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-12800 | 4 Consecutive Integers cannot be Square-Free | Let $n, n + 1, n + 2, n + 3$ be four consecutive positive integers.
At least one of these is not square-free. | Exactly one of $n, n + 1, n + 2, n + 3$ is divisible by $4 = 2^2$.
Thus, by definition, one of these is not square-free.
{{qed}} | Let $n, n + 1, n + 2, n + 3$ be four consecutive [[Definition:Positive Integer|positive integers]].
At least one of these is not [[Definition:Square-Free|square-free]]. | Exactly one of $n, n + 1, n + 2, n + 3$ is [[Definition:Divisor of Integer|divisible]] by $4 = 2^2$.
Thus, by definition, one of these is not [[Definition:Square-Free|square-free]].
{{qed}} | 4 Consecutive Integers cannot be Square-Free | https://proofwiki.org/wiki/4_Consecutive_Integers_cannot_be_Square-Free | https://proofwiki.org/wiki/4_Consecutive_Integers_cannot_be_Square-Free | [
"Square-Free Integers"
] | [
"Definition:Positive/Integer",
"Definition:Square-Free"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Square-Free"
] |
proofwiki-12801 | Sequence of 4 Consecutive Square-Free Triplets | The following sets of $4$ consecutive triplets of integers, with one integer between each triplet, are square-free:
:$29, 30, 31; 33, 34, 35; 37, 38, 39; 41, 42, 43$
:$101, 102, 103; 105, 106, 107; 109, 110, 111; 113, 114, 115$ | Note that $32, 36, 40$ and $104, 108, 112$ are all divisible by $4 = 2^2$, so are by definition not square-free.
Then inspecting each number in turn:
{{begin-eqn}}
{{eqn | l = 29
| o =
| c = is prime
}}
{{eqn | l = 30
| r = 2 \times 3 \times 5
| c = and so is square-free
}}
{{eqn | l = 31
... | The following sets of $4$ consecutive triplets of [[Definition:Integer|integers]], with one [[Definition:Integer|integer]] between each triplet, are [[Definition:Square-Free|square-free]]:
:$29, 30, 31; 33, 34, 35; 37, 38, 39; 41, 42, 43$
:$101, 102, 103; 105, 106, 107; 109, 110, 111; 113, 114, 115$ | Note that $32, 36, 40$ and $104, 108, 112$ are all [[Definition:Divisor of Integer|divisible]] by $4 = 2^2$, so are by definition not [[Definition:Square-Free|square-free]].
Then inspecting each number in turn:
{{begin-eqn}}
{{eqn | l = 29
| o =
| c = is [[Definition:Prime Number|prime]]
}}
{{eqn | l = ... | Sequence of 4 Consecutive Square-Free Triplets | https://proofwiki.org/wiki/Sequence_of_4_Consecutive_Square-Free_Triplets | https://proofwiki.org/wiki/Sequence_of_4_Consecutive_Square-Free_Triplets | [
"Square-Free Integers"
] | [
"Definition:Integer",
"Definition:Integer",
"Definition:Square-Free"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Square-Free",
"Definition:Prime Number",
"Definition:Square-Free",
"Definition:Prime Number",
"Definition:Square-Free",
"Definition:Square-Free",
"Definition:Square-Free",
"Definition:Prime Number",
"Definition:Square-Free",
"Definition:Square-... |
proofwiki-12802 | Image of Doubleton under Mapping | Let $S, T$ be sets.
Let $f: S \to T$ be a mapping.
Then:
:$\forall x, y \in S: f \sqbrk {\set {x, y} } = \set {\map f x, \map f y}$ | Let $x, y \in S$.
Thus
{{begin-eqn}}
{{eqn | l = f \sqbrk {\set {x, y} }
| r = f \sqbrk {\set x \cup \set y}
| c = Union of Unordered Tuples
}}
{{eqn | r = f \sqbrk {\set x} \cup f \sqbrk {\set y}
| c = Image of Union under Mapping
}}
{{eqn | r = \set {\map f x} \cup \set {\map f y}
| c = Image ... | Let $S, T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then:
:$\forall x, y \in S: f \sqbrk {\set {x, y} } = \set {\map f x, \map f y}$ | Let $x, y \in S$.
Thus
{{begin-eqn}}
{{eqn | l = f \sqbrk {\set {x, y} }
| r = f \sqbrk {\set x \cup \set y}
| c = [[Union of Unordered Tuples]]
}}
{{eqn | r = f \sqbrk {\set x} \cup f \sqbrk {\set y}
| c = [[Image of Union under Mapping]]
}}
{{eqn | r = \set {\map f x} \cup \set {\map f y}
| c... | Image of Doubleton under Mapping | https://proofwiki.org/wiki/Image_of_Doubleton_under_Mapping | https://proofwiki.org/wiki/Image_of_Doubleton_under_Mapping | [
"Images",
"Doubletons"
] | [
"Definition:Set",
"Definition:Mapping"
] | [
"Union of Unordered Tuples",
"Image of Union under Mapping",
"Image of Singleton under Mapping",
"Union of Unordered Tuples"
] |
proofwiki-12803 | Image under Increasing Mapping equal to Special Set is Complete Lattice | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $f: S \to S$ be an increasing mapping.
Let $P = \struct {M, \precsim}$ be an ordered subset of $L$ such that
:$M = \set {x \in S: x = \map f x}$
Then $P$ is complete lattice. | We will prove that
:$\forall X \subseteq M: \forall Y \subseteq S: Y = \left\{ {x \in S: x}\right.$ is upper bound for $\left.{X \land \map f x \preceq x}\right\} \implies \inf_L Y \in M$
Let $X \subseteq M$, $Y \subseteq S$ such that
:$Y = \left\{ {x \in S: x}\right.$ is upper bound for $\left.{X \land \map f x \prece... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $f: S \to S$ be an [[Definition:Increasing Mapping|increasing mapping]].
Let $P = \struct {M, \precsim}$ be an [[Definition:Ordered Subset|ordered subset]] of $L$ such that
:$M = \set {x \in S: x = \map f x}$
The... | We will prove that
:$\forall X \subseteq M: \forall Y \subseteq S: Y = \left\{ {x \in S: x}\right.$ is [[Definition:Upper Bound of Set|upper bound]] for $\left.{X \land \map f x \preceq x}\right\} \implies \inf_L Y \in M$
Let $X \subseteq M$, $Y \subseteq S$ such that
:$Y = \left\{ {x \in S: x}\right.$ is [[Definition... | Image under Increasing Mapping equal to Special Set is Complete Lattice | https://proofwiki.org/wiki/Image_under_Increasing_Mapping_equal_to_Special_Set_is_Complete_Lattice | https://proofwiki.org/wiki/Image_under_Increasing_Mapping_equal_to_Special_Set_is_Complete_Lattice | [
"Complete Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Increasing/Mapping",
"Definition:Ordered Subset",
"Definition:Complete Lattice"
] | [
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Definition:Increasing/Mapping",
"Definition:Transitive",
"Definition:Upper Bound of Set",
"Definition:Lower Bound of Set",
"Definition:U... |
proofwiki-12804 | Hensel's Lemma/First Form | Let $p$ be a prime number.
Let $k > 0$ be a positive integer.
Let $\map f X \in \Z \sqbrk X$ be a polynomial.
Let $x_k \in \Z$ such that:
{{begin-eqn}}
{{eqn | l = \map f {x_k}
| o = \equiv
| r = 0
| rr= \pmod {p^k}
}}
{{eqn | l = \map {f'} {x_k}
| o = \not \equiv
| r = 0
| rr= \pmod... | We use induction on $l$.
The base case $l = 0$ is trivial.
Let $l \ge 0$ be such that a solution $x_{k + l}$ exists and is unique up to a multiple of $p^{k + l}$.
Choose a solution $x_{k + l}$ satisfying:
{{begin-eqn}}
{{eqn | l = \map f {x_{k + l} }
| o = \equiv
| r = 0
| rr= \pmod {p^{k + l} }
}}
{{... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $k > 0$ be a [[Definition:Positive Integer|positive integer]].
Let $\map f X \in \Z \sqbrk X$ be a [[Definition:Polynomial (Abstract Algebra)|polynomial]].
Let $x_k \in \Z$ such that:
{{begin-eqn}}
{{eqn | l = \map f {x_k}
| o = \equiv
| r = 0
... | We use [[Principle of Mathematical Induction|induction]] on $l$.
The base case $l = 0$ is trivial.
Let $l \ge 0$ be such that a solution $x_{k + l}$ exists and is unique up to a [[Definition:Divisor of Integer|multiple]] of $p^{k + l}$.
Choose a solution $x_{k + l}$ satisfying:
{{begin-eqn}}
{{eqn | l = \map f {x_{k... | Hensel's Lemma/First Form | https://proofwiki.org/wiki/Hensel's_Lemma/First_Form | https://proofwiki.org/wiki/Hensel's_Lemma/First_Form | [
"Hensel's Lemma"
] | [
"Definition:Prime Number",
"Definition:Positive/Integer",
"Definition:Polynomial over Ring",
"Definition:Integer",
"Definition:Integer",
"Definition:Integer",
"Definition:Congruence (Number Theory)/Integers"
] | [
"Principle of Mathematical Induction",
"Definition:Divisor (Algebra)/Integer",
"Congruence by Divisor of Modulus",
"Taylor Expansion for Polynomials/Order 1",
"Principle of Mathematical Induction"
] |
proofwiki-12805 | Hensel's Lemma for Composite Numbers | Let $b \in \Z \setminus \set {-1, 0, 1}$ be an integer.
Let $k > 0$ be a positive integer.
Let $\map f X \in \Z \sqbrk X$ be a polynomial.
Let $x_k \in \Z$ such that:
:$\map f {x_k} \equiv 0 \pmod {b^k}$
:$\gcd \set {\map {f'} {x_k}, b} = 1$
Then for every integer $l \ge 0$, there exists an integer $x_{k + l}$ such tha... | We use induction on $l$.
The base case $l = 0$ is trivial.
Let $l \ge 0$ be such that a solution $x_{k + l}$ exists and is unique up to a multiple of $b^{k + l}$.
Choose a solution $x_{k + l}$.
Each solution $x_{k + l + 1}$ is also a solution of the previous congruence.
By uniqueness, it has to satisfy $x_{k + l + 1} \... | Let $b \in \Z \setminus \set {-1, 0, 1}$ be an [[Definition:Integer|integer]].
Let $k > 0$ be a [[Definition:Positive Integer|positive integer]].
Let $\map f X \in \Z \sqbrk X$ be a [[Definition:Polynomial (Abstract Algebra)|polynomial]].
Let $x_k \in \Z$ such that:
:$\map f {x_k} \equiv 0 \pmod {b^k}$
:$\gcd \set {... | We use [[Principle of Mathematical Induction|induction]] on $l$.
The base case $l = 0$ is trivial.
Let $l \ge 0$ be such that a solution $x_{k + l}$ exists and is unique up to a multiple of $b^{k + l}$.
Choose a solution $x_{k + l}$.
Each solution $x_{k + l + 1}$ is also a solution of the previous congruence.
By u... | Hensel's Lemma for Composite Numbers | https://proofwiki.org/wiki/Hensel's_Lemma_for_Composite_Numbers | https://proofwiki.org/wiki/Hensel's_Lemma_for_Composite_Numbers | [
"Hensel's Lemma"
] | [
"Definition:Integer",
"Definition:Positive/Integer",
"Definition:Polynomial over Ring",
"Definition:Integer",
"Definition:Integer",
"Definition:Integer",
"Definition:Congruence (Number Theory)/Integers"
] | [
"Principle of Mathematical Induction",
"Taylor Expansion for Polynomials/Order 1",
"Principle of Mathematical Induction"
] |
proofwiki-12806 | Divisor Sum of Power of 2 | Let $n \in \Z_{>0}$ be a power of $2$.
