
Hence, by Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals, they are linearly independent over $\Q$.
By Schanuel's Conjecture, the extens... | Let [[Schanuel's Conjecture]] be true.
Then [[Definition:Pi|$\pi$ (pi)]] and the [[Definition:Logarithm|logarithm]] of [[Definition:Pi|$\pi$ (pi)]]:
:$\ln \pi$
are [[Definition:Algebraically Independent|algebraically independent]] over the [[Definition:Rational Number|rational numbers $\Q$]]. | Assume the truth of [[Schanuel's Conjecture]].
Let $z_1 = \ln \pi$, $z_2 = i \pi$.
Note that $z_1$ is [[Definition:Wholly Real|wholly real]] and $z_2$ is [[Definition:Wholly Imaginary|wholly imaginary]].
Hence, by [[Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals]], they ar... | Schanuel's Conjecture Implies Algebraic Independence of Pi and Log of Pi over the Rationals | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Algebraic_Independence_of_Pi_and_Log_of_Pi_over_the_Rationals | https://proofwiki.org/wiki/Schanuel's_Conjecture_Implies_Algebraic_Independence_of_Pi_and_Log_of_Pi_over_the_Rationals | [
"Transcendental Numbers",
"Pi",
"Logarithms",
"Schanuel's Conjecture"
] | [
"Schanuel's Conjecture",
"Definition:Pi",
"Definition:Logarithm",
"Definition:Pi",
"Definition:Algebraically Independent",
"Definition:Rational Number"
] | [
"Schanuel's Conjecture",
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number/Wholly Imaginary",
"Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals",
"Definition:Linearly Independent/Set",
"Schanuel's Conjecture",
"Definition:Field Extension/Comple... |
proofwiki-12440 | Hermite-Lindemann-Weierstrass Theorem/Weaker/Corollary | Let $a$ be a algebraic number (possibly complex) which is neither $0$ nor $1$.
Then:
:every value of $\ln a$ is transcendental
where $\ln$ denotes complex natural logarithm. | {{AimForCont}} $\ln a$ is not transcendental.
Hence, by definition, it is algebraic.
Since $a$ is not $1$, $\ln a$ cannot be $0$.
Hence, by the Weaker Hermite-Lindemann-Weierstrass Theorem, $e^{\ln a} = a$ is transcendental.
This contradicts the assumption that $a$ is algebraic.
Hence, $\ln a$ must be transcendental.
{... | Let $a$ be a [[Definition:Algebraic Number|algebraic number]] (possibly [[Definition:Complex Number|complex]]) which is neither $0$ nor $1$.
Then:
:every value of $\ln a$ is [[Definition:Transcendental Number|transcendental]]
where $\ln$ denotes [[Definition:Complex Natural Logarithm|complex natural logarithm]]. | {{AimForCont}} $\ln a$ is not [[Definition:Transcendental Number|transcendental]].
Hence, by definition, it is [[Definition:Algebraic Number|algebraic]].
Since $a$ is not $1$, $\ln a$ cannot be $0$.
Hence, by the [[Weaker Hermite-Lindemann-Weierstrass Theorem]], $e^{\ln a} = a$ is [[Definition:Transcendental Number|... | Hermite-Lindemann-Weierstrass Theorem/Weaker/Corollary | https://proofwiki.org/wiki/Hermite-Lindemann-Weierstrass_Theorem/Weaker/Corollary | https://proofwiki.org/wiki/Hermite-Lindemann-Weierstrass_Theorem/Weaker/Corollary | [
"Hermite-Lindemann-Weierstrass Theorem"
] | [
"Definition:Algebraic Number",
"Definition:Complex Number",
"Definition:Transcendental Number",
"Definition:Natural Logarithm/Complex"
] | [
"Definition:Transcendental Number",
"Definition:Algebraic Number",
"Hermite-Lindemann-Weierstrass Theorem/Weaker",
"Definition:Transcendental Number",
"Definition:Contradiction",
"Definition:Algebraic Number",
"Definition:Transcendental Number",
"Category:Hermite-Lindemann-Weierstrass Theorem"
] |
proofwiki-12441 | Lines through Center Square of Order 3 Magic Square are in Arithmetic Sequence | Consider the order 3 magic square:
{{:Magic Square/Examples/Order 3}}
Each of the lines through the center cell contain $3$ integers in arithmetic sequence. | By observation:
:$\tuple {1, 5, 9}$: common difference $4$
:$\tuple {2, 5, 8}$: common difference $3$
:$\tuple {3, 5, 7}$: common difference $2$
:$\tuple {4, 5, 6}$: common difference $1$
{{qed}} | Consider the [[Definition:Order 3 Magic Square|order 3 magic square]]:
{{:Magic Square/Examples/Order 3}}
Each of the lines through the center cell contain $3$ [[Definition:Integer|integers]] in [[Definition:Arithmetic Sequence|arithmetic sequence]]. | By observation:
:$\tuple {1, 5, 9}$: [[Definition:Common Difference|common difference]] $4$
:$\tuple {2, 5, 8}$: [[Definition:Common Difference|common difference]] $3$
:$\tuple {3, 5, 7}$: [[Definition:Common Difference|common difference]] $2$
:$\tuple {4, 5, 6}$: [[Definition:Common Difference|common difference]] $1$... | Lines through Center Square of Order 3 Magic Square are in Arithmetic Sequence | https://proofwiki.org/wiki/Lines_through_Center_Square_of_Order_3_Magic_Square_are_in_Arithmetic_Sequence | https://proofwiki.org/wiki/Lines_through_Center_Square_of_Order_3_Magic_Square_are_in_Arithmetic_Sequence | [
"Magic Squares",
"Arithmetic Sequences"
] | [
"Magic Square/Examples/Order 3",
"Definition:Integer",
"Definition:Arithmetic Sequence"
] | [
"Definition:Arithmetic Sequence/Common Difference",
"Definition:Arithmetic Sequence/Common Difference",
"Definition:Arithmetic Sequence/Common Difference",
"Definition:Arithmetic Sequence/Common Difference"
] |
proofwiki-12442 | Sums of Squares in Lines of Order 3 Magic Square | Consider the order 3 magic square:
{{:Magic Square/Examples/Order 3}}
: The sums of the squares of the top and bottom rows are equal, and differ by $18$ from the sums of the squares of the middle row
: The sums of the squares of the left and right columns are equal , and differ by $18$ from the sums of the squares of t... | For the rows:
{{begin-eqn}}
{{eqn | l = 2^2 + 7^2 + 6^2
| r = 4 + 49 + 36
| c =
}}
{{eqn | r = 89
| c =
}}
{{eqn | l = 4^2 + 3^2 + 8^2
| r = 16 + 9 + 64
| c =
}}
{{eqn | r = 89
| c =
}}
{{eqn | l = 9^2 + 5^2 + 1^2
| r = 81 + 25 + 1
| c =
}}
{{eqn | r = 107
| c ... | Consider the [[Definition:Order 3 Magic Square|order 3 magic square]]:
{{:Magic Square/Examples/Order 3}}
: The sums of the [[Definition:Square (Algebra)|squares]] of the top and bottom rows are equal, and differ by $18$ from the sums of the [[Definition:Square (Algebra)|squares]] of the middle row
: The sums of the [... | For the rows:
{{begin-eqn}}
{{eqn | l = 2^2 + 7^2 + 6^2
| r = 4 + 49 + 36
| c =
}}
{{eqn | r = 89
| c =
}}
{{eqn | l = 4^2 + 3^2 + 8^2
| r = 16 + 9 + 64
| c =
}}
{{eqn | r = 89
| c =
}}
{{eqn | l = 9^2 + 5^2 + 1^2
| r = 81 + 25 + 1
| c =
}}
{{eqn | r = 107
| c... | Sums of Squares in Lines of Order 3 Magic Square | https://proofwiki.org/wiki/Sums_of_Squares_in_Lines_of_Order_3_Magic_Square | https://proofwiki.org/wiki/Sums_of_Squares_in_Lines_of_Order_3_Magic_Square | [
"Magic Squares"
] | [
"Magic Square/Examples/Order 3",
"Definition:Square/Function",
"Definition:Square/Function",
"Definition:Square/Function",
"Definition:Square/Function"
] | [] |
proofwiki-12443 | Omega Constant is Transcendental | The omega constant is transcendental. | From the definition of omega constant, it is the real number $\Omega$ such that:
:$\Omega \, e^\Omega = 1$
where $e$ denotes Euler's number.
{{AimForCont}} $\Omega$ is not transcendental.
Hence, by definition, $\Omega$ is algebraic.
Then $e^\Omega$ is also algebraic, because:
:$e^\Omega = \dfrac 1 \Omega$
However, by t... | The [[Definition:Omega Constant|omega constant]] is [[Definition:Transcendental Number|transcendental]]. | From the definition of [[Definition:Omega Constant|omega constant]], it is the [[Definition:Real Number|real number]] $\Omega$ such that:
:$\Omega \, e^\Omega = 1$
where $e$ denotes [[Definition:Euler's Number|Euler's number]].
{{AimForCont}} $\Omega$ is not [[Definition:Transcendental Number|transcendental]].
Hence... | Omega Constant is Transcendental | https://proofwiki.org/wiki/Omega_Constant_is_Transcendental | https://proofwiki.org/wiki/Omega_Constant_is_Transcendental | [
"Omega Constant",
"Lambert W Function",
"Transcendental Number Theory"
] | [
"Definition:Omega Constant",
"Definition:Transcendental Number"
] | [
"Definition:Omega Constant",
"Definition:Real Number",
"Definition:Euler's Number",
"Definition:Transcendental Number",
"Definition:Algebraic Number",
"Definition:Algebraic Number",
"Hermite-Lindemann-Weierstrass Theorem/Weaker",
"Definition:Transcendental Number",
"Definition:Contradiction",
"Def... |
proofwiki-12444 | Lambert W of Zero is Zero | Let $W_0$ denote principal branch of the Lambert W function.
Then:
:$W_0 \left({0}\right) = 0$ | From the definition of the principal branch of the Lambert W function:
:$y = W_0 \left({x}\right) \iff x = y e^y$
where $x \in \left[{-\dfrac 1 e \,.\,.\, \to}\right)$ and $y \in \left[{-1 \,.\,.\, \to}\right)$.
The result follows from substituting $x = 0$ and $y = 0$.
{{qed}}
Category:Lambert W Function
eqh9slzcxvkfb6... | Let $W_0$ denote [[Definition:Lambert W Function/Principal Branch|principal branch of the Lambert W function]].
Then:
:$W_0 \left({0}\right) = 0$ | From the definition of the [[Definition:Lambert W Function/Principal Branch|principal branch of the Lambert W function]]:
:$y = W_0 \left({x}\right) \iff x = y e^y$
where $x \in \left[{-\dfrac 1 e \,.\,.\, \to}\right)$ and $y \in \left[{-1 \,.\,.\, \to}\right)$.