Then:
:$\map {\sigma_1} n = 2 n - 1$ | Let $n = 2^k$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} n
| r = \dfrac {2^{k + 1} - 1} {2 - 1}
| c = Divisor Sum of Power of Prime
}}
{{eqn | r = 2 \times 2^k - 1
| c =
}}
{{eqn | r = 2 n - 1
| c =
}}
{{end-eqn}}
{{qed}}
Category:Divisor Sum Function
Category:2
5adh8ba8ttxo6k4732otbkyj3... | Let $n \in \Z_{>0}$ be a [[Definition:Integer Power|power]] of $2$.
Then:
:$\map {\sigma_1} n = 2 n - 1$ | Let $n = 2^k$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} n
| r = \dfrac {2^{k + 1} - 1} {2 - 1}
| c = [[Divisor Sum of Power of Prime]]
}}
{{eqn | r = 2 \times 2^k - 1
| c =
}}
{{eqn | r = 2 n - 1
| c =
}}
{{end-eqn}}
{{qed}}
[[Category:Divisor Sum Function]]
[[Category:2]]
5adh8ba8ttx... | Divisor Sum of Power of 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Power_of_2 | https://proofwiki.org/wiki/Divisor_Sum_of_Power_of_2 | [
"Divisor Sum Function",
"2"
] | [
"Definition:Power (Algebra)/Integer"
] | [
"Divisor Sum of Power of Prime",
"Category:Divisor Sum Function",
"Category:2"
] |
proofwiki-12807 | Equivalence of Definitions of Saturation Under Equivalence Relation | Let $\sim$ be an equivalence relation on a set $S$.
Let $T \subset S$ be a subset.
{{TFAE|def = Saturation (Equivalence Relation)|view = saturation}} | === Definitions 1 and 2 are equivalent ===
{{begin-eqn}}
{{eqn | l = \bigcup_{t \mathop \in T} \eqclass t \sim
| r = \set {s \in S: \exists t \in T: s \in \eqclass t \sim}
| c = {{Defof|Union of Family of Subsets}}
}}
{{eqn | r = \set {s \in S: \exists t \in T: s \sim t}
| c = {{Defof|Equivalence Clas... | Let $\sim$ be an [[Definition:Equivalence Relation|equivalence relation]] on a [[Definition:Set|set]] $S$.
Let $T \subset S$ be a [[Definition:Subset|subset]].
{{TFAE|def = Saturation (Equivalence Relation)|view = saturation}} | === Definitions 1 and 2 are equivalent ===
{{begin-eqn}}
{{eqn | l = \bigcup_{t \mathop \in T} \eqclass t \sim
| r = \set {s \in S: \exists t \in T: s \in \eqclass t \sim}
| c = {{Defof|Union of Family of Subsets}}
}}
{{eqn | r = \set {s \in S: \exists t \in T: s \sim t}
| c = {{Defof|Equivalence Cla... | Equivalence of Definitions of Saturation Under Equivalence Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Saturation_Under_Equivalence_Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Saturation_Under_Equivalence_Relation | [
"Equivalence Relations"
] | [
"Definition:Equivalence Relation",
"Definition:Set",
"Definition:Subset"
] | [] |
proofwiki-12808 | Equivalence of Definitions of Saturated Set Under Equivalence Relation | Let $\sim$ be an equivalence relation on a set $S$.
Let $T \subset S$ be a subset.
{{TFAE|def = Saturated Set (Equivalence Relation)|view = saturated set|context = Equivalence Relation}} | === 1 implies 2 ===
Let $T = \overline T$.
By definition of saturation:
:$T = \ds \bigcup_{t \mathop \in T} \eqclass t {}$
so we can take $U = T$.
{{qed}} | Let $\sim$ be an [[Definition:Equivalence Relation|equivalence relation]] on a [[Definition:Set|set]] $S$.
Let $T \subset S$ be a [[Definition:Subset|subset]].
{{TFAE|def = Saturated Set (Equivalence Relation)|view = saturated set|context = Equivalence Relation}} | === 1 implies 2 ===
Let $T = \overline T$.
By definition of [[Definition:Saturation (Equivalence Relation)|saturation]]:
:$T = \ds \bigcup_{t \mathop \in T} \eqclass t {}$
so we can take $U = T$.
{{qed}} | Equivalence of Definitions of Saturated Set Under Equivalence Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Saturated_Set_Under_Equivalence_Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Saturated_Set_Under_Equivalence_Relation | [
"Equivalence Relations"
] | [
"Definition:Equivalence Relation",
"Definition:Set",
"Definition:Subset"
] | [
"Definition:Saturation (Equivalence Relation)",
"Definition:Saturation (Equivalence Relation)",
"Definition:Saturation (Equivalence Relation)"
] |
proofwiki-12809 | Compact Subset is Bounded Below Join Semilattice | Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.
Let $P = \left({K\left({L}\right), \precsim}\right)$ be an ordered subset of $L$,
where $K\left({L}\right)$ denotes the compact subset of $L$.
Then $P$ is a bounded below join semilattice. | By Bottom is Compact:
:$\bot_L$ is a compact element,
where $\bot_L$ denotes the smallest element in $L$.
By definition of compact subset:
:$\bot_L \in K \left({L} \right)$
By definition of the smallest element:
:$\forall x \in K\left({L}\right): \bot_L \preceq x$
By definition of ordered subset:
:$\forall x \in K\left... | Let $L = \left({S, \vee, \preceq}\right)$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Let $P = \left({K\left({L}\right), \precsim}\right)$ be an [[Definition:Ordered Subset|ordered subset]] of $L$,
where $K\left({L}\right)$ denotes the [[Definition:Compact Sub... | By [[Bottom is Compact]]:
:$\bot_L$ is a [[Definition:Compact Element|compact element]],
where $\bot_L$ denotes the [[Definition:Smallest Element|smallest element]] in $L$.
By definition of [[Definition:Compact Subset of Lattice|compact subset]]:
:$\bot_L \in K \left({L} \right)$
By definition of the [[Definition:Sma... | Compact Subset is Bounded Below Join Semilattice | https://proofwiki.org/wiki/Compact_Subset_is_Bounded_Below_Join_Semilattice | https://proofwiki.org/wiki/Compact_Subset_is_Bounded_Below_Join_Semilattice | [
"Join and Meet Semilattices"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Ordered Subset",
"Definition:Compact Subset of Lattice",
"Definition:Bounded Below Set",
"Definition:Join Semilattice"
] | [
"Bottom is Compact",
"Definition:Compact Element",
"Definition:Smallest Element",
"Definition:Compact Subset of Lattice",
"Definition:Smallest Element",
"Definition:Ordered Subset",
"Definition:Bounded Below Set",
"Compact Subset is Join Subsemilattice",
"Definition:Join Semilattice"
] |
proofwiki-12810 | Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 2 | Let $\map h x : \closedint a b \to \R$ be continuously differentiable $\forall x \in \closedint a b$.
Suppose the function $\map h x$ satisfies the equation:
:$-\map {\dfrac \d {\d x} } {\paren {t P + \paren {1 - t} } h'} + t Q h = 0$
subject to the boundary conditions:
:$\map h {a, t} = \map h {b, t} = 0$
Then:
:$\ds ... | {{begin-eqn}}
{{eqn | l = 0
| r = \int_a^b \paren 0 h \rd x
}}
{{eqn | r = \int_a^b \paren {-\map {\frac \d {\d x} } {\paren {t P + \paren {1 - t} } h'} + t Q h } h \rd x
}}
{{eqn | r = \int_a^b t Q h^2 \rd x - \int_a^b h \map \d {\paren {t P + \paren {1 - t} } h'}
}}
{{eqn | r = \int_a^b t Q h^2 \rd x - h \bigin... | Let $\map h x : \closedint a b \to \R$ be [[Definition:Continuously Differentiable|continuously differentiable]] $\forall x \in \closedint a b$.
Suppose the function $\map h x$ satisfies the equation:
:$-\map {\dfrac \d {\d x} } {\paren {t P + \paren {1 - t} } h'} + t Q h = 0$
subject to the boundary conditions:
:$... | {{begin-eqn}}
{{eqn | l = 0
| r = \int_a^b \paren 0 h \rd x
}}
{{eqn | r = \int_a^b \paren {-\map {\frac \d {\d x} } {\paren {t P + \paren {1 - t} } h'} + t Q h } h \rd x
}}
{{eqn | r = \int_a^b t Q h^2 \rd x - \int_a^b h \map \d {\paren {t P + \paren {1 - t} } h'}
}}
{{eqn | r = \int_a^b t Q h^2 \rd x - h \bigin... | Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 2 | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Quadratic_Functional_to_be_Positive_Definite/Lemma_2 | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Quadratic_Functional_to_be_Positive_Definite/Lemma_2 | [
"Calculus of Variations"
] | [
"Definition:Continuously Differentiable"
] | [
"Integration by Parts"
] |
proofwiki-12811 | Jacobi's Necessary Condition | Let $J$ be a functional, such that:
:$J \sqbrk y = \ds \int_a^b \map F {x, y, y'} \rd x$
Let $\map y x$ correspond to the minimum of $J$.
Let:
:$F_{y'y'}>0$
along $\map y x$.
Then the open interval $\openint a b$ contains no points conjugate to $a$. | By Necessary Condition for Twice Differentiable Functional to have Minimum, $J$ is minimised by $y = \map {\hat y} x$ if:
:$\delta^2 J \sqbrk {\hat y; h} \ge 0$
for all admissable real functions $h$.
By lemma 1 of Legendre's Condition,
:$\ds \delta^2 J \sqbrk {y; h} = \int_a^b \paren {P h'^2 + Q h^2} \rd x$
where:
:$P... | Let $J$ be a [[Definition:Real Functional|functional]], such that:
:$J \sqbrk y = \ds \int_a^b \map F {x, y, y'} \rd x$
Let $\map y x$ correspond to the [[Definition:Minimum Value of Functional|minimum]] of $J$.
Let:
:$F_{y'y'}>0$
along $\map y x$.
Then the [[Definition:Open Real Interval|open interval]] $\openi... | By [[Necessary Condition for Twice Differentiable Functional to have Minimum]], $J$ is minimised by $y = \map {\hat y} x$ if:
:$\delta^2 J \sqbrk {\hat y; h} \ge 0$
for all admissable [[Definition:Real Function|real functions]] $h$.
By [[Legendre's Condition/Lemma 1|lemma 1 of Legendre's Condition]],
:$\ds \delta^... | Jacobi's Necessary Condition | https://proofwiki.org/wiki/Jacobi's_Necessary_Condition | https://proofwiki.org/wiki/Jacobi's_Necessary_Condition | [
"Calculus of Variations"
] | [
"Definition:Functional/Real",
"Definition:Minimum Value of Functional",
"Definition:Real Interval/Open",
"Definition:Point",
"Definition:Conjugate Point (Calculus of Variations)"
] | [
"Necessary Condition for Twice Differentiable Functional to have Minimum",
"Definition:Real Function",
"Legendre's Condition/Lemma 1",
"Nonnegative Quadratic Functional implies no Interior Conjugate Points",
"Definition:Conjugate Point (Calculus of Variations)"
] |
proofwiki-12812 | Stolz-Cesàro Theorem | Let $\sequence {a_n}$ be a sequence.
{{explain|Domain of $\sequence {a_n}$ -- $\R$ presumably but could it be $\C$?}}
Let $\sequence {b_n}$ be a sequence of (strictly) positive real numbers such that:
:$\ds \sum_{i \mathop = 0}^\infty b_n = \infty$
If:
:$\ds \lim_{n \mathop \to \infty} \dfrac {a_n} {b_n} = L \in \R$
th... | Define the following sums:
:$\ds A_n = \sum_{i \mathop = 1}^n a_i$
:$\ds B_n = \sum_{i \mathop = 1}^n b_i$
Let $\epsilon > 0$ and $\mu = \dfrac {\epsilon} 2$.