The result follows from substituting $x = 0$ and $y = 0$... | Lambert W of Zero is Zero | https://proofwiki.org/wiki/Lambert_W_of_Zero_is_Zero | https://proofwiki.org/wiki/Lambert_W_of_Zero_is_Zero | [
"Lambert W Function"
] | [
"Definition:Lambert W Function/Principal Branch/Real Valued"
] | [
"Definition:Lambert W Function/Principal Branch/Real Valued",
"Category:Lambert W Function"
] |
proofwiki-12445 | Sums of Squares of Lines of Order 3 Magic Square | Consider the order 3 magic square:
{{:Magic Square/Examples/Order 3}}
: The sums of the squares of the rows, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same rows of that same order 3 magic square when reflected in a vertical axis:
$\quad\begin{array}{|c|c|c|}
\hline 6 & 7... | For the rows:
{{begin-eqn}}
{{eqn | l = 276^2 + 951^2 + 438^2
| r = 76176 + 904401 + 191844
| c =
}}
{{eqn | r = 1172421
| c =
}}
{{eqn | l = 672^2 + 159^2 + 834^2
| r = 451584 + 25281 + 695556
| c =
}}
{{eqn | r = 1172421
| c =
}}
{{end-eqn}}
For the columns:
{{begin-eqn}}
{{eqn... | Consider the [[Definition:Order 3 Magic Square|order 3 magic square]]:
{{:Magic Square/Examples/Order 3}}
: The sums of the [[Definition:Square (Algebra)|squares]] of the [[Definition:Row of Matrix|rows]], when expressed as $3$-digit [[Definition:Decimal Notation|decimal numbers]], are equal to the sums of the [[Defin... | For the rows:
{{begin-eqn}}
{{eqn | l = 276^2 + 951^2 + 438^2
| r = 76176 + 904401 + 191844
| c =
}}
{{eqn | r = 1172421
| c =
}}
{{eqn | l = 672^2 + 159^2 + 834^2
| r = 451584 + 25281 + 695556
| c =
}}
{{eqn | r = 1172421
| c =
}}
{{end-eqn}}
For the columns:
{{begin-eqn}}
{... | Sums of Squares of Lines of Order 3 Magic Square | https://proofwiki.org/wiki/Sums_of_Squares_of_Lines_of_Order_3_Magic_Square | https://proofwiki.org/wiki/Sums_of_Squares_of_Lines_of_Order_3_Magic_Square | [
"Magic Squares"
] | [
"Magic Square/Examples/Order 3",
"Definition:Square/Function",
"Definition:Matrix/Row",
"Definition:Decimal Notation",
"Definition:Square/Function",
"Definition:Matrix/Row",
"Magic Square/Examples/Order 3",
"Definition:Square/Function",
"Definition:Matrix/Column",
"Definition:Decimal Notation",
... | [] |
proofwiki-12446 | 123456789 x 8 + 9 = 987654321 | {{begin-eqn}}
{{eqn | l = 1 \times 8 + 1
| r = 9
}}
{{eqn | l = 12 \times 8 + 2
| r = 98
}}
{{eqn | l = 123 \times 8 + 3
| r = 987
}}
{{eqn | l = 1234 \times 8 + 4
| r = 9876
}}
{{eqn | l = 12345 \times 8 + 5
| r = 98765
}}
{{eqn | l = 123456 \times 8 + 6
| r = 987654
}}
{{eqn | l = ... | The proof proceeds by induction.
Let $n, b \in \Z_{>0}$, where $b \ge 3$.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
:$\ds \paren {b - 2} \sum_{j \mathop = 1}^n j b^{n - j} + n = \sum_{j \mathop = 1}^n \paren {b - j} b^{n - j}$ | {{begin-eqn}}
{{eqn | l = 1 \times 8 + 1
| r = 9
}}
{{eqn | l = 12 \times 8 + 2
| r = 98
}}
{{eqn | l = 123 \times 8 + 3
| r = 987
}}
{{eqn | l = 1234 \times 8 + 4
| r = 9876
}}
{{eqn | l = 12345 \times 8 + 5
| r = 98765
}}
{{eqn | l = 123456 \times 8 + 6
| r = 987654
}}
{{eqn | l = ... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
Let $n, b \in \Z_{>0}$, where $b \ge 3$.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \paren {b - 2} \sum_{j \mathop = 1}^n j b^{n - j} + n = \sum_{j \mathop = 1}^n \paren {b - j} b^{n - j}$ | 123456789 x 8 + 9 = 987654321 | https://proofwiki.org/wiki/123456789_x_8_+_9_=_987654321 | https://proofwiki.org/wiki/123456789_x_8_+_9_=_987654321 | [
"Recreational Mathematics"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-12447 | 123456789 x 9 + 10 = 1111111111 | {{begin-eqn}}
{{eqn | l = 1 \times 9 + 2
| r = 11
}}
{{eqn | l = 12 \times 9 + 3
| r = 111
}}
{{eqn | l = 123 \times 9 + 4
| r = 1111
}}
{{eqn | l = 1234 \times 9 + 5
| r = 11111
}}
{{eqn | l = 12345 \times 9 + 6
| r = 111111
}}
{{eqn | l = 123456 \times 9 + 7
| r = 1111111
}}
{{eqn ... | The proof proceeds by induction.
Let $n, b \in \Z_{>0}$, where $b \ge 3$.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
:$\ds \paren {b - 1} \sum_{j \mathop = 1}^n j b^{n - j} + n + 1 = \sum_{j \mathop = 0}^n b^j$ | {{begin-eqn}}
{{eqn | l = 1 \times 9 + 2
| r = 11
}}
{{eqn | l = 12 \times 9 + 3
| r = 111
}}
{{eqn | l = 123 \times 9 + 4
| r = 1111
}}
{{eqn | l = 1234 \times 9 + 5
| r = 11111
}}
{{eqn | l = 12345 \times 9 + 6
| r = 111111
}}
{{eqn | l = 123456 \times 9 + 7
| r = 1111111
}}
{{eqn ... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
Let $n, b \in \Z_{>0}$, where $b \ge 3$.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \paren {b - 1} \sum_{j \mathop = 1}^n j b^{n - j} + n + 1 = \sum_{j \mathop = 0}^n b^j$ | 123456789 x 9 + 10 = 1111111111 | https://proofwiki.org/wiki/123456789_x_9_+_10_=_1111111111 | https://proofwiki.org/wiki/123456789_x_9_+_10_=_1111111111 | [
"Recreational Mathematics"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-12448 | Algebraic Numbers form Field | Let $\Bbb A$ denote the set of algebraic numbers in $\C$.
Then the algebraic structure $\struct {\Bbb A, +, \times}$ is a field.
In the above, $+$ and $\times$ denote complex addition and complex multiplication respectively. | {{ProofWanted}}
Category:Algebraic Numbers
Category:Examples of Fields
Category:Algebraic Number Theory
fq9uwwnem55uakz1yx3fs8skplq2ko0 | Let $\Bbb A$ denote the [[Definition:Set|set]] of [[Definition:Algebraic Number|algebraic numbers]] in $\C$.
Then the [[Definition:Algebraic Structure|algebraic structure]] $\struct {\Bbb A, +, \times}$ is a [[Definition:Field (Abstract Algebra)|field]].
In the above, $+$ and $\times$ denote [[Definition:Complex Ad... | {{ProofWanted}}
[[Category:Algebraic Numbers]]
[[Category:Examples of Fields]]
[[Category:Algebraic Number Theory]]
fq9uwwnem55uakz1yx3fs8skplq2ko0 | Algebraic Numbers form Field | https://proofwiki.org/wiki/Algebraic_Numbers_form_Field | https://proofwiki.org/wiki/Algebraic_Numbers_form_Field | [
"Algebraic Numbers",
"Examples of Fields",
"Algebraic Number Theory"
] | [
"Definition:Set",
"Definition:Algebraic Number",
"Definition:Algebraic Structure",
"Definition:Field (Abstract Algebra)",
"Definition:Addition/Complex Numbers",
"Definition:Multiplication/Complex Numbers"
] | [
"Category:Algebraic Numbers",
"Category:Examples of Fields",
"Category:Algebraic Number Theory"
] |
proofwiki-12449 | Linearly Independent over the Rational Numbers iff Linearly Independent over the Integers | Let $z_1, z_2, \ldots, z_n$ be complex numbers.
Then:
:$z_1, z_2, \ldots, z_n$ are linearly independent over the rational numbers $\Q$
{{iff}}:
:$z_1, z_2, \ldots, z_n$ are linearly independent over the integers $\Z$. | === Forward implication ===
Let $z_1, z_2, \ldots, z_n$ be linearly independent over the rational numbers $\Q$.
That is, if $q_1, q_2, \ldots, q_n$ are rational numbers such that:
:$q_1 z_1 + q_2 z_2 + \cdots + q_n z_n = 0$
then:
:$q_1 = q_2 = \cdots = q_n = 0$
Let $a_1, a_2, \ldots, a_n$ be integers such that:
:$a_1 z... | Let $z_1, z_2, \ldots, z_n$ be [[Definition:Complex Number|complex numbers]].
Then:
:$z_1, z_2, \ldots, z_n$ are [[Definition:Linearly Independent Set|linearly independent]] over the [[Definition:Rational Number|rational numbers $\Q$]]
{{iff}}:
:$z_1, z_2, \ldots, z_n$ are [[Definition:Linearly Independent Set|linear... | === Forward implication ===
Let $z_1, z_2, \ldots, z_n$ be [[Definition:Linearly Independent Set|linearly independent]] over the [[Definition:Rational Number|rational numbers $\Q$]].
That is, if $q_1, q_2, \ldots, q_n$ are [[Definition:Rational Number|rational numbers]] such that:
:$q_1 z_1 + q_2 z_2 + \cdots + q_n z... | Linearly Independent over the Rational Numbers iff Linearly Independent over the Integers | https://proofwiki.org/wiki/Linearly_Independent_over_the_Rational_Numbers_iff_Linearly_Independent_over_the_Integers | https://proofwiki.org/wiki/Linearly_Independent_over_the_Rational_Numbers_iff_Linearly_Independent_over_the_Integers | [
"Complex Analysis"
] | [
"Definition:Complex Number",
"Definition:Linearly Independent/Set",
"Definition:Rational Number",
"Definition:Linearly Independent/Set",
"Definition:Integer"
] | [
"Definition:Linearly Independent/Set",
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Integer",
"Integers form Subdomain of Rationals",
"Definition:Linearly Independent/Set",
"Definition:Integer",
"Definition:Linearly Independent/Set",
"Definition:Integer",
"Definition:Int... |
proofwiki-12450 | Sums of Squares of Diagonals of Order 3 Magic Square | Consider the order 3 magic square:
{{:Magic Square/Examples/Order 3}}
The sums of the squares of the diagonals, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same diagonals of that same order 3 magic square when reversed.
{{improve|Find a way to describe the "diagonals" accu... | For the top-left to bottom-right diagonals:
{{begin-eqn}}
{{eqn | l = 258^2 + 714^2 + 693^2
| r = 66564 + 509796 + 480249
| c =
}}
{{eqn | r = 1056609
| c =
}}
{{eqn | l = 852^2 + 417^2 + 396^2
| r = 725904 + 173889 + 156816
| c =
}}
{{eqn | r = 1056609
| c =
}}
{{end-eqn}}
For t... | Consider the [[Definition:Order 3 Magic Square|order 3 magic square]]:
{{:Magic Square/Examples/Order 3}}
The sums of the [[Definition:Square (Algebra)|squares]] of the [[Definition:Diagonal of Matrix|diagonals]], when expressed as $3$-digit [[Definition:Decimal Notation|decimal numbers]], are equal to the sums of the... | For the top-left to bottom-right diagonals:
{{begin-eqn}}
{{eqn | l = 258^2 + 714^2 + 693^2
| r = 66564 + 509796 + 480249
| c =
}}
{{eqn | r = 1056609
| c =
}}
{{eqn | l = 852^2 + 417^2 + 396^2
| r = 725904 + 173889 + 156816
| c =
}}
{{eqn | r = 1056609
| c =
}}
{{end-eqn}}
Fo... | Sums of Squares of Diagonals of Order 3 Magic Square | https://proofwiki.org/wiki/Sums_of_Squares_of_Diagonals_of_Order_3_Magic_Square | https://proofwiki.org/wiki/Sums_of_Squares_of_Diagonals_of_Order_3_Magic_Square | [
"Magic Squares"
] | [
"Magic Square/Examples/Order 3",
"Definition:Square/Function",
"Definition:Matrix/Diagonal",
"Definition:Decimal Notation",
"Definition:Square/Function",
"Definition:Matrix/Diagonal",
"Magic Square/Examples/Order 3"
] | [] |
proofwiki-12451 | Lambert W of Non-Zero Algebraic Number is Transcendental | Let $W$ denote the (general) Lambert $W$ function.