By the definition of convergent sequences, there exists $k \in \N$ such that:
:$\forall n > k: \paren {L - \mu} b_n < a_n < \paren {L + \mu} b_n$
Rewrite the su... | Let $\sequence {a_n}$ be a [[Definition:Sequence|sequence]].
{{explain|Domain of $\sequence {a_n}$ -- $\R$ presumably but could it be $\C$?}}
Let $\sequence {b_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]] such that:
:$\ds \sum_{i \mathop = ... | Define the following sums:
:$\ds A_n = \sum_{i \mathop = 1}^n a_i$
:$\ds B_n = \sum_{i \mathop = 1}^n b_i$
Let $\epsilon > 0$ and $\mu = \dfrac {\epsilon} 2$.
By the definition of [[Definition:Convergent Sequence (Analysis)|convergent sequences]], there exists $k \in \N$ such that:
:$\forall n > k: \paren {L - \mu}... | Stolz-Cesàro Theorem | https://proofwiki.org/wiki/Stolz-Cesàro_Theorem | https://proofwiki.org/wiki/Stolz-Cesàro_Theorem | [
"Analysis",
"Limits of Sequences"
] | [
"Definition:Sequence",
"Definition:Sequence",
"Definition:Strictly Positive/Real Number"
] | [
"Definition:Convergent Sequence/Analysis",
"Reciprocal of Null Sequence",
"Combination Theorem for Sequences",
"Definition:Sequence"
] |
proofwiki-12813 | Bottom is Compact | Let $L$ be a bounded below ordered set.
Then $\bot$ is a compact element
where $\bot$ is the smallest element in $L$. | By Bottom is Way Below Any Element:
:$\bot \ll \bot$
where $\ll$ denotes the way below relation.
Hence $\bot$ is a compact element.
{{qed}} | Let $L$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Ordered Set|ordered set]].
Then $\bot$ is a [[Definition:Compact Element|compact element]]
where $\bot$ is the [[Definition:Smallest Element|smallest element]] in $L$. | By [[Bottom is Way Below Any Element]]:
:$\bot \ll \bot$
where $\ll$ denotes the [[Definition:Element is Way Below|way below relation]].
Hence $\bot$ is a [[Definition:Compact Element|compact element]].
{{qed}} | Bottom is Compact | https://proofwiki.org/wiki/Bottom_is_Compact | https://proofwiki.org/wiki/Bottom_is_Compact | [
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Ordered Set",
"Definition:Compact Element",
"Definition:Smallest Element"
] | [
"Bottom is Way Below Any Element",
"Definition:Element is Way Below",
"Definition:Compact Element"
] |
proofwiki-12814 | Saturation Under Equivalence Relation in Terms of Graph | Let $\RR \subset S \times S$ be an equivalence relation on a set $S$.
Let $\pr_1, \pr_2 : S \times S \to S$ denote the projections.
Let $T\subset S$ be a subset.
Let $\overline T$ denote its saturation.
Then the following hold:
:$\overline T = \map {\pr_1} {\RR \cap \map {\pr_2^{-1} } T}$
:$\overline T = \map {\pr_2} {... | Let $s \in S$.
We have:
{{begin-eqn}}
{{eqn | o =
| r = s \in \map {\pr_1} {\RR \cap \map {\pr_2^{-1} } T}
| c =
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \exists t \in S: \tuple {s, t} \in \RR \cap \map {\pr_2^{-1} } T
| c =
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \ex... | Let $\RR \subset S \times S$ be an [[Definition:Equivalence Relation|equivalence relation]] on a [[Definition:Set|set]] $S$.
Let $\pr_1, \pr_2 : S \times S \to S$ denote the [[Definition:Projection (Mapping Theory)|projections]].
Let $T\subset S$ be a [[Definition:Subset|subset]].
Let $\overline T$ denote its [[Defi... | Let $s \in S$.
We have:
{{begin-eqn}}
{{eqn | o =
| r = s \in \map {\pr_1} {\RR \cap \map {\pr_2^{-1} } T}
| c =
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \exists t \in S: \tuple {s, t} \in \RR \cap \map {\pr_2^{-1} } T
| c =
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \... | Saturation Under Equivalence Relation in Terms of Graph | https://proofwiki.org/wiki/Saturation_Under_Equivalence_Relation_in_Terms_of_Graph | https://proofwiki.org/wiki/Saturation_Under_Equivalence_Relation_in_Terms_of_Graph | [
"Equivalence Relations"
] | [
"Definition:Equivalence Relation",
"Definition:Set",
"Definition:Projection (Mapping Theory)",
"Definition:Subset",
"Definition:Saturation (Equivalence Relation)"
] | [
"Category:Equivalence Relations"
] |
proofwiki-12815 | Nonnegative Quadratic Functional implies no Interior Conjugate Points | If the quadratic functional
:$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x$
where:
:$\forall x \in \closedint a b: \map P x > 0$
is nonnegative for all $\map h x$:
:$\map h a = \map h b = 0$
then the closed interval $\closedint a b$ contains no inside points conjugate to $a$.
In other words, the open interval $\openint ... | Consider the functional:
:$\forall t \in \closedint 0 1: \ds \int_a^b \paren {t \paren {P h^2 + Q h'^2} + \paren {1 - t} h'^2} \rd x$
By assumption:
:$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x \ge 0$
For $t = 1$, Euler's Equation reads:
:$\map {h''} x = 0$
which, along with condition $\map h a = 0$, is solved by:
:$\m... | If the [[Definition:Quadratic Functional|quadratic functional]]
:$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x$
where:
:$\forall x \in \closedint a b: \map P x > 0$
is nonnegative for all $\map h x$:
:$\map h a = \map h b = 0$
then the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ contains n... | Consider the [[Definition:Real Functional|functional]]:
:$\forall t \in \closedint 0 1: \ds \int_a^b \paren {t \paren {P h^2 + Q h'^2} + \paren {1 - t} h'^2} \rd x$
By assumption:
:$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x \ge 0$
For $t = 1$, [[Definition:Euler's Equation for Vanishing Variation|Euler's Equation]... | Nonnegative Quadratic Functional implies no Interior Conjugate Points | https://proofwiki.org/wiki/Nonnegative_Quadratic_Functional_implies_no_Interior_Conjugate_Points | https://proofwiki.org/wiki/Nonnegative_Quadratic_Functional_implies_no_Interior_Conjugate_Points | [
"Calculus of Variations"
] | [
"Definition:Quadratic Functional",
"Definition:Real Interval/Closed",
"Definition:Point",
"Definition:Conjugate Point (Calculus of Variations)",
"Definition:Real Interval/Open",
"Definition:Point",
"Definition:Conjugate Point (Calculus of Variations)"
] | [
"Definition:Functional/Real",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Conjugate Point (Calculus of Variations)",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Differential Equation/Solution",
"Definition:Conjugate Point (Calculus of Variations)",
"Necessar... |
proofwiki-12816 | Projection of Subset is Open iff Saturation is Open | Let $\sim$ be an equivalence relation on a topological space $\struct {X, \tau}$.
Let $p$ denote the quotient mapping induced by $\sim$.
Let $\tau_\sim$ be the quotient topology on $X / \sim$ by $p$.
Let $\struct {X / \sim, \tau_\sim}$ be the quotient space of $X$ by $\sim$.
Let $U \subset X$.
{{TFAE}}
{{begin-itemize}... | By definition of quotient topology, $\map p U$ is open in $\struct {X / \sim, \tau_\sim}$ {{iff}} $\map {p^{-1} } {\map p U}$ is open in $\struct {X, \tau}$.
{{qed}} | Let $\sim$ be an [[Definition:Equivalence Relation|equivalence relation]] on a [[Definition:Topological Space|topological space]] $\struct {X, \tau}$.
Let $p$ denote the [[Definition:Quotient Mapping|quotient mapping induced by $\sim$]].
Let $\tau_\sim$ be the [[Definition:Quotient Topology|quotient topology]] on $X ... | By definition of [[Definition:Quotient Topology|quotient topology]], $\map p U$ is [[Definition:Open Set (Topology)|open]] in $\struct {X / \sim, \tau_\sim}$ {{iff}} $\map {p^{-1} } {\map p U}$ is [[Definition:Open Set (Topology)|open]] in $\struct {X, \tau}$.
{{qed}} | Projection of Subset is Open iff Saturation is Open/Proof 1 | https://proofwiki.org/wiki/Projection_of_Subset_is_Open_iff_Saturation_is_Open | https://proofwiki.org/wiki/Projection_of_Subset_is_Open_iff_Saturation_is_Open/Proof_1 | [
"Projection of Subset is Open iff Saturation is Open",
"Quotient Spaces (Topology)"
] | [
"Definition:Equivalence Relation",
"Definition:Topological Space",
"Definition:Quotient Mapping",
"Definition:Quotient Topology",
"Definition:Quotient Topology/Quotient Space",
"Definition:Open Set/Topology",
"Definition:Saturation (Equivalence Relation)",
"Definition:Open Set/Topology"
] | [
"Definition:Quotient Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] |
proofwiki-12817 | Compact Closure of Element is Principal Ideal on Compact Subset iff Element is Compact | Let $L = \struct {S, \vee, \preceq}$ be a bounded below algebraic join semilattice.
Let $P = \struct {\map K L, \precsim}$ be an ordered subset of $L$
where $\map K L$ denotes the compact subset of $L$.
Let $x \in S$.
Then $x^{\mathrm{compact} }$ is principal ideal in $P$ {{iff}} $x$ is a compact element. | === Sufficient Condition ===
Assume that
:$x^{\mathrm{compact} }$ is principal ideal in $P$.
By definitions of compact subset and compact closure:
:$x^{\mathrm{compact} } \subseteq \map K L$
By definition of principal ideal:
:$\exists y \in x^{\mathrm{compact} }: y$ is upper bound for $x^{\mathrm{compact} }$ in $P$.
By... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Join Semilattice|join semilattice]].
Let $P = \struct {\map K L, \precsim}$ be an [[Definition:Ordered Subset|ordered subset]] of $L$
where $\map K L$ denotes the [[Defi... | === Sufficient Condition ===
Assume that
:$x^{\mathrm{compact} }$ is [[Definition:Principal Ideal of Preordered Set|principal ideal]] in $P$.
By definitions of [[Definition:Compact Subset of Lattice|compact subset]] and [[Definition:Compact Closure|compact closure]]:
:$x^{\mathrm{compact} } \subseteq \map K L$
By de... | Compact Closure of Element is Principal Ideal on Compact Subset iff Element is Compact | https://proofwiki.org/wiki/Compact_Closure_of_Element_is_Principal_Ideal_on_Compact_Subset_iff_Element_is_Compact | https://proofwiki.org/wiki/Compact_Closure_of_Element_is_Principal_Ideal_on_Compact_Subset_iff_Element_is_Compact | [
"Principal Ideals of Preordered Sets",
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Join Semilattice",
"Definition:Ordered Subset",
"Definition:Compact Subset of Lattice",
"Definition:Principal Ideal of Preordered Set",
"Definition:Compact Element"
] | [
"Definition:Principal Ideal of Preordered Set",
"Definition:Compact Subset of Lattice",
"Definition:Compact Closure",
"Definition:Principal Ideal of Preordered Set",
"Definition:Upper Bound of Set",
"Definition:Ordered Subset",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set",
"Definiti... |
proofwiki-12818 | Multiple of 6 is Semiperfect | Let $n \in \Z_{>0}$ be a multiple of $6$.
Then $n$ is semiperfect. | Let $n = 6 k$.
Then:
:$n = 2 \times 3 k$
and so $3 k$ is a factor of $n$.
:$n = 3 \times 2 k$
and so $2 k$ is a factor of $n$.
:$n = 6 \times k$
and so $k$ is a factor of $n$.
But:
:$n = k + 2 k + 3 k$
and so is the sum of a subset of its factors
Hence the result by definition of semiperfect.