Let $a$ be a non-zero algebraic number.
Then $\map W a$ is transcendental. | From the definition of Lambert $W$ function:
:$a = \map W a e^{\map W a}$
{{AimForCont}} $\map W a$ is not transcendental.
Hence, $\map W a$ is algebraic.
From the Weaker Hermite-Lindemann-Weierstrass Theorem, $e^{\map W a}$ is transcendental.
However, from the equation above, it is also equal to:
:$\dfrac a {\map W a}... | Let $W$ denote the [[Definition:Lambert W Function|(general) Lambert $W$ function]].
Let $a$ be a non-zero [[Definition:Algebraic Number|algebraic number]].
Then $\map W a$ is [[Definition:Transcendental|transcendental]]. | From the definition of [[Definition:Lambert W Function|Lambert $W$ function]]:
:$a = \map W a e^{\map W a}$
{{AimForCont}} $\map W a$ is not [[Definition:Transcendental|transcendental]].
Hence, $\map W a$ is [[Definition:Algebraic Number|algebraic]].
From the [[Weaker Hermite-Lindemann-Weierstrass Theorem]], $e^{\... | Lambert W of Non-Zero Algebraic Number is Transcendental | https://proofwiki.org/wiki/Lambert_W_of_Non-Zero_Algebraic_Number_is_Transcendental | https://proofwiki.org/wiki/Lambert_W_of_Non-Zero_Algebraic_Number_is_Transcendental | [
"Lambert W Function",
"Transcendental Number Theory"
] | [
"Definition:Lambert W Function",
"Definition:Algebraic Number",
"Definition:Transcendental"
] | [
"Definition:Lambert W Function",
"Definition:Transcendental",
"Definition:Algebraic Number",
"Hermite-Lindemann-Weierstrass Theorem/Weaker",
"Definition:Transcendental",
"Definition:Algebraic Number",
"Definition:Algebraic Number",
"Definition:Contradiction",
"Definition:Transcendental",
"Definiti... |
proofwiki-12452 | Pi Squared is Irrational | Pi squared ($\pi^2$) is irrational. | {{AimForCont}} $\pi^2$ is rational.
We establish a lemma:
=== Lemma ===
{{:Pi Squared is Irrational/Proof 1/Lemma}}{{qed|lemma}}
We will use the definition of $A_n$ from the lemma.
Then we will deduce that $A_n$ is an integer for all $n$.
First we confirm by direct integration that $A_0$ and $A_1$ are integers:
{{begin... | [[Definition:Pi|Pi]] [[Definition:Square (Algebra)|squared]] ($\pi^2$) is [[Definition:Irrational Number|irrational]]. | {{AimForCont}} $\pi^2$ is [[Definition:Rational Number|rational]].
We establish a [[Definition:Lemma|lemma]]:
=== [[Pi Squared is Irrational/Proof 1/Lemma|Lemma]] ===
{{:Pi Squared is Irrational/Proof 1/Lemma}}{{qed|lemma}}
We will use the definition of $A_n$ from the [[Pi Squared is Irrational/Proof 1/Lemma|lemma... | Pi Squared is Irrational/Proof 1 | https://proofwiki.org/wiki/Pi_Squared_is_Irrational | https://proofwiki.org/wiki/Pi_Squared_is_Irrational/Proof_1 | [
"Pi Squared is Irrational",
"Pi"
] | [
"Definition:Pi",
"Definition:Square/Function",
"Definition:Irrational Number"
] | [
"Definition:Rational Number",
"Definition:Lemma",
"Pi Squared is Irrational/Proof 1/Lemma",
"Pi Squared is Irrational/Proof 1/Lemma",
"Definition:Integer",
"Definition:Integer",
"Area under Arc of Sine Function",
"Linear Combination of Integrals",
"Primitive of Power of x by Sine of a x",
"Cosine ... |
proofwiki-12453 | Pi Squared is Irrational | Pi squared ($\pi^2$) is irrational. | {{AimForCont}} $\pi^2$ is rational.
Then $\pi^2 = \dfrac p q$ where $p$ and $q$ are integers and $q \ne 0$.
Let us define the following even function as a polynomial of degree $2 n$:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \frac {\paren {1 - x^2}^n} {n!}
| c =
}}
{{eqn | r = \frac {\paren {1 + x}^n \paren... | [[Definition:Pi|Pi]] [[Definition:Square (Algebra)|squared]] ($\pi^2$) is [[Definition:Irrational Number|irrational]]. | {{AimForCont}} $\pi^2$ is [[Definition:Rational Number|rational]].
Then $\pi^2 = \dfrac p q$ where $p$ and $q$ are [[Definition:Integer|integers]] and $q \ne 0$.
Let us define the following [[Definition:Even Function|even function]] as a [[Definition:Polynomial|polynomial]] of [[Definition:Degree of Polynomial|degree... | Pi Squared is Irrational/Proof 2 | https://proofwiki.org/wiki/Pi_Squared_is_Irrational | https://proofwiki.org/wiki/Pi_Squared_is_Irrational/Proof_2 | [
"Pi Squared is Irrational",
"Pi"
] | [
"Definition:Pi",
"Definition:Square/Function",
"Definition:Irrational Number"
] | [
"Definition:Rational Number",
"Definition:Integer",
"Definition:Even Function",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Difference of Two Squares",
"Taylor's Theorem",
"Definition:Real Function",
"Definition:Real Interval/Closed",
"Definition:Derivative/Higher Derivatives/High... |
proofwiki-12454 | Pi Squared is Irrational | Pi squared ($\pi^2$) is irrational. | {{AimForCont}} $\pi^2$ is rational.
We establish a lemma:
=== Lemma ===
{{:Pi Squared is Irrational/Proof 3/Lemma}}{{qed|lemma}}
We will use the definition of $A_n$ from the lemma.
Then we will deduce that $A_n$ is an integer for all $n$.
First we confirm by direct integration that $A_0$ and $A_1$ are integers:
{{begin... | [[Definition:Pi|Pi]] [[Definition:Square (Algebra)|squared]] ($\pi^2$) is [[Definition:Irrational Number|irrational]]. | {{AimForCont}} $\pi^2$ is [[Definition:Rational Number|rational]].
We establish a [[Definition:Lemma|lemma]]:
=== [[Pi Squared is Irrational/Proof 3/Lemma|Lemma]] ===
{{:Pi Squared is Irrational/Proof 3/Lemma}}{{qed|lemma}}
We will use the definition of $A_n$ from the [[Pi Squared is Irrational/Proof 3/Lemma|lemma... | Pi Squared is Irrational/Proof 3 | https://proofwiki.org/wiki/Pi_Squared_is_Irrational | https://proofwiki.org/wiki/Pi_Squared_is_Irrational/Proof_3 | [
"Pi Squared is Irrational",
"Pi"
] | [
"Definition:Pi",
"Definition:Square/Function",
"Definition:Irrational Number"
] | [
"Definition:Rational Number",
"Definition:Lemma",
"Pi Squared is Irrational/Proof 3/Lemma",
"Pi Squared is Irrational/Proof 3/Lemma",
"Definition:Integer",
"Definition:Integer",
"Zeroth Power of Real Number equals One",
"Factorial/Examples/0",
"Linear Combination of Integrals",
"Primitive of Power... |
proofwiki-12455 | Way Below in Lattice of Power Set | Let $X$ be a set.
Let $L = \struct {\powerset X, \cup, \cap, \preceq}$ be a lattice of power set of $X$ where $\mathord\preceq = \mathord\subseteq \cap \paren {\powerset X \times \powerset X}$
Let $x, y \in \powerset X$.
Then $x \ll y$ {{iff}}
:for every a set $Y$ of subsets of $X$ such that $y \subseteq \bigcup Y$
::t... | === Sufficient Condition ===
Let $x \ll y$
Let $Y$ be a set of subsets of $X$ such that
:$y \subseteq \bigcup Y$
By definitions of power set and subset:
:$Y \subseteq \powerset X$
By Power Set is Complete Lattice:
:$\bigcup Y = \sup Y$
By definition of $\preceq$:
:$y \preceq \sup Y$
By Way Below in Complete Lattice:
:t... | Let $X$ be a [[Definition:Set|set]].
Let $L = \struct {\powerset X, \cup, \cap, \preceq}$ be a [[Definition:Lattice (Order Theory)|lattice]] of [[Definition:Power Set|power set]] of $X$ where $\mathord\preceq = \mathord\subseteq \cap \paren {\powerset X \times \powerset X}$
Let $x, y \in \powerset X$.
Then $x \ll y... | === Sufficient Condition ===
Let $x \ll y$
Let $Y$ be a [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $X$ such that
:$y \subseteq \bigcup Y$
By definitions of [[Definition:Power Set|power set]] and [[Definition:Subset|subset]]:
:$Y \subseteq \powerset X$
By [[Power Set is Complete Lattice]]:
:$... | Way Below in Lattice of Power Set | https://proofwiki.org/wiki/Way_Below_in_Lattice_of_Power_Set | https://proofwiki.org/wiki/Way_Below_in_Lattice_of_Power_Set | [
"Way Below Relation",
"Power Set"
] | [
"Definition:Set",
"Definition:Lattice (Order Theory)",
"Definition:Power Set",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Finite Subset",
"Definition:Element is Way Below"
] | [
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Power Set",
"Definition:Subset",
"Power Set is Complete Lattice",
"Way Below in Complete Lattice",
"Definition:Finite Subset",
"Power Set is Complete Lattice",
"Definition:Finite Subset",
"Definition:Set of Sets",
"Definition:Subset",
... |
proofwiki-12456 | Divisibility by 10 | An integer $N$ expressed in decimal notation is divisible by $10$ {{iff}} the {{LSD}} of $N$ is $0$.
That is:
:$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $10$
{{iff}}:
:$a_0 = 0$ | Let $N$ be divisible by $10$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod {10}
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod {10}
}}
{{eqn | ll= \leadstoandfrom
| l = a_0 + 10 \sum_{k \mathop = 1}^n a_k 10^{k - 1}
... | An [[Definition:Integer|integer]] $N$ expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $10$ {{iff}} the {{LSD}} of $N$ is $0$.
That is:
:$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is [[Definition:Divisor of In... | Let $N$ be [[Definition:Divisor of Integer|divisible]] by $10$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod {10}
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod {10}
}}
{{eqn | ll= \leadstoandfrom
| l = a_0 + 10 \sum_{... | Divisibility by 10 | https://proofwiki.org/wiki/Divisibility_by_10 | https://proofwiki.org/wiki/Divisibility_by_10 | [
"Divisibility Tests",
"10"
] | [
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-12457 | Divisibility by Power of 10 | Let $r \in \Z_{\ge 1}$ be a strictly positive integer.
An integer $N$ expressed in decimal notation is divisible by $10^r$ {{iff}} the last $r$ digits of $N$ are all $0$.
That is:
:$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $10^r$
{{iff}}:
:$a_0 + a_1 10 + ... | Let $N$ be divisible by $10^r$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod {10^r}
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod {10^r}
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^r a_k 10^r + \sum_{k \m... | Let $r \in \Z_{\ge 1}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
An [[Definition:Integer|integer]] $N$ expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $10^r$ {{iff}} the last $r$ [[Definition:Digit|digits]] of $N$ are all $0... | Let $N$ be [[Definition:Divisor of Integer|divisible]] by $10^r$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod {10^r}
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod {10^r}
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \... | Divisibility by Power of 10 | https://proofwiki.org/wiki/Divisibility_by_Power_of_10 | https://proofwiki.org/wiki/Divisibility_by_Power_of_10 | [
"Divisibility Tests",
"10"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-12458 | Difference of Two Squares cannot equal 2 modulo 4 | Let $n \in \Z_{>0}$ be of the form $4 k + 2$ for some $k \in \Z$.
Then $n$ cannot be expressed in the form:
:$n = a^2 - b^2$
for $a, b \in \Z$. | Let $n = a^2 - b^2$ for some $a, b \in \Z$.
By Square Modulo 4, both $a$ and $b$ are of the form $4 k$ or $4 k + 1$ for some integer $k$.
There are $4$ cases:
;$a \equiv b \equiv 0 \pmod 4$:
Then:
:$a^2 - b^2 \equiv 0 \pmod 4$
and so $n$ is in the form $4 k$.
;$a \equiv 0 \pmod 4$, $b \equiv 1 \pmod 4$:
Then:
:$a^2 - b... | Let $n \in \Z_{>0}$ be of the form $4 k + 2$ for some $k \in \Z$.
Then $n$ cannot be expressed in the form:
:$n = a^2 - b^2$
for $a, b \in \Z$. | Let $n = a^2 - b^2$ for some $a, b \in \Z$.
By [[Square Modulo 4]], both $a$ and $b$ are of the form $4 k$ or $4 k + 1$ for some [[Definition:Integer|integer]] $k$.
There are $4$ cases:
;$a \equiv b \equiv 0 \pmod 4$:
Then:
:$a^2 - b^2 \equiv 0 \pmod 4$
and so $n$ is in the form $4 k$.
;$a \equiv 0 \pmod 4$, $... | Difference of Two Squares cannot equal 2 modulo 4/Proof 1 | https://proofwiki.org/wiki/Difference_of_Two_Squares_cannot_equal_2_modulo_4 | https://proofwiki.org/wiki/Difference_of_Two_Squares_cannot_equal_2_modulo_4/Proof_1 | [
"Square Numbers",
"Difference of Two Squares cannot equal 2 modulo 4"
] | [] | [
"Square Modulo 4",
"Definition:Integer"
] |
proofwiki-12459 | Difference of Two Squares cannot equal 2 modulo 4 | Let $n \in \Z_{>0}$ be of the form $4 k + 2$ for some $k \in \Z$.
Then $n$ cannot be expressed in the form:
:$n = a^2 - b^2$
for $a, b \in \Z$. | Let $n_0 = c^2 - d^2$ for some $c, d \in \Z$.
Then:
:$n_0 = \paren {c + d} \paren {c - d}$
and so:
:$\paren {c + d} - \paren {c - d} = 2 d$
Therefore $n$ must be expressible as a product of two integers whose difference is even.
Now consider the integer $n \in \Z$ that satisfies $n \equiv 2 \pmod 4$.
$n$ is an even num... | Let $n \in \Z_{>0}$ be of the form $4 k + 2$ for some $k \in \Z$.
Then $n$ cannot be expressed in the form:
:$n = a^2 - b^2$
for $a, b \in \Z$. | Let $n_0 = c^2 - d^2$ for some $c, d \in \Z$.
Then:
:$n_0 = \paren {c + d} \paren {c - d}$
and so:
:$\paren {c + d} - \paren {c - d} = 2 d$
Therefore $n$ must be expressible as a [[Definition:Multiplication|product]] of two [[Definition:Integer|integers]] whose [[Definition:Difference (Subtraction)|difference]] is [... | Difference of Two Squares cannot equal 2 modulo 4/Proof 2 | https://proofwiki.org/wiki/Difference_of_Two_Squares_cannot_equal_2_modulo_4 | https://proofwiki.org/wiki/Difference_of_Two_Squares_cannot_equal_2_modulo_4/Proof_2 | [
"Square Numbers",
"Difference of Two Squares cannot equal 2 modulo 4"
] | [] | [
"Definition:Multiplication",
"Definition:Integer",
"Definition:Subtraction/Difference",
"Definition:Even Integer",
"Definition:Integer",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Odd Integer",
... |
proofwiki-12460 | Compact Closure is Set of Finite Subsets in Lattice of Power Set | Let $X$ be a set.
Let $L = \struct {\powerset X, \cup, \cap, \preceq}$ be the lattice of power set of $X$ where $\mathord\preceq = \mathord\subseteq \cap \powerset X \times \powerset X$
Let $x \in \powerset X$.
Then $x^{\mathrm{compact} } = \map {\operatorname{Fin} } x$
where $\map {\operatorname{Fin} } x$ denotes the ... | === $\subseteq$ ===
Let $y \in x^{\mathrm{compact} }$.
By definition of compact closure:
:$y \preceq x$ and $y$ is compact.
By definition of $\preceq$:
:$y \subseteq x$
By Element is Finite iff Element is Compact in Lattice of Power Set:
"$y$ is a finite set.
Thus by definition of $\operatorname{Fin}$:
:$y \in \map {\o... | Let $X$ be a [[Definition:Set|set]].
Let $L = \struct {\powerset X, \cup, \cap, \preceq}$ be the [[Definition:Lattice (Order Theory)|lattice]] of [[Definition:Power Set|power set]] of $X$ where $\mathord\preceq = \mathord\subseteq \cap \powerset X \times \powerset X$
Let $x \in \powerset X$.
Then $x^{\mathrm{compac... | === $\subseteq$ ===
Let $y \in x^{\mathrm{compact} }$.
By definition of [[Definition:Compact Closure|compact closure]]:
:$y \preceq x$ and $y$ is [[Definition:Compact Element|compact]].
By definition of $\preceq$:
:$y \subseteq x$
By [[Element is Finite iff Element is Compact in Lattice of Power Set]]:
"$y$ is a [[... | Compact Closure is Set of Finite Subsets in Lattice of Power Set | https://proofwiki.org/wiki/Compact_Closure_is_Set_of_Finite_Subsets_in_Lattice_of_Power_Set | https://proofwiki.org/wiki/Compact_Closure_is_Set_of_Finite_Subsets_in_Lattice_of_Power_Set | [
"Way Below Relation",
"Power Set"
] | [
"Definition:Set",
"Definition:Lattice (Order Theory)",
"Definition:Power Set",
"Definition:Set of Sets",
"Definition:Finite Subset"
] | [
"Definition:Compact Closure",
"Definition:Compact Element",
"Element is Finite iff Element is Compact in Lattice of Power Set",
"Definition:Finite Set",
"Definition:Finite Set",
"Element is Finite iff Element is Compact in Lattice of Power Set",
"Definition:Compact Element",
"Definition:Compact Closur... |
proofwiki-12461 | 10 is Only Triangular Number that is Sum of Consecutive Odd Squares | $10$ is the only triangular number which is the sum of two consecutive odd squares:
:$10 = 1^2 + 3^2$ | {{:Closed Form for Triangular Numbers}}
for $n \in \Z_{\ge 0}$.
The expression for the $n$th odd square number is:
:$4 n^2 + 4 n + 1$
again, for $n \in \Z_{\ge 0}$.
Therefore the closed-form expression for the $n$th sum of two consecutive odd squares is:
:$4 n^2 + 4 n + 1 + 4 \paren {n + 1}^2 + 4 \paren {n + 1} + 1$
Th... | $10$ is the only [[Definition:Triangular Number|triangular number]] which is the sum of two consecutive [[Definition:Odd Integer|odd]] [[Definition:Square Number|squares]]:
:$10 = 1^2 + 3^2$ | {{:Closed Form for Triangular Numbers}}
for $n \in \Z_{\ge 0}$.
The expression for the $n$th [[Definition:Odd Integer|odd]] [[Definition:Square Number|square number]] is:
:$4 n^2 + 4 n + 1$
again, for $n \in \Z_{\ge 0}$.
Therefore the [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th sum of... | 10 is Only Triangular Number that is Sum of Consecutive Odd Squares | https://proofwiki.org/wiki/10_is_Only_Triangular_Number_that_is_Sum_of_Consecutive_Odd_Squares | https://proofwiki.org/wiki/10_is_Only_Triangular_Number_that_is_Sum_of_Consecutive_Odd_Squares | [
"10",
"Odd Squares",
"Triangular Numbers"
] | [
"Definition:Triangular Number",
"Definition:Odd Integer",
"Definition:Square Number"
] | [
"Definition:Odd Integer",
"Definition:Square Number",
"Definition:Closed Form Expression",
"Definition:Odd Integer",
"Definition:Square Number",
"Solutions to Diophantine Equation 16x^2+32x+20 = y^2+y"
] |
proofwiki-12462 | 10 Consecutive Integers contain Coprime Integer | Let $n \in \Z$ be an integer.
Let $S := \set {n, n + 1, n + 2, \ldots, n + 9}$ be the set of $10$ consecutive integers starting from $n$.
Then at least one element of $S$ is coprime to every other element of $S$. | Consider $2$ elements $a, b$ of $S$ which share a common divisor $d$.
Then $d \divides \size {a - b}$ and so $d < 10$.
Now from the Fundamental Theorem of Arithmetic, $d$ must have a prime factor which is strictly less than $10$.
So for $a$ and $b$ to have a common divisor, at least one such common divisor is in $\set ... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Let $S := \set {n, n + 1, n + 2, \ldots, n + 9}$ be the [[Definition:Set|set]] of $10$ consecutive [[Definition:Integer|integers]] starting from $n$.
Then at least one [[Definition:Element|element]] of $S$ is [[Definition:Coprime Integers|coprime]] to every other ... | Consider $2$ [[Definition:Element|elements]] $a, b$ of $S$ which share a [[Definition:Common Divisor|common divisor]] $d$.
Then $d \divides \size {a - b}$ and so $d < 10$.
Now from the [[Fundamental Theorem of Arithmetic]], $d$ must have a [[Definition:Prime Factor|prime factor]] which is strictly less than $10$.
So... | 10 Consecutive Integers contain Coprime Integer | https://proofwiki.org/wiki/10_Consecutive_Integers_contain_Coprime_Integer | https://proofwiki.org/wiki/10_Consecutive_Integers_contain_Coprime_Integer | [
"Coprime Integers",
"10"
] | [
"Definition:Integer",
"Definition:Set",
"Definition:Integer",
"Definition:Element",
"Definition:Coprime/Integers",
"Definition:Element"
] | [
"Definition:Element",
"Definition:Common Divisor",
"Fundamental Theorem of Arithmetic",
"Definition:Prime Factor",
"Definition:Common Divisor",
"Definition:Common Divisor",
"Definition:Element",
"Definition:Common Divisor",
"Definition:Element",
"Definition:Common Divisor",
"Definition:Odd Integ... |
proofwiki-12463 | Two Fifths as Pandigital Fraction | There are $3$ ways $\dfrac 2 5$ can be expressed as a pandigital fraction:
:$\dfrac 2 5 = \dfrac {6894} {17235}$
:$\dfrac 2 5 = \dfrac {8694} {21735}$
:$\dfrac 2 5 = \dfrac {9486} {23715}$ | Can be verified by brute force.
Category:Pandigital Fractions
055zswhtnrwnvqeqdxkph8z2yd2qqse | There are $3$ ways $\dfrac 2 5$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
:$\dfrac 2 5 = \dfrac {6894} {17235}$
:$\dfrac 2 5 = \dfrac {8694} {21735}$
:$\dfrac 2 5 = \dfrac {9486} {23715}$ | Can be verified by brute force.