{{qed}}
Category:Semiperfe... | Let $n \in \Z_{>0}$ be a [[Definition:Multiple of Integer|multiple]] of $6$.
Then $n$ is [[Definition:Semiperfect Number|semiperfect]]. | Let $n = 6 k$.
Then:
:$n = 2 \times 3 k$
and so $3 k$ is a [[Definition:Divisor of Integer|factor]] of $n$.
:$n = 3 \times 2 k$
and so $2 k$ is a [[Definition:Divisor of Integer|factor]] of $n$.
:$n = 6 \times k$
and so $k$ is a [[Definition:Divisor of Integer|factor]] of $n$.
But:
:$n = k + 2 k + 3 k$
and so is th... | Multiple of 6 is Semiperfect | https://proofwiki.org/wiki/Multiple_of_6_is_Semiperfect | https://proofwiki.org/wiki/Multiple_of_6_is_Semiperfect | [
"Semiperfect Numbers",
"6"
] | [
"Definition:Multiple/Integer",
"Definition:Semiperfect Number"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Subset",
"Definition:Divisor (Algebra)/Integer",
"Definition:Semiperfect Number",
"Category:Semiperfect Numbers",
"Category:6"
] |
proofwiki-12819 | Open Projection and Closed Graph Implies Quotient is Hausdorff | Let $\RR \subseteq X \times X$ be an equivalence relation on a topological space $\struct {X, \tau}$.
Let $X / \RR$ be the quotient space.
Let $p$ denote the quotient mapping.
Let:
:$\RR$ be closed in $X \times X$
:$p$ be an open mapping.
Then $X / \RR$ is Hausdorff. | Let $\eqclass x \RR, \eqclass y \RR \in X / \RR$ such that $\eqclass x \RR \ne \eqclass y \RR$.
Then $\tuple {x, y} \notin \RR$.
Since $\RR$ is closed, $\paren {X \times X} \setminus \RR$ is open.
Let:
:$\BB = \set { U \times V : U, V \in \tau}$
By Natural Basis of Product Topology, $\BB$ is a basis for the product top... | Let $\RR \subseteq X \times X$ be an [[Definition:Equivalence Relation|equivalence relation]] on a [[Definition:Topological Space|topological space]] $\struct {X, \tau}$.
Let $X / \RR$ be the [[Definition:Quotient Space (Topology)|quotient space]].
Let $p$ denote the [[Definition:Quotient Mapping|quotient mapping]].
... | Let $\eqclass x \RR, \eqclass y \RR \in X / \RR$ such that $\eqclass x \RR \ne \eqclass y \RR$.
Then $\tuple {x, y} \notin \RR$.
Since $\RR$ is [[Definition:Closed Set (Topology)|closed]], $\paren {X \times X} \setminus \RR$ is [[Definition:Open Set (Topology)|open]].
Let:
:$\BB = \set { U \times V : U, V \in \tau}$... | Open Projection and Closed Graph Implies Quotient is Hausdorff | https://proofwiki.org/wiki/Open_Projection_and_Closed_Graph_Implies_Quotient_is_Hausdorff | https://proofwiki.org/wiki/Open_Projection_and_Closed_Graph_Implies_Quotient_is_Hausdorff | [
"Hausdorff Spaces",
"Quotient Spaces (Topology)"
] | [
"Definition:Equivalence Relation",
"Definition:Topological Space",
"Definition:Quotient Topology/Quotient Space",
"Definition:Quotient Mapping",
"Definition:Closed Set/Topology",
"Definition:Open Mapping",
"Definition:T2 Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Natural Basis of Product Topology",
"Definition:Basis (Topology)",
"Definition:Product Topology",
"Definition:Open Mapping",
"Definition:Open Set/Topology",
"Definition:Contradiction"
] |
proofwiki-12820 | Subgroup is Closed iff Quotient is Hausdorff | Let $G$ be a topological group.
Let $H \le G$ be a subgroup.
Let $G / H$ be their quotient.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$H$ is closed in $G$}}
{{item|(2):|$G / H$ is Hausdorff}}
{{end-itemize}} | {{ProofWanted|use Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open and Open Projection and Closed Graph Implies Quotient is Hausdorff}}
Category:Topological Groups
Category:Hausdorff Spaces
q2a4yzdbsj9cz236j6j2e36yywa9a9o | Let $G$ be a [[Definition:Topological Group|topological group]].
Let $H \le G$ be a [[Definition:Subgroup|subgroup]].
Let $G / H$ be their [[Definition:Quotient of Topological Groups|quotient]].
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$H$ is [[Definition:Closed Set (Topology)|closed]] in $G$}}
{{item|(2):|$G / H$ is... | {{ProofWanted|use [[Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open]] and [[Open Projection and Closed Graph Implies Quotient is Hausdorff]]}}
[[Category:Topological Groups]]
[[Category:Hausdorff Spaces]]
q2a4yzdbsj9cz236j6j2e36yywa9a9o | Subgroup is Closed iff Quotient is Hausdorff | https://proofwiki.org/wiki/Subgroup_is_Closed_iff_Quotient_is_Hausdorff | https://proofwiki.org/wiki/Subgroup_is_Closed_iff_Quotient_is_Hausdorff | [
"Topological Groups",
"Hausdorff Spaces"
] | [
"Definition:Topological Group",
"Definition:Subgroup",
"Definition:Quotient of Topological Groups",
"Definition:Closed Set/Topology",
"Definition:T2 Space"
] | [
"Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open",
"Open Projection and Closed Graph Implies Quotient is Hausdorff",
"Category:Topological Groups",
"Category:Hausdorff Spaces"
] |
proofwiki-12821 | Higher Homotopy Groups are Abelian | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $x_0 \in S$.
Let $n \ge 2$ be a integer.
Let $\pi_n \left({T, x_0}\right)$ be the $n$th homotopy group with base point $x_0$.
Then $\pi_n \left({T, x_0}\right)$ is abelian. | {{proof wanted}}
Category:Homotopy Theory
Category:Algebraic Topology
g4qe2zuesl24pqvto0uiq5uikf4igaz | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $x_0 \in S$.
Let $n \ge 2$ be a [[Definition:Integer|integer]].
Let $\pi_n \left({T, x_0}\right)$ be the $n$th [[Definition:Homotopy Group|homotopy group]] with base point $x_0$.
Then $\pi_n \left({T, x_0}\right)$ is [[De... | {{proof wanted}}
[[Category:Homotopy Theory]]
[[Category:Algebraic Topology]]
g4qe2zuesl24pqvto0uiq5uikf4igaz | Higher Homotopy Groups are Abelian | https://proofwiki.org/wiki/Higher_Homotopy_Groups_are_Abelian | https://proofwiki.org/wiki/Higher_Homotopy_Groups_are_Abelian | [
"Homotopy Theory",
"Algebraic Topology"
] | [
"Definition:Topological Space",
"Definition:Integer",
"Definition:Homotopy Group",
"Definition:Abelian Group"
] | [
"Category:Homotopy Theory",
"Category:Algebraic Topology"
] |
proofwiki-12822 | Multiple of Semiperfect Number is Semiperfect | Let $n \in \Z_{>0}$ be a semiperfect number.
Let $k \in \Z_{>0}$ be a (strictly) positive integer.
Then $k n$ is also a semiperfect number. | Let $P$ be a subset of the divisors of $n$ such that the sum of the elements of $P$ equals $n$.
Let $\sigma = \ds \sum_{p \mathop \in P} p$ be the sum of the elements of $P$.
Let:
:$Q = \set {k p: p \in P}$
be the set of elements of $P$ multiplied by $k$.
We have by definition that:
:$\forall p \in P: p \divides P$
whe... | Let $n \in \Z_{>0}$ be a [[Definition:Semiperfect Number|semiperfect number]].
Let $k \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then $k n$ is also a [[Definition:Semiperfect Number|semiperfect number]]. | Let $P$ be a [[Definition:Subset|subset]] of the [[Definition:Divisor of Integer|divisors]] of $n$ such that the [[Definition:Integer Addition|sum]] of the [[Definition:Element|elements]] of $P$ equals $n$.
Let $\sigma = \ds \sum_{p \mathop \in P} p$ be the [[Definition:Integer Addition|sum]] of the [[Definition:Eleme... | Multiple of Semiperfect Number is Semiperfect | https://proofwiki.org/wiki/Multiple_of_Semiperfect_Number_is_Semiperfect | https://proofwiki.org/wiki/Multiple_of_Semiperfect_Number_is_Semiperfect | [
"Semiperfect Numbers"
] | [
"Definition:Semiperfect Number",
"Definition:Strictly Positive/Integer",
"Definition:Semiperfect Number"
] | [
"Definition:Subset",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Element",
"Definition:Addition/Integers",
"Definition:Element",
"Definition:Set",
"Definition:Element",
"Definition:Divisor (Algebra)/Integer",
"Definition:Subset",
"Definition:Divisor (Algeb... |
proofwiki-12823 | Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset | Let $L = \struct {S, \preceq}$ be an algebric lattice.
Let $c: S \to S$ be a closure operator that preserves directed suprema.
Then:
:$c \sqbrk {\map K L} \subseteq \map K {\struct {c \sqbrk S, \precsim} }$
where:
:$\map K L$ denotes the compact subset of $L$
:$c \sqbrk S$ denotes the image of $S$ under $c$
:$\mathord ... | Let $x \in c \sqbrk {\map K L}$.
By definition of image of set:
:$\exists y \in \map K L: x = \map c y$
and
:$x \in c \sqbrk S$
By definition of compact subset:
:$y$ is compact in $L$.
By definition of compact element:
:$y \ll y$
where $\ll$ denotes the way below relation.
Define $P = \struct {c \sqbrk S, \precsim}$ as... | Let $L = \struct {S, \preceq}$ be an [[Definition:Algebraic Ordered Set|algebric]] [[Definition:Lattice (Order Theory)|lattice]].
Let $c: S \to S$ be a [[Definition:Closure Operator|closure operator]] that [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
Then:
:$c \sqbrk {\map K L} \sub... | Let $x \in c \sqbrk {\map K L}$.
By definition of [[Definition:Image of Subset under Mapping|image of set]]:
:$\exists y \in \map K L: x = \map c y$
and
:$x \in c \sqbrk S$
By definition of [[Definition:Compact Subset of Lattice|compact subset]]:
:$y$ is [[Definition:Compact Element|compact]] in $L$.
By definition o... | Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset | https://proofwiki.org/wiki/Image_of_Compact_Subset_under_Directed_Suprema_Preserving_Closure_Operator_is_Subset_of_Compact_Subset | https://proofwiki.org/wiki/Image_of_Compact_Subset_under_Directed_Suprema_Preserving_Closure_Operator_is_Subset_of_Compact_Subset | [
"Continuous Lattices",
"Way Below Relation"
] | [
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Closure Operator",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Compact Subset of Lattice",
"Definition:Image (Set Theory)/Mapping/Subset"
] | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Compact Subset of Lattice",
"Definition:Compact Element",
"Definition:Compact Element",
"Definition:Element is Way Below",
"Definition:Ordered Subset",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Ordered Subset",... |
proofwiki-12824 | Integers such that all Coprime and Less are Prime | The following positive integers have the property that all positive integers less than and coprime to it, excluding $1$, are prime:
:$1, 2, 3, 4, 6, 8, 12, 18, 24, 30$
{{OEIS|A048597}}
There are no other positive integers with this property. | Let $S_n$ denote the set of all positive integers less than and coprime to $n$, excluding $1$.
Let $\map P n$ denote the propositional function:
:All positive integers less than and coprime to $n$, excluding $1$, are prime.
We establish that $\map P n = \T$ for all the positive integers given:
{{begin-eqn}}
{{eqn | l =... | The following [[Definition:Positive Integer|positive integers]] have the property that all [[Definition:Positive Integer|positive integers]] less than and [[Definition:Coprime Integers|coprime]] to it, excluding $1$, are [[Definition:Prime Number|prime]]:
:$1, 2, 3, 4, 6, 8, 12, 18, 24, 30$
{{OEIS|A048597}}
There are ... | Let $S_n$ denote the [[Definition:Set|set]] of all [[Definition:Positive Integer|positive integers]] less than and [[Definition:Coprime Integers|coprime]] to $n$, excluding $1$.