[[Category:Pandigital Fractions]]
055zswhtnrwnvqeqdxkph8z2yd2qqse | Two Fifths as Pandigital Fraction | https://proofwiki.org/wiki/Two_Fifths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Two_Fifths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12464 | Two Sevenths as Pandigital Fraction | There are $6$ ways $\dfrac 2 7$ can be expressed as a pandigital fraction:
:$\dfrac 2 7 = \dfrac {3654} {12789}$
:$\dfrac 2 7 = \dfrac {3674} {12859}$
:$\dfrac 2 7 = \dfrac {5342} {18697}$
:$\dfrac 2 7 = \dfrac {7418} {25963}$
:$\dfrac 2 7 = \dfrac {9786} {34251}$
:$\dfrac 2 7 = \dfrac {9862} {34517}$ | Can be verified by brute force.
Category:Pandigital Fractions
3vg8lm7xffdosxq7mvyuxv48daojqy2 | There are $6$ ways $\dfrac 2 7$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
:$\dfrac 2 7 = \dfrac {3654} {12789}$
:$\dfrac 2 7 = \dfrac {3674} {12859}$
:$\dfrac 2 7 = \dfrac {5342} {18697}$
:$\dfrac 2 7 = \dfrac {7418} {25963}$
:$\dfrac 2 7 = \dfrac {9786} {34251}$
:$\dfrac 2 7 ... | Can be verified by brute force.
[[Category:Pandigital Fractions]]
3vg8lm7xffdosxq7mvyuxv48daojqy2 | Two Sevenths as Pandigital Fraction | https://proofwiki.org/wiki/Two_Sevenths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Two_Sevenths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12465 | Two Ninths as Pandigital Fraction | There are $2$ ways $\dfrac 2 9$ can be expressed as a pandigital fraction:
{{begin-eqn}}
{{eqn | l = \dfrac 2 9
| r = \dfrac {3924} {17658}
}}
{{eqn | l = \dfrac 2 9
| r = \dfrac {7596} {34182}
}}
{{end-eqn}} | Can be verified by brute force.
Category:Pandigital Fractions
aw7pqsuqjxc9s094hdo4vcrm83po897 | There are $2$ ways $\dfrac 2 9$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
{{begin-eqn}}
{{eqn | l = \dfrac 2 9
| r = \dfrac {3924} {17658}
}}
{{eqn | l = \dfrac 2 9
| r = \dfrac {7596} {34182}
}}
{{end-eqn}} | Can be verified by brute force.
[[Category:Pandigital Fractions]]
aw7pqsuqjxc9s094hdo4vcrm83po897 | Two Ninths as Pandigital Fraction | https://proofwiki.org/wiki/Two_Ninths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Two_Ninths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12466 | Closed Form for Tetrahedral Numbers | The closed-form expression for the $n$th tetrahedral number is:
:$H_n = \dfrac {n \paren {n + 1} \paren {n + 2} } 6$ | {{begin-eqn}}
{{eqn | l = H_n
| r = \sum_{r \mathop = 1}^n T_r
| c = {{Defof|Tetrahedral Number}}
}}
{{eqn | r = \sum_{r \mathop = 1}^n \frac {r \paren {r + 1} } 2
| c = Closed Form for Triangular Numbers
}}
{{eqn | r = \sum_{r \mathop = 1}^n \paren {\frac 1 2 r^2 + \frac 1 2 r}
| c =
}}
{{eqn ... | The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Tetrahedral Number|tetrahedral number]] is:
:$H_n = \dfrac {n \paren {n + 1} \paren {n + 2} } 6$ | {{begin-eqn}}
{{eqn | l = H_n
| r = \sum_{r \mathop = 1}^n T_r
| c = {{Defof|Tetrahedral Number}}
}}
{{eqn | r = \sum_{r \mathop = 1}^n \frac {r \paren {r + 1} } 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | r = \sum_{r \mathop = 1}^n \paren {\frac 1 2 r^2 + \frac 1 2 r}
| c =
}}
{{... | Closed Form for Tetrahedral Numbers/Proof 2 | https://proofwiki.org/wiki/Closed_Form_for_Tetrahedral_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Tetrahedral_Numbers/Proof_2 | [
"Closed Form for Tetrahedral Numbers",
"Tetrahedral Numbers",
"Closed Forms"
] | [
"Definition:Closed Form Expression",
"Definition:Tetrahedral Number"
] | [
"Closed Form for Triangular Numbers",
"Summation is Linear",
"Sum of Sequence of Squares",
"Closed Form for Triangular Numbers"
] |
proofwiki-12467 | Factorial as Product of Two Factorials | Apart from the general pattern, following directly from the definition of the factorial:
:$\paren {n!}! = n! \paren {n! - 1}!$
the only known factorial which is the product of two factorials is:
:$10! = 6! \, 7!$ | {{begin-eqn}}
{{eqn | l = 10!
| r = 7! \times 8 \times 9 \times 10
| c = {{Defof|Factorial}}
}}
{{eqn | r = 7! \times \paren {2 \times 4} \times \paren {3 \times 3} \times \paren {2 \times 5}
}}
{{eqn | r = 7! \times 2 \times 4 \times 3 \times \paren {3 \times 2} \times 5
}}
{{eqn | r = 7! \times 2 \times 3... | Apart from the general pattern, following directly from the definition of the [[Definition:Factorial|factorial]]:
:$\paren {n!}! = n! \paren {n! - 1}!$
the only known [[Definition:Factorial|factorial]] which is the product of two [[Definition:Factorial|factorials]] is:
:$10! = 6! \, 7!$ | {{begin-eqn}}
{{eqn | l = 10!
| r = 7! \times 8 \times 9 \times 10
| c = {{Defof|Factorial}}
}}
{{eqn | r = 7! \times \paren {2 \times 4} \times \paren {3 \times 3} \times \paren {2 \times 5}
}}
{{eqn | r = 7! \times 2 \times 4 \times 3 \times \paren {3 \times 2} \times 5
}}
{{eqn | r = 7! \times 2 \times 3... | Factorial as Product of Two Factorials | https://proofwiki.org/wiki/Factorial_as_Product_of_Two_Factorials | https://proofwiki.org/wiki/Factorial_as_Product_of_Two_Factorials | [
"Factorials",
"10"
] | [
"Definition:Factorial",
"Definition:Factorial",
"Definition:Factorial"
] | [] |
proofwiki-12468 | Factorial as Product of Three Factorials | This general pattern can be used to find a factorial which is the product of three factorials:
:$\paren {\paren {n!}!}! = n! \paren {n! - 1}! \paren {\paren {n!}! - 1}!$
while there are instances of factorials which do not fit that pattern. | {{begin-eqn}}
{{eqn | l = \paren {\paren {n!}!}!
| r = \paren {n!}! \times \paren {\paren {n!}! - 1}!
| c = Factorial as Product of Two Factorials
}}
{{eqn | r = n! \times \paren {n! - 1}! \times \paren {\paren {n!}! - 1}!
| c = Factorial as Product of Two Factorials
}}
{{end-eqn}}
{{qed}} | This general pattern can be used to find a [[Definition:Factorial|factorial]] which is the product of three [[Definition:Factorial|factorials]]:
:$\paren {\paren {n!}!}! = n! \paren {n! - 1}! \paren {\paren {n!}! - 1}!$
while there are instances of [[Definition:Factorial|factorials]] which do not fit that pattern. | {{begin-eqn}}
{{eqn | l = \paren {\paren {n!}!}!
| r = \paren {n!}! \times \paren {\paren {n!}! - 1}!
| c = [[Factorial as Product of Two Factorials]]
}}
{{eqn | r = n! \times \paren {n! - 1}! \times \paren {\paren {n!}! - 1}!
| c = [[Factorial as Product of Two Factorials]]
}}
{{end-eqn}}
{{qed}} | Factorial as Product of Three Factorials | https://proofwiki.org/wiki/Factorial_as_Product_of_Three_Factorials | https://proofwiki.org/wiki/Factorial_as_Product_of_Three_Factorials | [
"Factorials",
"Factorial as Product of Three Factorials"
] | [
"Definition:Factorial",
"Definition:Factorial",
"Definition:Factorial"
] | [
"Factorial as Product of Two Factorials",
"Factorial as Product of Two Factorials"
] |
proofwiki-12469 | Two Thirds as Pandigital Fraction | $\dfrac 2 3$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
5itdiplla5qizaah8gv9ev3z2sozpk9 | $\dfrac 2 3$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
5itdiplla5qizaah8gv9ev3z2sozpk9 | Two Thirds as Pandigital Fraction | https://proofwiki.org/wiki/Two_Thirds_as_Pandigital_Fraction | https://proofwiki.org/wiki/Two_Thirds_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12470 | Three Quarters as Pandigital Fraction | $\dfrac 3 4$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
s57iaxw90tqkcob960zu0fhvdy8ue5s | $\dfrac 3 4$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
s57iaxw90tqkcob960zu0fhvdy8ue5s | Three Quarters as Pandigital Fraction | https://proofwiki.org/wiki/Three_Quarters_as_Pandigital_Fraction | https://proofwiki.org/wiki/Three_Quarters_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12471 | Three Fifths as Pandigital Fraction | $\dfrac 3 5$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
t2wttw0dnqbst5g9t3c17e2qgc53t5p | $\dfrac 3 5$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
t2wttw0dnqbst5g9t3c17e2qgc53t5p | Three Fifths as Pandigital Fraction | https://proofwiki.org/wiki/Three_Fifths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Three_Fifths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12472 | Three Sevenths as Pandigital Fraction | $\dfrac 3 7$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
9as5b4nk5h9f4g64bx10vunxe6pm2sl | $\dfrac 3 7$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
9as5b4nk5h9f4g64bx10vunxe6pm2sl | Three Sevenths as Pandigital Fraction | https://proofwiki.org/wiki/Three_Sevenths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Three_Sevenths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12473 | Three Eighths as Pandigital Fraction | $\dfrac 3 8$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
i2ruxvt2pmp3tnlu4isqnkejpfbe1ry | $\dfrac 3 8$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
i2ruxvt2pmp3tnlu4isqnkejpfbe1ry | Three Eighths as Pandigital Fraction | https://proofwiki.org/wiki/Three_Eighths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Three_Eighths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12474 | Four Sevenths as Pandigital Fraction | $\dfrac 4 7$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
ekzrdi64qubrwjuffq00ynvvqr31xzy | $\dfrac 4 7$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
ekzrdi64qubrwjuffq00ynvvqr31xzy | Four Sevenths as Pandigital Fraction | https://proofwiki.org/wiki/Four_Sevenths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Four_Sevenths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12475 | Five Sevenths as Pandigital Fraction | $\dfrac 5 7$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
kmgdxe5yibpy0lk6s7whooizqedd8tk | $\dfrac 5 7$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
kmgdxe5yibpy0lk6s7whooizqedd8tk | Five Sevenths as Pandigital Fraction | https://proofwiki.org/wiki/Five_Sevenths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Five_Sevenths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12476 | Six Sevenths as Pandigital Fraction | $\dfrac 6 7$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
7j9wpx06dq7cikndvmfc4ckkfnyj1km | $\dfrac 6 7$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
7j9wpx06dq7cikndvmfc4ckkfnyj1km | Six Sevenths as Pandigital Fraction | https://proofwiki.org/wiki/Six_Sevenths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Six_Sevenths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12477 | Five Sixths as Pandigital Fraction | $\dfrac 5 6$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
8wmrbiarp95of35q9y7yfswbm6lup6m | $\dfrac 5 6$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
8wmrbiarp95of35q9y7yfswbm6lup6m | Five Sixths as Pandigital Fraction | https://proofwiki.org/wiki/Five_Sixths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Five_Sixths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12478 | Five Eighths as Pandigital Fraction | $\dfrac 5 8$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
msdo74yu6ouq74zjz79o04aqzfa96k8 | $\dfrac 5 8$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
msdo74yu6ouq74zjz79o04aqzfa96k8 | Five Eighths as Pandigital Fraction | https://proofwiki.org/wiki/Five_Eighths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Five_Eighths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12479 | Seven Eighths as Pandigital Fraction | $\dfrac 7 8$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
75q6t9qdl4n76fgzmne9at71y5yhwc7 | $\dfrac 7 8$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
75q6t9qdl4n76fgzmne9at71y5yhwc7 | Seven Eighths as Pandigital Fraction | https://proofwiki.org/wiki/Seven_Eighths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Seven_Eighths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12480 | Four Ninths as Pandigital Fraction | $\dfrac 4 9$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
rb2cdg84i5nvjusdmsp187c9sqxjlma | $\dfrac 4 9$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
rb2cdg84i5nvjusdmsp187c9sqxjlma | Four Ninths as Pandigital Fraction | https://proofwiki.org/wiki/Four_Ninths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Four_Ninths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12481 | Five Ninths as Pandigital Fraction | $\dfrac 5 9$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
2wf5bsw1mdpjj1o554kpuzfui1u2u4y | $\dfrac 5 9$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
2wf5bsw1mdpjj1o554kpuzfui1u2u4y | Five Ninths as Pandigital Fraction | https://proofwiki.org/wiki/Five_Ninths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Five_Ninths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12482 | Seven Ninths as Pandigital Fraction | $\dfrac 7 9$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
r6s00mwho0u132y6az41r1wjeere5d3 | $\dfrac 7 9$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
r6s00mwho0u132y6az41r1wjeere5d3 | Seven Ninths as Pandigital Fraction | https://proofwiki.org/wiki/Seven_Ninths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Seven_Ninths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12483 | Eight Ninths as Pandigital Fraction | $\dfrac 8 9$ cannot be expressed as a pandigital fraction. | Can be verified by brute force.