Let $\map P n$ denote the [[Definition:Propositional Function|propositional function]]:
:All [[Definition:Positive Integer|positive integers]... | Integers such that all Coprime and Less are Prime | https://proofwiki.org/wiki/Integers_such_that_all_Coprime_and_Less_are_Prime | https://proofwiki.org/wiki/Integers_such_that_all_Coprime_and_Less_are_Prime | [
"Euler Phi Function"
] | [
"Definition:Positive/Integer",
"Definition:Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Prime Number",
"Definition:Positive/Integer"
] | [
"Definition:Set",
"Definition:Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Propositional Function",
"Definition:Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Prime Number",
"Definition:Positive/Integer",
"Definition:Prime Number",
"Definition:Prime Number",
"Defi... |
proofwiki-12825 | Pythagorean Triangles whose Area equal their Perimeter | There exist exactly $2$ Pythagorean triples which define a Pythagorean triangle whose area equals its perimeter:
:$(1): \quad \tuple {6, 8, 10}$, leading to an area and perimeter of $24$
:$(2): \quad \tuple {5, 12, 13}$, leading to an area and perimeter of $30$. | From Area of Right Triangle, the area $\AA$ is:
:$\AA = \dfrac {a b} 2$
where $a$ and $b$ are the legs.
$(1): \quad$ The area of the $\tuple {6, 8, 10}$ triangle is $\dfrac {6 \times 8} 2 = 24$.
Its perimeter equals $6 + 8 + 10 = 24$.
$(2): \quad$ The area of the $\tuple {5, 12, 13}$ triangle is $\dfrac {5 \times 12} 2... | There exist exactly $2$ [[Definition:Pythagorean Triple|Pythagorean triples]] which define a [[Definition:Pythagorean Triangle|Pythagorean triangle]] whose [[Definition:Area|area]] equals its [[Definition:Perimeter|perimeter]]:
:$(1): \quad \tuple {6, 8, 10}$, leading to an [[Definition:Area|area]] and [[Definition:Pe... | From [[Area of Right Triangle]], the [[Definition:Area|area]] $\AA$ is:
:$\AA = \dfrac {a b} 2$
where $a$ and $b$ are the [[Definition:Leg of Right Triangle|legs]].
$(1): \quad$ The [[Definition:Area|area]] of the $\tuple {6, 8, 10}$ [[Definition:Pythagorean Triangle|triangle]] is $\dfrac {6 \times 8} 2 = 24$.
Its [[... | Pythagorean Triangles whose Area equal their Perimeter | https://proofwiki.org/wiki/Pythagorean_Triangles_whose_Area_equal_their_Perimeter | https://proofwiki.org/wiki/Pythagorean_Triangles_whose_Area_equal_their_Perimeter | [
"Pythagorean Triangles",
"24",
"30"
] | [
"Definition:Pythagorean Triple",
"Definition:Pythagorean Triangle",
"Definition:Area",
"Definition:Perimeter",
"Definition:Area",
"Definition:Perimeter",
"Definition:Area",
"Definition:Perimeter"
] | [
"Area of Right Triangle",
"Definition:Area",
"Definition:Triangle (Geometry)/Right-Angled/Legs",
"Definition:Area",
"Definition:Pythagorean Triangle",
"Definition:Perimeter",
"Definition:Area",
"Definition:Pythagorean Triangle",
"Definition:Perimeter",
"Definition:Linear Measure/Length",
"Defini... |
proofwiki-12826 | Separable Extension is Contained in Galois Extension | Let $E/F$ be a separable finite field extension.
Then there exists a finite field extension $L/E$ such that $L/F$ is Galois. | {{ProofWanted}}
Category:Galois Theory
lh89mtvcdjlsiei1e74gdkpyl4lil3k | Let $E/F$ be a [[Definition:Separable Extension|separable]] [[Definition:Finite Field Extension|finite field extension]].
Then there exists a [[Definition:Finite Field Extension|finite field extension]] $L/E$ such that $L/F$ is [[Definition:Galois Extension|Galois]]. | {{ProofWanted}}
[[Category:Galois Theory]]
lh89mtvcdjlsiei1e74gdkpyl4lil3k | Separable Extension is Contained in Galois Extension | https://proofwiki.org/wiki/Separable_Extension_is_Contained_in_Galois_Extension | https://proofwiki.org/wiki/Separable_Extension_is_Contained_in_Galois_Extension | [
"Galois Theory"
] | [
"Definition:Separable Extension",
"Definition:Field Extension/Degree/Finite",
"Definition:Field Extension/Degree/Finite",
"Definition:Galois Extension"
] | [
"Category:Galois Theory"
] |
proofwiki-12827 | Galois Extension is Galois over Intermediate Field | Let $L / F$ be a Galois Extension.
Let $E$ be an intermediate field.
Then $L / E$ is Galois. | {{ProofWanted}}
Category:Galois Theory
s3zuqy3xgnauel2aduazzo1ksnwjq25 | Let $L / F$ be a [[Definition:Galois Extension|Galois Extension]].
Let $E$ be an [[Definition:Intermediate Field|intermediate field]].
Then $L / E$ is [[Definition:Galois Extension|Galois]]. | {{ProofWanted}}
[[Category:Galois Theory]]
s3zuqy3xgnauel2aduazzo1ksnwjq25 | Galois Extension is Galois over Intermediate Field | https://proofwiki.org/wiki/Galois_Extension_is_Galois_over_Intermediate_Field | https://proofwiki.org/wiki/Galois_Extension_is_Galois_over_Intermediate_Field | [
"Galois Theory"
] | [
"Definition:Galois Extension",
"Definition:Intermediate Field",
"Definition:Galois Extension"
] | [
"Category:Galois Theory"
] |
proofwiki-12828 | Frobenius Endomorphism on Field is Injective | Let $p$ be a prime number.
Let $F$ be a field of characteristic $p$.
Then the Frobenius endomorphism $\Frob: F \to F$ is injective. | We have:
:$\map \Frob 1 = 1$
By Ring Homomorphism from Field is Monomorphism or Zero Homomorphism, $\Frob$ is injective.
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $F$ be a [[Definition:Field (Abstract Algebra)|field]] of [[Definition:Characteristic of Field|characteristic]] $p$.
Then the [[Definition:Frobenius Endomorphism|Frobenius endomorphism]] $\Frob: F \to F$ is [[Definition:Injection|injective]]. | We have:
:$\map \Frob 1 = 1$
By [[Ring Homomorphism from Field is Monomorphism or Zero Homomorphism]], $\Frob$ is [[Definition:Injection|injective]].
{{qed}} | Frobenius Endomorphism on Field is Injective | https://proofwiki.org/wiki/Frobenius_Endomorphism_on_Field_is_Injective | https://proofwiki.org/wiki/Frobenius_Endomorphism_on_Field_is_Injective | [
"Frobenius Endomorphisms"
] | [
"Definition:Prime Number",
"Definition:Field (Abstract Algebra)",
"Definition:Characteristic of Field",
"Definition:Frobenius Endomorphism",
"Definition:Injection"
] | [
"Ring Homomorphism from Field is Monomorphism or Zero Homomorphism",
"Definition:Injection"
] |
proofwiki-12829 | Galois Field is Perfect | Let $\GF$ be a Galois field.
Then $\GF$ is perfect. | By Characteristic of Galois Field is Prime, $\Char \GF$ is a prime number, say $p$.
By Frobenius Endomorphism on Field is Injective, $\Frob$ is injective.
By Injection from Finite Set to Itself is Surjection, $\Frob$ is bijective.
By Bijective Ring Homomorphism is Isomorphism, $\Frob$ is an automorphism.
{{qed}}
Catego... | Let $\GF$ be a [[Definition:Galois Field|Galois field]].
Then $\GF$ is [[Definition:Perfect Field|perfect]]. | By [[Characteristic of Galois Field is Prime]], $\Char \GF$ is a [[Definition:Prime Number|prime number]], say $p$.
By [[Frobenius Endomorphism on Field is Injective]], $\Frob$ is [[Definition:Injection|injective]].
By [[Injection from Finite Set to Itself is Surjection]], $\Frob$ is [[Definition:Bijection|bijective]... | Galois Field is Perfect | https://proofwiki.org/wiki/Galois_Field_is_Perfect | https://proofwiki.org/wiki/Galois_Field_is_Perfect | [
"Galois Fields",
"Perfect Fields"
] | [
"Definition:Galois Field",
"Definition:Perfect Field"
] | [
"Characteristic of Galois Field is Prime",
"Definition:Prime Number",
"Frobenius Endomorphism on Field is Injective",
"Definition:Injection",
"Injection from Finite Set to Itself is Surjection",
"Definition:Bijection",
"Bijective Ring Homomorphism is Isomorphism",
"Definition:Field Automorphism",
"C... |
proofwiki-12830 | Algebraically Closed Field is Perfect | Let $F$ be an algebraically closed field.
Then $F$ is perfect. | Let $E / F$ be any algebraic extension.
Since $F$ is an algebraically closed field, $E = F$.
By Field is Separable over itself, $E$ is separable over $F$.
Hence $F$ is perfect.
{{qed}}
Category:Field Extensions
Category:Perfect Fields
oe9rzy5e66rnq45gbo0dhdf23qfa3hz | Let $F$ be an [[Definition:Algebraically Closed Field|algebraically closed field]].
Then $F$ is [[Definition:Perfect Field|perfect]]. | Let $E / F$ be any [[Definition:Algebraic Extension|algebraic extension]].
Since $F$ is an [[Definition:Algebraically Closed Field|algebraically closed field]], $E = F$.
By [[Field is Separable over itself]], $E$ is [[Definition:Separable Extension|separable]] over $F$.
Hence $F$ is [[Definition:Perfect Field|perfec... | Algebraically Closed Field is Perfect | https://proofwiki.org/wiki/Algebraically_Closed_Field_is_Perfect | https://proofwiki.org/wiki/Algebraically_Closed_Field_is_Perfect | [
"Field Extensions",
"Perfect Fields"
] | [
"Definition:Algebraically Closed Field",
"Definition:Perfect Field"
] | [
"Definition:Algebraic Extension",
"Definition:Algebraically Closed Field",
"Field is Separable over itself",
"Definition:Separable Extension",
"Definition:Perfect Field",
"Category:Field Extensions",
"Category:Perfect Fields"
] |
proofwiki-12831 | Area of Smallest Rectangle accommodating Re-Entrant Knight's Tour | The area of the smallest rectangular chessboard on which a re-entrant knight's tour is possible is $30$ squares.
This can be configured either as a $5 \times 6$ chessboard or a $3 \times 10$ chessboard. | {{ProofWanted|Haven't even started the definitions yet for chess problems}} | The [[Definition:Area|area]] of the smallest [[Definition:Rectangle|rectangular]] [[Definition:Chessboard|chessboard]] on which a [[Definition:Re-Entrant Knight's Tour|re-entrant knight's tour]] is possible is $30$ squares.
This can be configured either as a $5 \times 6$ [[Definition:Chessboard|chessboard]] or a $3 \... | {{ProofWanted|Haven't even started the definitions yet for chess problems}} | Area of Smallest Rectangle accommodating Re-Entrant Knight's Tour | https://proofwiki.org/wiki/Area_of_Smallest_Rectangle_accommodating_Re-Entrant_Knight's_Tour | https://proofwiki.org/wiki/Area_of_Smallest_Rectangle_accommodating_Re-Entrant_Knight's_Tour | [
"30",
"Knight's Tours"
] | [
"Definition:Area",
"Definition:Quadrilateral/Rectangle",
"Definition:Chess/Chessboard",
"Definition:Re-Entrant Knight's Tour",
"Definition:Chess/Chessboard",
"Definition:Chess/Chessboard"
] | [] |
proofwiki-12832 | Area of Smallest Square accommodating Re-Entrant Knight's Tour | The area of the smallest square chessboard on which a re-entrant knight's tour is possible is $6 \times 6 = 36$ squares. | {{ProofWanted|Haven't even started the definitions yet for chess problems}} | The [[Definition:Area|area]] of the smallest [[Definition:Square (Geometry)|square]] [[Definition:Chessboard|chessboard]] on which a [[Definition:Re-Entrant Knight's Tour|re-entrant knight's tour]] is possible is $6 \times 6 = 36$ squares. | {{ProofWanted|Haven't even started the definitions yet for chess problems}} | Area of Smallest Square accommodating Re-Entrant Knight's Tour | https://proofwiki.org/wiki/Area_of_Smallest_Square_accommodating_Re-Entrant_Knight's_Tour | https://proofwiki.org/wiki/Area_of_Smallest_Square_accommodating_Re-Entrant_Knight's_Tour | [
"Recreational Mathematics",
"6",
"36",
"Knight's Tours"
] | [
"Definition:Area",
"Definition:Quadrilateral/Square",
"Definition:Chess/Chessboard",
"Definition:Re-Entrant Knight's Tour"
] | [] |
proofwiki-12833 | Subgroup of Index Least Prime Divisor is Normal | Let $G$ be a finite group of order $n>1$.
Let $p$ be the least prime divisor of $n$.
Let $H$ be a subgroup of $G$ such that:
:$\index G H = p$
where $\index G H$ denotes the index of $H$ in $G$.
Then $H$ is normal. | Let $G / H$ denote the left coset space of $G$ modulo $H$.
Consider the group action $G \curvearrowright G / H$, defined as:
:$\forall g \in G, \forall x H \in G / H: g * \paren {x H} := \paren {g x} H$
{{explain|Expand group action as defined on {{ProofWiki}} to explain the notation $G \curvearrowright G / H$}}
Let $\... | Let $G$ be a [[Definition:Finite Group|finite group]] of [[Definition:Order of Group|order]] $n>1$.
Let $p$ be the least [[Definition:Prime Divisor|prime divisor]] of $n$.
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$ such that:
:$\index G H = p$
where $\index G H$ denotes the [[Definition:Index of Subgroup|in... | Let $G / H$ denote the [[Definition:Left Coset Space|left coset space of $G$ modulo $H$]].
Consider the [[Definition:Group Action on Coset Space|group action]] $G \curvearrowright G / H$, defined as:
:$\forall g \in G, \forall x H \in G / H: g * \paren {x H} := \paren {g x} H$
{{explain|Expand [[Definition:Group Acti... | Subgroup of Index Least Prime Divisor is Normal | https://proofwiki.org/wiki/Subgroup_of_Index_Least_Prime_Divisor_is_Normal | https://proofwiki.org/wiki/Subgroup_of_Index_Least_Prime_Divisor_is_Normal | [
"Finite Groups"
] | [
"Definition:Finite Group",
"Definition:Order of Structure",
"Definition:Prime Factor",
"Definition:Subgroup",
"Definition:Index of Subgroup",
"Definition:Normal Subgroup"
] | [
"Definition:Coset Space/Left Coset Space",
"Definition:Group Action on Coset Space",
"Definition:Group Action",
"Definition:Kernel of Group Action",
"Definition:Kernel of Group Action",
"Kernel of Group Action is Normal Subgroup",
"Definition:Normal Subgroup",
"Stabilizer of Coset Action on Set of Sub... |
proofwiki-12834 | Alternating Group is Simple except on 4 Letters | Let $n$ be an integer such that $n \ne 4$.
Then the $n$th alternating group $A_n$ is simple. | Recall that a group is simple if its normal subgroups are itself and the trivial subgroup.
Let $n < 4$.
$A_2$ is the trivial group and hence simple.
$A_3$ is the cyclic group of order $3$, hence a prime group.
By Prime Group is Simple, $A_3$ is simple.
{{qed|lemma}}
We note that $A_4$ is a special case.
From Normality ... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ne 4$.
Then the $n$th [[Definition:Alternating Group|alternating group]] $A_n$ is [[Definition:Simple Group|simple]]. | Recall that a [[Definition:Group|group]] is [[Definition:Simple Group|simple]] if its [[Definition:Normal Subgroup|normal subgroups]] are itself and the [[Definition:Trivial Subgroup|trivial subgroup]].
Let $n < 4$.
$A_2$ is the [[Definition:Trivial Group|trivial group]] and hence [[Definition:Simple Group|simple]].
... | Alternating Group is Simple except on 4 Letters | https://proofwiki.org/wiki/Alternating_Group_is_Simple_except_on_4_Letters | https://proofwiki.org/wiki/Alternating_Group_is_Simple_except_on_4_Letters | [
"Alternating Group is Simple except on 4 Letters",
"Alternating Groups",
"Simple Groups",
"4"
] | [
"Definition:Integer",
"Definition:Alternating Group",
"Definition:Simple Group"
] | [
"Definition:Group",
"Definition:Simple Group",
"Definition:Normal Subgroup",
"Definition:Trivial Subgroup",
"Definition:Trivial Group",
"Definition:Simple Group",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Prime Group",
"Prime Group is Simple",
"Definition:Simple Gro... |
proofwiki-12835 | Universal Property of Quotient Ring | Let $R, S$ be commutative rings.
Let $I \trianglelefteq R$ be an ideal of $R$.
Let $\pi : R \to R / I$ be the quotient epimorphism.
Let $f: R \to S$ be a ring homomorphism with $\map f I = \set 0$.
Then there exists a unique ring homomorphism $\overline f: R / I \to S$ such that $f = \overline f \circ \pi$.
:<nowiki>$\... | {{MissingLinks|justification for the statements and steps made in the chain of reasoning}}
Define $\overline f: R / I \to S$ by:
:$\forall r \in R: \map {\overline f} {r + I} = \map f r$
Since $f$ is a ring homomorphism, $f$ is well-defined.
Suppose for some $r_1, r_2 \in R$ that:
:$r_1 + I = r_2 + I$
Since $I$ is an... | Let $R, S$ be [[Definition:Commutative Ring|commutative rings]].
Let $I \trianglelefteq R$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
Let $\pi : R \to R / I$ be the [[Definition:Quotient Epimorphism|quotient epimorphism]].
Let $f: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]] with $\map f I ... | {{MissingLinks|justification for the statements and steps made in the chain of reasoning}}
Define $\overline f: R / I \to S$ by:
:$\forall r \in R: \map {\overline f} {r + I} = \map f r$
Since $f$ is a [[Definition:Ring Homomorphism|ring homomorphism]], $f$ is [[Definition:Well-Defined|well-defined]].
Suppose for ... | Universal Property of Quotient Ring | https://proofwiki.org/wiki/Universal_Property_of_Quotient_Ring | https://proofwiki.org/wiki/Universal_Property_of_Quotient_Ring | [
"Quotient Rings",
"Universal Properties"
] | [
"Definition:Commutative Ring",
"Definition:Ideal of Ring",
"Definition:Quotient Epimorphism",
"Definition:Ring Homomorphism",
"Definition:Unique",
"Definition:Ring Homomorphism"
] | [
"Definition:Ring Homomorphism",
"Definition:Well-Defined",
"Definition:Ideal of Ring",
"Definition:Ring Zero",
"Definition:Well-Defined",
"Quotient Ring is Ring/Quotient Ring Addition is Well-Defined",
"Definition:Ring Homomorphism",
"Definition:Unique",
"Definition:Ring Homomorphism",
"Definition... |
proofwiki-12836 | Universal Property of Quotient Space | Let $X$ and $Y$ be topological spaces.
Let $\sim$ be an equivalence relation on $X$.
Let $\pi : X \to X / {\sim}$ be the quotient mapping.
Let $f : X \to Y$ be continuous and $\sim$-invariant.
Then there exists a unique continuous map $\overline f : X / {\sim} \to Y$ such that $f = \overline f \circ \pi$. | {{proofread}} | Let $X$ and $Y$ be [[Definition:Topological Space|topological spaces]].
Let $\sim$ be an [[Definition:Equivalence Relation|equivalence relation]] on $X$.
Let $\pi : X \to X / {\sim}$ be the [[Definition:Quotient Mapping|quotient mapping]].
Let $f : X \to Y$ be [[Definition:Continuous Mapping (Topology)|continuous]] ... | {{proofread}} | Universal Property of Quotient Space | https://proofwiki.org/wiki/Universal_Property_of_Quotient_Space | https://proofwiki.org/wiki/Universal_Property_of_Quotient_Space | [
"Quotient Spaces (Topology)",
"Universal Properties"
] | [
"Definition:Topological Space",
"Definition:Equivalence Relation",
"Definition:Quotient Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Invariant Mapping Under Equivalence Relation",
"Definition:Continuous Mapping (Topology)"
] | [] |
proofwiki-12837 | Universal Property of Quotient Group | Let $G$ and $H$ be groups.
Let $N \trianglelefteq G$ be an normal subgroup.
Let $\pi: G \to G / N$ be the quotient epimorphism.
Let $f: G \to H$ be a group homomorphism with $N \subseteq \ker f$.
Then there exists a unique group homomorphism $\overline f: G / N \to H$ such that $f = \overline f \circ \pi$.
$\xymatrix{
... | === Existence ===
{{proofread}}
Let $\sim$ denote (left) congruence modulo $N$.
From Congruence Modulo Subgroup is Equivalence Relation, $\sim$ is an equivalence relation on $X$.
For all $g \in G$, let $\eqclass g \sim$ denote the equivalence class of $g$ under $\sim$.
Note that Group Homomorphism is Invariant under Co... | Let $G$ and $H$ be [[Definition:Group|groups]].
Let $N \trianglelefteq G$ be an [[Definition:Normal Subgroup|normal subgroup]].
Let $\pi: G \to G / N$ be the [[Definition:Quotient Epimorphism|quotient epimorphism]].
Let $f: G \to H$ be a [[Definition:Group Homomorphism|group homomorphism]] with $N \subseteq \ker f$.... | === Existence ===
{{proofread}}
Let $\sim$ denote [[Definition:Left Congruence Modulo Subgroup|(left) congruence modulo $N$]].
From [[Congruence Modulo Subgroup is Equivalence Relation]], $\sim$ is an [[Definition:Equivalence Relation|equivalence relation]] on $X$.
For all $g \in G$, let $\eqclass g \sim$ denote th... | Universal Property of Quotient Group | https://proofwiki.org/wiki/Universal_Property_of_Quotient_Group | https://proofwiki.org/wiki/Universal_Property_of_Quotient_Group | [
"Quotient Groups",
"Universal Properties"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Quotient Epimorphism",
"Definition:Group Homomorphism",
"Definition:Group Homomorphism"
] | [
"Definition:Congruence Modulo Subgroup/Left Congruence",
"Congruence Modulo Subgroup is Equivalence Relation",
"Definition:Equivalence Relation",
"Definition:Equivalence Class",
"Group Homomorphism is Invariant under Congruence Modulo Kernel",
"Definition:Invariant Mapping Under Equivalence Relation",
"... |
proofwiki-12838 | Arithmetic Sequence of 4 Terms with 3 Distinct Prime Factors | The arithmetic sequence:
:$30, 66, 102, 138$
is the smallest of $4$ terms which consists entirely of positive integers each with $3$ distinct prime factors. | We demonstrate that this is indeed an arithmetic sequence:
{{begin-eqn}}
{{eqn | l = 66 - 30
| r = 36
}}
{{eqn | l = 102 - 66
| r = 36
}}
{{eqn | l = 138 - 102
| r = 36
}}
{{end-eqn}}
demonstrating the common difference of $36$.