Category:Pandigital Fractions
01jl5wmi3om94ivfpddlhdg7pv9khuu | $\dfrac 8 9$ cannot be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]. | Can be verified by brute force.
[[Category:Pandigital Fractions]]
01jl5wmi3om94ivfpddlhdg7pv9khuu | Eight Ninths as Pandigital Fraction | https://proofwiki.org/wiki/Eight_Ninths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Eight_Ninths_as_Pandigital_Fraction | [
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [
"Category:Pandigital Fractions"
] |
proofwiki-12484 | Generating Function for Triangular Numbers | Let $T_n$ denote the $n$th triangular number.
Then the generating function for $\sequence {T_n}$ is given as:
:$\ds \map G z = \frac z {\paren {1 - z}^3}$ | {{begin-eqn}}
{{eqn | l = \frac z {\paren {1 - z}^3}
| r = z \paren {1 - z}^{-3}
| c = Exponent Combination Laws for Negative Power
}}
{{eqn | r = z \sum_{n \mathop = 0}^\infty \dbinom {-3} n \paren {-z}^n
| c = General Binomial Theorem
}}
{{eqn | r = z \sum_{n \mathop = 0}^\infty \paren {-1}^n \dbino... | Let $T_n$ denote the $n$th [[Definition:Triangular Number|triangular number]].
Then the [[Definition:Generating Function|generating function]] for $\sequence {T_n}$ is given as:
:$\ds \map G z = \frac z {\paren {1 - z}^3}$ | {{begin-eqn}}
{{eqn | l = \frac z {\paren {1 - z}^3}
| r = z \paren {1 - z}^{-3}
| c = [[Exponent Combination Laws/Negative Power|Exponent Combination Laws for Negative Power]]
}}
{{eqn | r = z \sum_{n \mathop = 0}^\infty \dbinom {-3} n \paren {-z}^n
| c = [[General Binomial Theorem]]
}}
{{eqn | r = z... | Generating Function for Triangular Numbers | https://proofwiki.org/wiki/Generating_Function_for_Triangular_Numbers | https://proofwiki.org/wiki/Generating_Function_for_Triangular_Numbers | [
"Triangular Numbers",
"Examples of Generating Functions"
] | [
"Definition:Triangular Number",
"Definition:Generating Function"
] | [
"Exponent Combination Laws/Negative Power",
"Binomial Theorem/General Binomial Theorem",
"Negated Upper Index of Binomial Coefficient",
"Symmetry Rule for Binomial Coefficients",
"Category:Triangular Numbers",
"Category:Examples of Generating Functions"
] |
proofwiki-12485 | Lattice of Power Set is Algebraic | Let $X$ be a set.
Let $L = \struct {\powerset X, \cup, \cap, \preceq}$ be the lattice of power set of $X$
where $\mathord \preceq = \mathord \subseteq \cap \paren {\powerset X \times \powerset X}$
Then $L$ is algebraic. | We will prove that
:$\forall x \in \powerset X: x^{\mathrm{compact} }$ is directed.
Let $x \in \powerset X$.
By Empty Set is Bottom of Lattice of Power Set:
:$\O = \bot$
where $\bot$ denotes the bottom of $L$.
By Bottom is Way Below Any Element:
:$\bot \ll \bot$
where $\ll$ is the way below relation.
By definition:
:$\... | Let $X$ be a [[Definition:Set|set]].
Let $L = \struct {\powerset X, \cup, \cap, \preceq}$ be the [[Definition:Lattice (Order Theory)|lattice]] of [[Definition:Power Set|power set]] of $X$
where $\mathord \preceq = \mathord \subseteq \cap \paren {\powerset X \times \powerset X}$
Then $L$ is [[Definition:Algebraic Ord... | We will prove that
:$\forall x \in \powerset X: x^{\mathrm{compact} }$ is [[Definition:Directed Subset|directed]].
Let $x \in \powerset X$.
By [[Empty Set is Bottom of Lattice of Power Set]]:
:$\O = \bot$
where $\bot$ denotes the [[Definition:Bottom of Lattice|bottom]] of $L$.
By [[Bottom is Way Below Any Element]]:... | Lattice of Power Set is Algebraic | https://proofwiki.org/wiki/Lattice_of_Power_Set_is_Algebraic | https://proofwiki.org/wiki/Lattice_of_Power_Set_is_Algebraic | [
"Continuous Lattices",
"Power Set"
] | [
"Definition:Set",
"Definition:Lattice (Order Theory)",
"Definition:Power Set",
"Definition:Algebraic Ordered Set"
] | [
"Definition:Directed Subset",
"Empty Set is Bottom of Lattice of Power Set",
"Definition:Bottom of Lattice",
"Bottom is Way Below Any Element",
"Definition:Element is Way Below",
"Definition:Compact Element",
"Definition:Smallest Element",
"Definition:Compact Closure",
"Definition:Non-Empty Set",
... |
proofwiki-12486 | Difference of Squares of Sum and Difference | :$\forall a, b \in \R: \paren {a + b}^2 - \paren {a - b}^2 = 4 a b$ | {{begin-eqn}}
{{eqn | o =
| r = \left({a + b}\right)^2 - \left({a - b}\right)^2
| c =
}}
{{eqn | r = \left({a^2 + 2 a b + b^2}\right) - \left({a^2 - 2 a b + b^2}\right)
| c = Square of Sum and Square of Difference
}}
{{eqn | r = a^2 + 2 a b + b^2 - a^2 + 2 a b - b^2
| c =
}}
{{eqn | r = 2 a b... | :$\forall a, b \in \R: \paren {a + b}^2 - \paren {a - b}^2 = 4 a b$ | {{begin-eqn}}
{{eqn | o =
| r = \left({a + b}\right)^2 - \left({a - b}\right)^2
| c =
}}
{{eqn | r = \left({a^2 + 2 a b + b^2}\right) - \left({a^2 - 2 a b + b^2}\right)
| c = [[Square of Sum]] and [[Square of Difference]]
}}
{{eqn | r = a^2 + 2 a b + b^2 - a^2 + 2 a b - b^2
| c =
}}
{{eqn | r... | Difference of Squares of Sum and Difference/Algebraic Proof | https://proofwiki.org/wiki/Difference_of_Squares_of_Sum_and_Difference | https://proofwiki.org/wiki/Difference_of_Squares_of_Sum_and_Difference/Algebraic_Proof | [
"Difference of Squares of Sum and Difference",
"Algebra",
"Squares"
] | [] | [
"Square of Sum",
"Square of Difference"
] |
proofwiki-12487 | Non-Empty Compact Closure is Directed | Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice.
Let $x \in S$ such that
:$x^{\mathrm{compact} }$ is a non-empty set,
where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.
Then :$x^{\mathrm{compact} }$ is directed. | Thus by assumption:
:$x^{\mathrm{compact} }$ is a non-empty set.
Let $y, z \in x^{\mathrm{compact} }$.
By definition of compact closure:
:$y \preceq x$, $z \preceq x$, and $y$ and $z$ are compact.
By definition of compact element:
:$y \ll y$ and $z \ll z$
where $\ll$ denotes the way below relation.
By Way Below is Cong... | Let $L = \left({S, \vee, \preceq}\right)$ be a [[Definition:Join Semilattice|join semilattice]].
Let $x \in S$ such that
:$x^{\mathrm{compact} }$ is a [[Definition:Non-Empty Set|non-empty set]],
where $x^{\mathrm{compact} }$ denotes the [[Definition:Compact Closure|compact closure]] of $x$.
Then :$x^{\mathrm{compact}... | Thus by assumption:
:$x^{\mathrm{compact} }$ is a [[Definition:Non-Empty Set|non-empty set]].
Let $y, z \in x^{\mathrm{compact} }$.
By definition of [[Definition:Compact Closure|compact closure]]:
:$y \preceq x$, $z \preceq x$, and $y$ and $z$ are [[Definition:Compact Element|compact]].
By definition of [[Definition... | Non-Empty Compact Closure is Directed | https://proofwiki.org/wiki/Non-Empty_Compact_Closure_is_Directed | https://proofwiki.org/wiki/Non-Empty_Compact_Closure_is_Directed | [
"Way Below Relation",
"Order Theory"
] | [
"Definition:Join Semilattice",
"Definition:Non-Empty Set",
"Definition:Compact Closure",
"Definition:Directed Subset"
] | [
"Definition:Non-Empty Set",
"Definition:Compact Closure",
"Definition:Compact Element",
"Definition:Compact Element",
"Definition:Element is Way Below",
"Way Below is Congruent for Join",
"Definition:Compact Element",
"Definition:Supremum of Set",
"Definition:Compact Closure",
"Join Succeeds Opera... |
proofwiki-12488 | Set of Cardinality not Greater than Cardinality of Finite Set is Finite | Let $X, Y$ be sets such that
:$\card X \le \card Y$
and
:$Y$ is finite,
where $\card X$ denotes the cardinality of $X$.
Then $X$ is finite. | By Finite iff Cardinality Less than Aleph Zero:
:$\card Y < \aleph_0$
Then
:$\card X < \aleph_0$
Thus by Finite iff Cardinality Less than Aleph Zero:
:$X$ is a finite set.
{{qed}} | Let $X, Y$ be [[Definition:Set|sets]] such that
:$\card X \le \card Y$
and
:$Y$ is [[Definition:Finite Set|finite]],
where $\card X$ denotes the [[Definition:Cardinality|cardinality]] of $X$.
Then $X$ is [[Definition:Finite Set|finite]]. | By [[Finite iff Cardinality Less than Aleph Zero]]:
:$\card Y < \aleph_0$
Then
:$\card X < \aleph_0$
Thus by [[Finite iff Cardinality Less than Aleph Zero]]:
:$X$ is a [[Definition:Finite Set|finite set]].
{{qed}} | Set of Cardinality not Greater than Cardinality of Finite Set is Finite | https://proofwiki.org/wiki/Set_of_Cardinality_not_Greater_than_Cardinality_of_Finite_Set_is_Finite | https://proofwiki.org/wiki/Set_of_Cardinality_not_Greater_than_Cardinality_of_Finite_Set_is_Finite | [
"Cardinals"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Finite Set"
] | [
"Finite iff Cardinality Less than Aleph Zero",
"Finite iff Cardinality Less than Aleph Zero",
"Definition:Finite Set"
] |
proofwiki-12489 | Finite iff Cardinality Less than Aleph Zero | Let $X$ be a set.