Then we note:
{{begin-eqn}}
{{eqn | l = 30
| r = 2 \times 3 \times ... | The [[Definition:Arithmetic Sequence|arithmetic sequence]]:
:$30, 66, 102, 138$
is the smallest of $4$ terms which consists entirely of [[Definition:Positive Integer|positive integers]] each with $3$ [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]]. | We demonstrate that this is indeed an [[Definition:Arithmetic Sequence|arithmetic sequence]]:
{{begin-eqn}}
{{eqn | l = 66 - 30
| r = 36
}}
{{eqn | l = 102 - 66
| r = 36
}}
{{eqn | l = 138 - 102
| r = 36
}}
{{end-eqn}}
demonstrating the [[Definition:Common Difference|common difference]] of $36$.
T... | Arithmetic Sequence of 4 Terms with 3 Distinct Prime Factors | https://proofwiki.org/wiki/Arithmetic_Sequence_of_4_Terms_with_3_Distinct_Prime_Factors | https://proofwiki.org/wiki/Arithmetic_Sequence_of_4_Terms_with_3_Distinct_Prime_Factors | [
"Arithmetic Sequences"
] | [
"Definition:Arithmetic Sequence",
"Definition:Positive/Integer",
"Definition:Distinct",
"Definition:Prime Factor"
] | [
"Definition:Arithmetic Sequence",
"Definition:Arithmetic Sequence/Common Difference",
"Definition:Integer Sequence",
"Definition:Length of Sequence",
"Definition:Term of Sequence",
"Definition:Arithmetic Sequence",
"Definition:Addition/Integers",
"Definition:Smallest Element",
"Definition:Integer Se... |
proofwiki-12839 | Sum of Successive Powers in 2 ways | $31$ and $8191$ can be expressed as the sum of successive powers starting from $1$ in in $2$ different ways. | {{begin-eqn}}
{{eqn | l = 31
| r = 1 + 5 + 5^2
| c =
}}
{{eqn | r = 1 + 2 + 2^2 + 2^3 + 2^4
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 8191
| r = 1 + 90 + 90^2
| c =
}}
{{eqn | r = 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10}+ 2^{11}+ 2^{12}
| c =
}}
{{end-e... | $31$ and $8191$ can be expressed as the [[Definition:Integer Addition|sum]] of successive [[Definition:Integer Power|powers]] starting from $1$ in in $2$ different ways. | {{begin-eqn}}
{{eqn | l = 31
| r = 1 + 5 + 5^2
| c =
}}
{{eqn | r = 1 + 2 + 2^2 + 2^3 + 2^4
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 8191
| r = 1 + 90 + 90^2
| c =
}}
{{eqn | r = 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10}+ 2^{11}+ 2^{12}
| c =
}}
{{end... | Sum of Successive Powers in 2 ways | https://proofwiki.org/wiki/Sum_of_Successive_Powers_in_2_ways | https://proofwiki.org/wiki/Sum_of_Successive_Powers_in_2_ways | [
"Sums of Sequences",
"Powers",
"31",
"8191"
] | [
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer"
] | [] |
proofwiki-12840 | 31 is Smallest Prime whose Reciprocal has Odd Period | $31$ is the smallest prime number to have a decimal expansion of the reciprocal with an odd period greater than $1$:
:$\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$ | From Reciprocal of $31$:
{{:Reciprocal of 31}}
Counting the digits, it is seen that this has a period of recurrence of $15$, an odd integer.
The prime numbers less than $31$ are $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$.
We investigate the reciprocal of each of these:
{{begin-eqn}}
{{eqn | l = \dfrac 1 2
... | $31$ is the smallest [[Definition:Prime Number|prime number]] to have a [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] with an [[Definition:Odd Integer|odd]] [[Definition:Period of Recurrence|period]] greater than $1$:
:$\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 16... | From [[Reciprocal of 31|Reciprocal of $31$]]:
{{:Reciprocal of 31}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $15$, an [[Definition:Odd Integer|odd integer]].
The [[Definition:Prime Number|prime numbers]] less than $31$ are $2$, $3$, $5$, $7$, $11$, $1... | 31 is Smallest Prime whose Reciprocal has Odd Period | https://proofwiki.org/wiki/31_is_Smallest_Prime_whose_Reciprocal_has_Odd_Period | https://proofwiki.org/wiki/31_is_Smallest_Prime_whose_Reciprocal_has_Odd_Period | [
"31",
"Examples of Reciprocals"
] | [
"Definition:Prime Number",
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Odd Integer",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Reciprocal of 31",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Odd Integer",
"Definition:Prime Number",
"Definition:Reciprocal",
"Reciprocal of 2",
"Reciprocal of 3",
"Definition:Basis Expansion/Recurrence/Period",
"Reciprocal of 5",
"Reciprocal of 7",
"Definition:Basis Expansio... |
proofwiki-12841 | Properties of Periodic Part of Reciprocal of 31 | We have from Reciprocal of $31$ that the decimal expansion of the reciprocal of $31$ is:
{{:Reciprocal of 31}}
Then:
{{begin-eqn}}
{{eqn | l = 032258 \times 2
| r = 64 \, 516
}}
{{eqn | l = 032258 \times 4
| r = 129 \, 032
}}
{{eqn | l = 032258 \times 5
| r = 161 \, 290
}}
{{eqn | l = 032258 \times 7
... | Verified by calculation. | We have from [[Reciprocal of 31|Reciprocal of $31$]] that the [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $31$ is:
{{:Reciprocal of 31}}
Then:
{{begin-eqn}}
{{eqn | l = 032258 \times 2
| r = 64 \, 516
}}
{{eqn | l = 032258 \times 4
| r = 129 \, 032
}}
{... | Verified by calculation. | Properties of Periodic Part of Reciprocal of 31 | https://proofwiki.org/wiki/Properties_of_Periodic_Part_of_Reciprocal_of_31 | https://proofwiki.org/wiki/Properties_of_Periodic_Part_of_Reciprocal_of_31 | [
"31",
"Examples of Reciprocals"
] | [
"Reciprocal of 31",
"Definition:Decimal Expansion",
"Definition:Reciprocal"
] | [] |
proofwiki-12842 | Smallest Adjacent Happy Numbers | The smallest adjacent happy numbers are $31$ and $32$. | This can be determined by testing all the positive integers in succession for happiness.
Checking $31$ and $32$:
{{begin-eqn}}
{{eqn | l = 31
| o = \to
| r = 3^2 + 1^2 = 9 + 1 = 10
| c =
}}
{{eqn | o = \to
| r = 1^2 = 1
| c = and so $31$ is happy.
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = ... | The smallest adjacent [[Definition:Happy Number|happy numbers]] are $31$ and $32$. | This can be determined by testing all the [[Definition:Positive Integer|positive integers]] in succession for [[Definition:Happy Number|happiness]].
Checking $31$ and $32$:
{{begin-eqn}}
{{eqn | l = 31
| o = \to
| r = 3^2 + 1^2 = 9 + 1 = 10
| c =
}}
{{eqn | o = \to
| r = 1^2 = 1
| c = ... | Smallest Adjacent Happy Numbers | https://proofwiki.org/wiki/Smallest_Adjacent_Happy_Numbers | https://proofwiki.org/wiki/Smallest_Adjacent_Happy_Numbers | [
"Happy Numbers"
] | [
"Definition:Happy Number"
] | [
"Definition:Positive/Integer",
"Definition:Happy Number",
"Definition:Happy Number",
"Definition:Happy Number"
] |
proofwiki-12843 | Increasing Mapping Preserves Lower Bounds | Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be ordered sets.
Let $f:S \to S'$ be an increasing mapping.
Let $x \in S$, $X \subseteq S$ such that
:$x$ is lower bound for $X$.
Then $f \left({x}\right)$ is lower bound for $f \left[{X}\right]$. | Let $y \in f\left[{X}\right]$.
By definition of image of set:
:$\exists z \in X: y = f \left({z}\right)$
By definition of lower bound:
:$x \preceq z$
Thus by definition of increasing mapping:
:$f \left({x}\right) \preceq' y$
{{qed}} | Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be [[Definition:Ordered Set|ordered sets]].
Let $f:S \to S'$ be an [[Definition:Increasing Mapping|increasing mapping]].
Let $x \in S$, $X \subseteq S$ such that
:$x$ is [[Definition:Lower Bound of Set|lower bound]] for $X$.
Then $f \left({x}\r... | Let $y \in f\left[{X}\right]$.
By definition of [[Definition:Image of Subset under Mapping|image of set]]:
:$\exists z \in X: y = f \left({z}\right)$
By definition of [[Definition:Lower Bound of Set|lower bound]]:
:$x \preceq z$
Thus by definition of [[Definition:Increasing Mapping|increasing mapping]]:
:$f \left({x... | Increasing Mapping Preserves Lower Bounds | https://proofwiki.org/wiki/Increasing_Mapping_Preserves_Lower_Bounds | https://proofwiki.org/wiki/Increasing_Mapping_Preserves_Lower_Bounds | [
"Increasing Mappings"
] | [
"Definition:Ordered Set",
"Definition:Increasing/Mapping",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set"
] | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Lower Bound of Set",
"Definition:Increasing/Mapping"
] |
proofwiki-12844 | Order Isomorphism Preserves Lower Bounds | Let $L = \struct {S, \preceq}$, $L' = \struct {S', \preceq'}$ be ordered sets.
Let $f: S \to S'$ be an order isomorphism between $L$ and $L'$.
Let $x \in S$, $X \subseteq S$.
Then $x$ is lower bound for $X$ {{iff}} $\map f x$ is lower bound for $f \sqbrk X$. | By definition of order isomorphism:
:$f$ is an order embedding. | Let $L = \struct {S, \preceq}$, $L' = \struct {S', \preceq'}$ be [[Definition:Ordered Set|ordered sets]].
Let $f: S \to S'$ be an [[Definition:Order Isomorphism|order isomorphism]] between $L$ and $L'$.
Let $x \in S$, $X \subseteq S$.
Then $x$ is [[Definition:Lower Bound of Set|lower bound]] for $X$ {{iff}} $\map f... | By definition of [[Definition:Order Isomorphism|order isomorphism]]:
:$f$ is an [[Definition:Order Embedding|order embedding]]. | Order Isomorphism Preserves Lower Bounds | https://proofwiki.org/wiki/Order_Isomorphism_Preserves_Lower_Bounds | https://proofwiki.org/wiki/Order_Isomorphism_Preserves_Lower_Bounds | [
"Order Isomorphisms"
] | [
"Definition:Ordered Set",
"Definition:Order Isomorphism",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set"
] | [
"Definition:Order Isomorphism",
"Definition:Order Embedding",
"Definition:Order Embedding"
] |
proofwiki-12845 | Order Embedding is Increasing Mapping | Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be ordered sets.
Let $f:S_1 \to S_2$ be an order embedding.
Then $f$ is increasing mapping. | By definition of order embedding:
:$\forall x, y \in S_1: x \preceq_1 y \implies f\left({x}\right) \preceq_2 f\left({y}\right)$
Hence $f$ is an increasing mapping.
{{qed}}
Category:Order Embeddings
Category:Increasing Mappings
2dys8l0sul24o8x0jrg12yha84gkk4f | Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be [[Definition:Ordered Set|ordered sets]].
Let $f:S_1 \to S_2$ be an [[Definition:Order Embedding|order embedding]].
Then $f$ is [[Definition:Increasing Mapping|increasing mapping]]. | By definition of [[Definition:Order Embedding|order embedding]]:
:$\forall x, y \in S_1: x \preceq_1 y \implies f\left({x}\right) \preceq_2 f\left({y}\right)$
Hence $f$ is an [[Definition:Increasing Mapping|increasing mapping]].