Then $X$ is finite {{iff}} $\card X < \aleph_0$
where:
:$\card X$ denotes the cardinality of $X$
:$\aleph_0 = \card \N$ by Aleph Zero equals Cardinality of Naturals. | === Sufficient Condition ===
Let $X$ be finite.
By definition of finite set:
:$\exists n \in \N: X \sim \N_n$
where:
:$\sim$ denotes the set equivalence
:$\N_n$ denotes the initial segment of natural numbers less than $n$.
By the von Neumann construction of natural numbers:
:$\N_n = n$
By definition of cardinality:
:$\... | Let $X$ be a [[Definition:Set|set]].
Then $X$ is [[Definition:Finite Set|finite]] {{iff}} $\card X < \aleph_0$
where:
:$\card X$ denotes the [[Definition:Cardinality|cardinality]] of $X$
:$\aleph_0 = \card \N$ by [[Aleph Zero equals Cardinality of Naturals]]. | === Sufficient Condition ===
Let $X$ be [[Definition:Finite Set|finite]].
By definition of [[Definition:Finite Set|finite set]]:
:$\exists n \in \N: X \sim \N_n$
where:
:$\sim$ denotes the [[Definition:Set Equivalence|set equivalence]]
:$\N_n$ denotes the [[Definition:Initial Segment of Natural Numbers|initial segmen... | Finite iff Cardinality Less than Aleph Zero | https://proofwiki.org/wiki/Finite_iff_Cardinality_Less_than_Aleph_Zero | https://proofwiki.org/wiki/Finite_iff_Cardinality_Less_than_Aleph_Zero | [
"Cardinals",
"Aleph Mapping"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Cardinality",
"Aleph Zero equals Cardinality of Naturals"
] | [
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Set Equivalence",
"Definition:Initial Segment of Natural Numbers",
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Cardinality",
"Definition:Natural Numbers/Von Neumann Construction",
"Subset implies Cardinal Inequality",... |
proofwiki-12490 | P-adic Valuation of Difference of Powers with Coprime Exponent | Let $x, y \in \Z$ be distinct integers.
Let $n \ge 1$ be a natural number.
Let $p$ be a prime number.
Let:
:$p \divides x - y$
and:
:$p \nmid x y n$.
Then
:$\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x - y}$ | From Difference of Two Powers:
:$x^n - y^n = \paren {x - y} \paren {x^{n - 1} + \cdots + y^{n - 1} }$
We have to show that:
:$p \nmid x^{n - 1} + \cdots + y^{n - 1}$
Because $x \equiv y \pmod p$:
:$x^{n - 1} + \cdots + y^{n - 1} \equiv x^{n - 1} + x^{n - 1} + \cdots + x^{n - 1} = n x^{n - 1} \pmod p$
Because $p \nmid x... | Let $x, y \in \Z$ be [[Definition:Distinct|distinct]] [[Definition:Integer|integers]].
Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
Let $p$ be a [[Definition:Prime Number|prime number]].
Let:
:$p \divides x - y$
and:
:$p \nmid x y n$.
Then
:$\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x - y}$ | From [[Difference of Two Powers]]:
:$x^n - y^n = \paren {x - y} \paren {x^{n - 1} + \cdots + y^{n - 1} }$
We have to show that:
:$p \nmid x^{n - 1} + \cdots + y^{n - 1}$
Because $x \equiv y \pmod p$:
:$x^{n - 1} + \cdots + y^{n - 1} \equiv x^{n - 1} + x^{n - 1} + \cdots + x^{n - 1} = n x^{n - 1} \pmod p$
Because $p ... | P-adic Valuation of Difference of Powers with Coprime Exponent/Proof 1 | https://proofwiki.org/wiki/P-adic_Valuation_of_Difference_of_Powers_with_Coprime_Exponent | https://proofwiki.org/wiki/P-adic_Valuation_of_Difference_of_Powers_with_Coprime_Exponent/Proof_1 | [
"Lifting The Exponent Lemma",
"P-adic Valuation of Difference of Powers with Coprime Exponent"
] | [
"Definition:Distinct",
"Definition:Integer",
"Definition:Natural Numbers",
"Definition:Prime Number"
] | [
"Difference of Two Powers"
] |
proofwiki-12491 | P-adic Valuation of Difference of Powers with Coprime Exponent | Let $x, y \in \Z$ be distinct integers.
Let $n \ge 1$ be a natural number.
Let $p$ be a prime number.
Let:
:$p \divides x - y$
and:
:$p \nmid x y n$.
Then
:$\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x - y}$ | From Difference of Two Powers:
:$x^n - y^n = \paren {x - y} \paren {x^{n - 1} + \cdots + y^{n - 1} }$
We have to prove that:
:$p \nmid x^{n - 1} + \cdots + y^{n - 1}$
Let $\map P u = u^n - y^n$.
If $p \divides x^{n - 1} + \cdots + y^{n - 1}$, then $x$ would be a double root of $P$ modulo $p$.
By Double Root of Polynomi... | Let $x, y \in \Z$ be [[Definition:Distinct|distinct]] [[Definition:Integer|integers]].
Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
Let $p$ be a [[Definition:Prime Number|prime number]].
Let:
:$p \divides x - y$
and:
:$p \nmid x y n$.
Then
:$\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x - y}$ | From [[Difference of Two Powers]]:
:$x^n - y^n = \paren {x - y} \paren {x^{n - 1} + \cdots + y^{n - 1} }$
We have to prove that:
:$p \nmid x^{n - 1} + \cdots + y^{n - 1}$
Let $\map P u = u^n - y^n$.
If $p \divides x^{n - 1} + \cdots + y^{n - 1}$, then $x$ would be a double root of $P$ modulo $p$.
By [[Double Root o... | P-adic Valuation of Difference of Powers with Coprime Exponent/Proof 2 | https://proofwiki.org/wiki/P-adic_Valuation_of_Difference_of_Powers_with_Coprime_Exponent | https://proofwiki.org/wiki/P-adic_Valuation_of_Difference_of_Powers_with_Coprime_Exponent/Proof_2 | [
"Lifting The Exponent Lemma",
"P-adic Valuation of Difference of Powers with Coprime Exponent"
] | [
"Definition:Distinct",
"Definition:Integer",
"Definition:Natural Numbers",
"Definition:Prime Number"
] | [
"Difference of Two Powers",
"Double Root of Polynomial is Root of Derivative"
] |
proofwiki-12492 | Fundamental Theorem of Line Integrals | Let $\CC$ be a smooth curve given by the vector function $\map {\mathbf r} t$ for $a \le t \le b$.
Let $f$ be a differentiable function of two or three variables whose gradient vector $\nabla f$ is continuous on $\CC$.
Then:
:$\ds \int_\CC \nabla f \cdot d \mathbf r = \map f {\map {\mathbf r} b} - \map f {\map {\mathbf... | {{begin-eqn}}
{{eqn | l = \int_\CC \nabla f \cdot \rd \mathbf r
| r = \int_a^b \nabla f \cdot \map {\mathbf r'} t \rd t
}}
{{eqn | r = \int_a^b \frac {\partial f} {\partial x} \frac {\rd x} {\rd t} + \frac {\partial f} {\partial y} \frac {\rd y} {\rd t} + \frac {\partial f} {\partial z} \frac {\rd z} {\rd t} \rd ... | Let $\CC$ be a smooth curve given by the vector function $\map {\mathbf r} t$ for $a \le t \le b$.
Let $f$ be a differentiable function of two or three variables whose gradient vector $\nabla f$ is continuous on $\CC$.
Then:
:$\ds \int_\CC \nabla f \cdot d \mathbf r = \map f {\map {\mathbf r} b} - \map f {\map {\ma... | {{begin-eqn}}
{{eqn | l = \int_\CC \nabla f \cdot \rd \mathbf r
| r = \int_a^b \nabla f \cdot \map {\mathbf r'} t \rd t
}}
{{eqn | r = \int_a^b \frac {\partial f} {\partial x} \frac {\rd x} {\rd t} + \frac {\partial f} {\partial y} \frac {\rd y} {\rd t} + \frac {\partial f} {\partial z} \frac {\rd z} {\rd t} \rd ... | Fundamental Theorem of Line Integrals | https://proofwiki.org/wiki/Fundamental_Theorem_of_Line_Integrals | https://proofwiki.org/wiki/Fundamental_Theorem_of_Line_Integrals | [
"Vector Calculus",
"Fundamental Theorems"
] | [] | [
"Chain Rule for Real-Valued Functions",
"Fundamental Theorem of Calculus",
"Category:Vector Calculus",
"Category:Fundamental Theorems"
] |
proofwiki-12493 | 11 is Only Palindromic Prime with Even Number of Digits | $11$ is the only palindromic prime with an even number of digits when expressed in decimal notation. | Let $P$ be a palindromic number with $2 n$ digits:
:$P = \sqbrk {a_{2 n - 1} a_{2 n - 2} \ldots a_2 a_1 a_0}_{10}$
Thus:
:$P = \ds \sum_{j \mathop = 0}^{n - 1} a_j + 10^{2 n - 1 - j}$
Consider the summation:
:$S = \ds \sum_{k \mathop = 0}^{2 n - 1} \paren {-1}^k a_k$
As $a_k = a_{2 n - 1 - k}$ we have:
{{begin-eqn}}
{{... | $11$ is the only [[Definition:Palindromic Prime|palindromic prime]] with an [[Definition:Even Integer|even number]] of [[Definition:Digit|digits]] when expressed in [[Definition:Decimal Notation|decimal notation]]. | Let $P$ be a [[Definition:Palindromic Number|palindromic number]] with $2 n$ [[Definition:Digit|digits]]:
:$P = \sqbrk {a_{2 n - 1} a_{2 n - 2} \ldots a_2 a_1 a_0}_{10}$
Thus:
:$P = \ds \sum_{j \mathop = 0}^{n - 1} a_j + 10^{2 n - 1 - j}$
Consider the [[Definition:Summation|summation]]:
:$S = \ds \sum_{k \mathop = 0... | 11 is Only Palindromic Prime with Even Number of Digits | https://proofwiki.org/wiki/11_is_Only_Palindromic_Prime_with_Even_Number_of_Digits | https://proofwiki.org/wiki/11_is_Only_Palindromic_Prime_with_Even_Number_of_Digits | [
"Palindromic Primes",
"11"
] | [
"Definition:Palindromic Prime",
"Definition:Even Integer",
"Definition:Digit",
"Definition:Decimal Notation"
] | [
"Definition:Palindromic Number",
"Definition:Digit",
"Definition:Summation",
"Definition:Odd Integer",
"Definition:Parity of Integer",
"Divisibility by 11",
"Definition:Divisor (Algebra)/Integer",
"Definition:Palindromic Number",
"Definition:Even Integer",
"Definition:Digit",
"Definition:Prime N... |
proofwiki-12494 | Ratio of Consecutive Lucas Numbers | For $n \in \N$, let $L_n$ be the $n$th Lucas number.
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {L_{n + 1} } {L_n} = \phi$
where $\phi = \dfrac {1 + \sqrt 5} 2$ is the golden mean. | {{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \frac {L_{n + 1} } {L_n}
| r = \lim_{n \mathop \to \infty} \frac {\phi^{n + 1} + \paren {-\phi^{-1} }^{n + 1} } {\phi^n + \paren {-\phi^{-1} }^n}
| c = Closed Form for Lucas Numbers
}}
{{eqn | r = \lim_{n \mathop \to \infty} \frac {\phi - \phi^{-1} \pare... | For $n \in \N$, let $L_n$ be the $n$th [[Definition:Lucas Number|Lucas number]].
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {L_{n + 1} } {L_n} = \phi$
where $\phi = \dfrac {1 + \sqrt 5} 2$ is the [[Definition:Golden Mean|golden mean]]. | {{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \frac {L_{n + 1} } {L_n}
| r = \lim_{n \mathop \to \infty} \frac {\phi^{n + 1} + \paren {-\phi^{-1} }^{n + 1} } {\phi^n + \paren {-\phi^{-1} }^n}
| c = [[Closed Form for Lucas Numbers]]
}}
{{eqn | r = \lim_{n \mathop \to \infty} \frac {\phi - \phi^{-1} \... | Ratio of Consecutive Lucas Numbers | https://proofwiki.org/wiki/Ratio_of_Consecutive_Lucas_Numbers | https://proofwiki.org/wiki/Ratio_of_Consecutive_Lucas_Numbers | [
"Lucas Numbers",
"Golden Mean"
] | [
"Definition:Lucas Number",
"Definition:Golden Mean"
] | [
"Closed Form for Lucas Numbers",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Combination Theorem for Limits of Functions/Real/Quotient Rule"
] |
proofwiki-12495 | Ordered Set of All Mappings is Ordered Set | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $X$ be a set.
Then $L^X$ is also an ordered set. | By definition of ordered set of all mappings:
:$L^X = \struct {S^X, \precsim}$
where
:$\forall f, g \in S^X: f \precsim g \iff f \preceq g$
:$\preceq$ denotes the ordering on mappings,
:$S^X$ denotes the set of all mappings from $X$ into $S$. | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $X$ be a [[Definition:Set|set]].
Then $L^X$ is also an [[Definition:Ordered Set|ordered set]]. | By definition of [[Definition:Ordered Set of All Mappings|ordered set of all mappings]]:
:$L^X = \struct {S^X, \precsim}$
where
:$\forall f, g \in S^X: f \precsim g \iff f \preceq g$
:$\preceq$ denotes the [[Definition:Ordering on Mappings|ordering on mappings]],
:$S^X$ denotes the [[Definition:Set of All Mappings|set ... | Ordered Set of All Mappings is Ordered Set | https://proofwiki.org/wiki/Ordered_Set_of_All_Mappings_is_Ordered_Set | https://proofwiki.org/wiki/Ordered_Set_of_All_Mappings_is_Ordered_Set | [
"Ordered Sets"
] | [
"Definition:Ordered Set",
"Definition:Set",
"Definition:Ordered Set"
] | [
"Definition:Ordered Set of All Mappings",
"Definition:Ordering on Mappings",
"Definition:Set of All Mappings",
"Definition:Ordering on Mappings",
"Definition:Ordering on Mappings",
"Definition:Ordering on Mappings",
"Definition:Ordering on Mappings"
] |
proofwiki-12496 | Closed Form for Lucas Numbers | The Lucas numbers have a closed-form solution:
:$L_n = \phi^n + \paren {1 - \phi}^n = \paren {\dfrac {1 + \sqrt 5} 2}^n + \paren {\dfrac {1 - \sqrt 5} 2}^n$
where $\phi$ is the golden mean.
Putting $\hat \phi = 1 - \phi = -\dfrac 1 \phi$ this can be written:
:$L_n = \phi^n + \hat \phi^n$ | Proof by induction:
For all $n \in \N$, let $\map P n$ be the proposition:
:$L_n = \phi^n + \paren {1 - \phi}^n = \paren {\dfrac {1 + \sqrt 5} 2})^n + \paren {\dfrac {1 - \sqrt 5} 2}^n = \phi^n + \hat \phi^n$ | The [[Definition:Lucas Number|Lucas numbers]] have a [[Definition:Closed-Form Solution|closed-form solution]]:
:$L_n = \phi^n + \paren {1 - \phi}^n = \paren {\dfrac {1 + \sqrt 5} 2}^n + \paren {\dfrac {1 - \sqrt 5} 2}^n$
where $\phi$ is the [[Definition:Golden Mean|golden mean]].
Putting $\hat \phi = 1 - \phi = -\dfr... | Proof by [[Second Principle of Mathematical Induction|induction]]:
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$L_n = \phi^n + \paren {1 - \phi}^n = \paren {\dfrac {1 + \sqrt 5} 2})^n + \paren {\dfrac {1 - \sqrt 5} 2}^n = \phi^n + \hat \phi^n$ | Closed Form for Lucas Numbers | https://proofwiki.org/wiki/Closed_Form_for_Lucas_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Lucas_Numbers | [
"Lucas Numbers",
"Golden Mean",
"Closed Forms"
] | [
"Definition:Lucas Number",
"Definition:Closed Form Solution",
"Definition:Golden Mean"
] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Second Principle of Mathematical Induction",
"Second Principle of Mathematical Induction",
"Second Principle of Mathematical Induction",
"Second Principle of Mathematical Induction"
] |
proofwiki-12497 | Fibonacci Number 2n equals Fibonacci Number n by Lucas Number n | Let $F_n$ denote the $n$th Fibonacci number.
Let $L_n$ denote the $n$th Lucas number.
Then:
:$F_{2 n} = F_n L_n$ | Let:
:$\phi = \dfrac {1 + \sqrt 5} 2$
:$\hat \phi = \dfrac {1 - \sqrt 5} 2$
Then:
{{begin-eqn}}
{{eqn | l = F_{2 n}
| r = \dfrac {\phi^{2 n} - \hat \phi^{2 n} } {\sqrt 5}
| c = Euler-Binet Formula
}}
{{eqn | r = \dfrac {\paren {\phi^n + \hat \phi^n} \paren {\phi^n - \hat \phi^n} } {\sqrt 5}
| c = Diff... | Let $F_n$ denote the $n$th [[Definition:Fibonacci Number|Fibonacci number]].
Let $L_n$ denote the $n$th [[Definition:Lucas Number|Lucas number]].
Then:
:$F_{2 n} = F_n L_n$ | Let:
:$\phi = \dfrac {1 + \sqrt 5} 2$
:$\hat \phi = \dfrac {1 - \sqrt 5} 2$
Then:
{{begin-eqn}}
{{eqn | l = F_{2 n}
| r = \dfrac {\phi^{2 n} - \hat \phi^{2 n} } {\sqrt 5}
| c = [[Euler-Binet Formula]]
}}
{{eqn | r = \dfrac {\paren {\phi^n + \hat \phi^n} \paren {\phi^n - \hat \phi^n} } {\sqrt 5}
| c ... | Fibonacci Number 2n equals Fibonacci Number n by Lucas Number n | https://proofwiki.org/wiki/Fibonacci_Number_2n_equals_Fibonacci_Number_n_by_Lucas_Number_n | https://proofwiki.org/wiki/Fibonacci_Number_2n_equals_Fibonacci_Number_n_by_Lucas_Number_n | [
"Fibonacci Numbers",
"Lucas Numbers"
] | [
"Definition:Fibonacci Number",
"Definition:Lucas Number"
] | [
"Euler-Binet Formula",
"Difference of Two Squares",
"Closed Form for Lucas Numbers",
"Euler-Binet Formula"
] |
proofwiki-12498 | Fibonacci Number 3n in terms of Fibonacci Number n and Lucas Number 2n | Let $F_n$ denote the $n$th Fibonacci number.
Let $L_n$ denote the $n$th Lucas number.
Then:
:$F_{3 n} = F_n \paren {L_{2 n} + \paren {-1}^n}$ | Let:
:$\phi = \dfrac {1 + \sqrt 5} 2$
:$\hat \phi = \dfrac {1 - \sqrt 5} 2$
Then:
{{begin-eqn}}
{{eqn | l = F_{3 n}
| r = \dfrac {\phi^{3 n} - \hat \phi^{3 n} } {\sqrt 5}
| c = Euler-Binet Formula
}}
{{eqn | r = \dfrac {\paren {\phi^n - \hat \phi^n} \paren {\phi^{2 n} + \phi^n \hat \phi^n + \hat \phi^{2 n} ... | Let $F_n$ denote the $n$th [[Definition:Fibonacci Number|Fibonacci number]].
Let $L_n$ denote the $n$th [[Definition:Lucas Number|Lucas number]].
Then:
:$F_{3 n} = F_n \paren {L_{2 n} + \paren {-1}^n}$ | Let:
:$\phi = \dfrac {1 + \sqrt 5} 2$
:$\hat \phi = \dfrac {1 - \sqrt 5} 2$
Then:
{{begin-eqn}}
{{eqn | l = F_{3 n}
| r = \dfrac {\phi^{3 n} - \hat \phi^{3 n} } {\sqrt 5}
| c = [[Euler-Binet Formula]]
}}
{{eqn | r = \dfrac {\paren {\phi^n - \hat \phi^n} \paren {\phi^{2 n} + \phi^n \hat \phi^n + \hat \phi^... | Fibonacci Number 3n in terms of Fibonacci Number n and Lucas Number 2n | https://proofwiki.org/wiki/Fibonacci_Number_3n_in_terms_of_Fibonacci_Number_n_and_Lucas_Number_2n | https://proofwiki.org/wiki/Fibonacci_Number_3n_in_terms_of_Fibonacci_Number_n_and_Lucas_Number_2n | [
"Fibonacci Numbers",
"Lucas Numbers"
] | [
"Definition:Fibonacci Number",
"Definition:Lucas Number"
] | [
"Euler-Binet Formula",
"Difference of Two Powers/Examples/Difference of Two Cubes",
"Euler-Binet Formula",
"Closed Form for Lucas Numbers",
"Difference of Two Squares"
] |
proofwiki-12499 | Relation between Square of Fibonacci Number and Square of Lucas Number | Let $F_n$ denote the $n$th Fibonacci number.
Let $L_n$ denote the $n$th Lucas number.
Then:
:$5 {F_n}^2 + 4 \paren {-1}^n = {L_n}^2$ | Let:
:$\phi = \dfrac {1 + \sqrt 5} 2$
:$\hat \phi = \dfrac {1 - \sqrt 5} 2$
Note that we have:
{{begin-eqn}}
{{eqn | l = \phi \hat \phi
| r = \dfrac {1 + \sqrt 5} 2 \dfrac {1 - \sqrt 5} 2
| c =
}}
{{eqn | r = \dfrac {1 - 5} 4
| c = Difference of Two Squares
}}
{{eqn | r = -1
| c =
}}
{{end-eqn... | Let $F_n$ denote the $n$th [[Definition:Fibonacci Number|Fibonacci number]].
Let $L_n$ denote the $n$th [[Definition:Lucas Number|Lucas number]].
Then:
:$5 {F_n}^2 + 4 \paren {-1}^n = {L_n}^2$ | Let:
:$\phi = \dfrac {1 + \sqrt 5} 2$
:$\hat \phi = \dfrac {1 - \sqrt 5} 2$
Note that we have:
{{begin-eqn}}
{{eqn | l = \phi \hat \phi
| r = \dfrac {1 + \sqrt 5} 2 \dfrac {1 - \sqrt 5} 2
| c =
}}
{{eqn | r = \dfrac {1 - 5} 4
| c = [[Difference of Two Squares]]
}}
{{eqn | r = -1
| c =
}}
{{e... | Relation between Square of Fibonacci Number and Square of Lucas Number | https://proofwiki.org/wiki/Relation_between_Square_of_Fibonacci_Number_and_Square_of_Lucas_Number | https://proofwiki.org/wiki/Relation_between_Square_of_Fibonacci_Number_and_Square_of_Lucas_Number | [
"Fibonacci Numbers",
"Lucas Numbers"
] | [
"Definition:Fibonacci Number",
"Definition:Lucas Number"
] | [
"Difference of Two Squares",
"Euler-Binet Formula",
"Closed Form for Lucas Numbers"
] |
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