{{qed}}
[[Category:Order Embeddings]]
[[Category:Increasing Mappings]]
2dys8l0sul24o8x0jr... | Order Embedding is Increasing Mapping | https://proofwiki.org/wiki/Order_Embedding_is_Increasing_Mapping | https://proofwiki.org/wiki/Order_Embedding_is_Increasing_Mapping | [
"Order Embeddings",
"Increasing Mappings"
] | [
"Definition:Ordered Set",
"Definition:Order Embedding",
"Definition:Increasing/Mapping"
] | [
"Definition:Order Embedding",
"Definition:Increasing/Mapping",
"Category:Order Embeddings",
"Category:Increasing Mappings"
] |
proofwiki-12846 | Order Isomorphism Preserves Upper Bounds | Let $L = \struct {S, \preceq}$, $L' = \struct {S', \preceq'}$ be ordered sets.
Let $f: S \to S'$ be an order isomorphism between $L$ and $L'$.
Let $x \in S$, $X \subseteq S$.
Then:
:$x$ is an upper bound for $X$
{{iff}}:
:$\map f x$ is an upper bound for $f \sqbrk X$. | By definition of order isomorphism:
:$f$ is an order embedding. | Let $L = \struct {S, \preceq}$, $L' = \struct {S', \preceq'}$ be [[Definition:Ordered Set|ordered sets]].
Let $f: S \to S'$ be an [[Definition:Order Isomorphism|order isomorphism]] between $L$ and $L'$.
Let $x \in S$, $X \subseteq S$.
Then:
:$x$ is an [[Definition:Upper Bound of Set|upper bound]] for $X$
{{iff}}:
:... | By definition of [[Definition:Order Isomorphism|order isomorphism]]:
:$f$ is an [[Definition:Order Embedding|order embedding]]. | Order Isomorphism Preserves Upper Bounds | https://proofwiki.org/wiki/Order_Isomorphism_Preserves_Upper_Bounds | https://proofwiki.org/wiki/Order_Isomorphism_Preserves_Upper_Bounds | [
"Order Isomorphisms"
] | [
"Definition:Ordered Set",
"Definition:Order Isomorphism",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set"
] | [
"Definition:Order Isomorphism",
"Definition:Order Embedding",
"Definition:Order Embedding"
] |
proofwiki-12847 | Increasing Mapping Preserves Upper Bounds | Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be ordered sets.
Let $f:S \to S'$ be an increasing mapping.
Let $x \in S$, $X \subseteq S$ such that
:$x$ is upper bound for $X$.
Then $f\left({x}\right)$ is upper bound for $f\left[{X}\right]$. | Let $y \in f\left[{X}\right]$.
By definition of image of set:
:$\exists z \in X: y = f\left({z}\right)$
By definition of upper bound:
:$z \preceq x$
Thus by definition of increasing mapping:
:$y \preceq' f\left({x}\right)$
{{qed}} | Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be [[Definition:Ordered Set|ordered sets]].
Let $f:S \to S'$ be an [[Definition:Increasing Mapping|increasing mapping]].
Let $x \in S$, $X \subseteq S$ such that
:$x$ is [[Definition:Upper Bound of Set|upper bound]] for $X$.
Then $f\left({x}\ri... | Let $y \in f\left[{X}\right]$.
By definition of [[Definition:Image of Subset under Mapping|image of set]]:
:$\exists z \in X: y = f\left({z}\right)$
By definition of [[Definition:Upper Bound of Set|upper bound]]:
:$z \preceq x$
Thus by definition of [[Definition:Increasing Mapping|increasing mapping]]:
:$y \preceq' ... | Increasing Mapping Preserves Upper Bounds | https://proofwiki.org/wiki/Increasing_Mapping_Preserves_Upper_Bounds | https://proofwiki.org/wiki/Increasing_Mapping_Preserves_Upper_Bounds | [
"Increasing Mappings"
] | [
"Definition:Ordered Set",
"Definition:Increasing/Mapping",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set"
] | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Upper Bound of Set",
"Definition:Increasing/Mapping"
] |
proofwiki-12848 | Order Isomorphism Preserves Infima and Suprema | Let $L = \struct {S, \preceq}$ and $L' = \struct {S', \preceq'}$ be ordered sets.
Let $f: S \to S'$ be an order isomorphism between $L$ and $L'$.
Then $f$ preserves infima and suprema. | === $f$ preserves infima ===
Let $X$ be a subset of $S$ such that
:$X$ admits an infimum in $L$.
By definition of infimum:
:$\inf X$ is lower bound for $X$.
Thus by Order Isomorphism Preserves Lower Bounds:
:$\map f {\inf X}$ is lower bound for $f \sqbrk X$.
We will prove that
:$\forall x \in S': x$ is lower bound for ... | Let $L = \struct {S, \preceq}$ and $L' = \struct {S', \preceq'}$ be [[Definition:Ordered Set|ordered sets]].
Let $f: S \to S'$ be an [[Definition:Order Isomorphism|order isomorphism]] between $L$ and $L'$.
Then $f$ [[Definition:Mapping Preserves Infimum/All|preserves infima]] and [[Definition:Mapping Preserves Supre... | === $f$ [[Definition:Mapping Preserves Infimum/All|preserves infima]] ===
Let $X$ be a [[Definition:Subset|subset]] of $S$ such that
:$X$ admits an [[Definition:Infimum of Set|infimum]] in $L$.
By definition of [[Definition:Infimum of Set|infimum]]:
:$\inf X$ is [[Definition:Lower Bound of Set|lower bound]] for $X$.
... | Order Isomorphism Preserves Infima and Suprema | https://proofwiki.org/wiki/Order_Isomorphism_Preserves_Infima_and_Suprema | https://proofwiki.org/wiki/Order_Isomorphism_Preserves_Infima_and_Suprema | [
"Order Isomorphisms"
] | [
"Definition:Ordered Set",
"Definition:Order Isomorphism",
"Definition:Mapping Preserves Infimum/All",
"Definition:Mapping Preserves Supremum/All"
] | [
"Definition:Mapping Preserves Infimum/All",
"Definition:Subset",
"Definition:Infimum of Set",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Order Isomorphism Preserves Lower Bounds",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
... |
proofwiki-12849 | Moser's Circle Problem | Let $n$ points be marked on the circumference of a circle $C$.
Let chords be drawn between each pair of these points.
For each $n$, the maximum number $\map C n$ of regions into which $C$ can be divided is as follows:
:{| border="1"
|-
! align="right" style = "padding: 2px 10px" | $n$
! align="right" style = "padding:... | :300px $\quad$ 300px $\quad$ 300px
:300px $\quad$ 300px $\quad$ 300px
{{proof wanted}}
{{Namedfor|Leo Moser|cat = Moser}} | Let $n$ [[Definition:Point|points]] be marked on the [[Definition:Circumference of Circle|circumference]] of a [[Definition:Circle|circle]] $C$.
Let [[Definition:Chord of Circle|chords]] be drawn between each pair of these [[Definition:Point|points]].
For each $n$, the maximum number $\map C n$ of [[Definition:Region... | :[[File:CircleDividedByChord1.png|300px]] $\quad$ [[File:CircleDividedByChord2.png|300px]] $\quad$ [[File:CircleDividedByChords3.png|300px]]
:[[File:CircleDividedByChords4.png|300px]] $\quad$ [[File:CircleDividedByChords5.png|300px]] $\quad$ [[File:CircleDividedByChords6.png|300px]]
{{proof wanted}}
{{Namedfor|Leo M... | Moser's Circle Problem | https://proofwiki.org/wiki/Moser's_Circle_Problem | https://proofwiki.org/wiki/Moser's_Circle_Problem | [
"Moser's Circle Problem",
"Chords of Circles",
"Circles"
] | [
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Circle/Chord",
"Definition:Point",
"Definition:Region/Plane"
] | [
"File:CircleDividedByChord1.png",
"File:CircleDividedByChord2.png",
"File:CircleDividedByChords3.png",
"File:CircleDividedByChords4.png",
"File:CircleDividedByChords5.png",
"File:CircleDividedByChords6.png"
] |
proofwiki-12850 | Prime Factors of 2^64 - 1 | The prime decomposition of $2^{64} - 1$ is given by:
:$2^{64} - 1 = 3 \times 5 \times 17 \times 257 \times 641 \times 65 \, 537 \times 6 \, 700 \, 417$ | {{begin-eqn}}
{{eqn | l = 2^{64} - 1
| r = \paren {2^{32} }^2 - 1
| c = Power of Power
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{32} - 1}
| c = Difference of Two Squares
}}
{{eqn | r = \paren {2^{32} + 1} \paren {\paren {2^{16} }^2 - 1}
| c = Power of Power
}}
{{eqn | r = \paren {2^{32} + 1}... | The [[Definition:Prime Decomposition|prime decomposition]] of $2^{64} - 1$ is given by:
:$2^{64} - 1 = 3 \times 5 \times 17 \times 257 \times 641 \times 65 \, 537 \times 6 \, 700 \, 417$ | {{begin-eqn}}
{{eqn | l = 2^{64} - 1
| r = \paren {2^{32} }^2 - 1
| c = [[Power of Power]]
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{32} - 1}
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \paren {2^{32} + 1} \paren {\paren {2^{16} }^2 - 1}
| c = [[Power of Power]]
}}
{{eqn | r = \paren ... | Prime Factors of 2^64 - 1 | https://proofwiki.org/wiki/Prime_Factors_of_2^64_-_1 | https://proofwiki.org/wiki/Prime_Factors_of_2^64_-_1 | [
"18,446,744,073,709,551,615"
] | [
"Definition:Prime Decomposition"
] | [
"Exponent Combination Laws/Power of Power",
"Difference of Two Squares",
"Exponent Combination Laws/Power of Power",
"Difference of Two Squares",
"Exponent Combination Laws/Power of Power",
"Difference of Two Squares",
"Exponent Combination Laws/Power of Power",
"Difference of Two Squares",
"Exponen... |
proofwiki-12851 | Prime Decomposition of 5th Fermat Number | The prime decomposition of the $5$th Fermat number is given by:
{{begin-eqn}}
{{eqn | l = 2^{\paren {2^5} } + 1
| r = 4 \, 294 \, 967 \, 297
| c = Sequence of Fermat Numbers
}}
{{eqn | r = 641 \times 6 \, 700 \, 417
| c =
}}
{{eqn | r = \paren {5 \times 2^7 + 1} \times \paren {3 \times 17449 \times 2... | From Divisor of Fermat Number, if $2^{\paren {2^n} } + 1$ has a divisor, it will be in the form:
:$k \, 2^{n + 2} + 1$
In the case of $n = 5$, a divisor of $2^{\paren {2^n} } + 1$ is then of the form:
:$k \, 2^7 + 1 = k \times 128 + 1$
Further, such a number will (for small $k$ at least) be prime, otherwise it will its... | The [[Definition:Prime Decomposition|prime decomposition]] of the $5$th [[Definition:Fermat Number|Fermat number]] is given by:
{{begin-eqn}}
{{eqn | l = 2^{\paren {2^5} } + 1
| r = 4 \, 294 \, 967 \, 297
| c = [[Definition:Fermat Number/Sequence|Sequence of Fermat Numbers]]
}}
{{eqn | r = 641 \times 6 \, ... | From [[Divisor of Fermat Number]], if $2^{\paren {2^n} } + 1$ has a [[Definition:Divisor of Integer|divisor]], it will be in the form:
:$k \, 2^{n + 2} + 1$
In the case of $n = 5$, a [[Definition:Divisor of Integer|divisor]] of $2^{\paren {2^n} } + 1$ is then of the form:
:$k \, 2^7 + 1 = k \times 128 + 1$
Further, s... | Prime Decomposition of 5th Fermat Number/Proof 1 | https://proofwiki.org/wiki/Prime_Decomposition_of_5th_Fermat_Number | https://proofwiki.org/wiki/Prime_Decomposition_of_5th_Fermat_Number/Proof_1 | [
"4,294,967,297",
"641",
"6,700,417",
"Fermat Numbers",
"Prime Decomposition of 5th Fermat Number"
] | [
"Definition:Prime Decomposition",
"Definition:Fermat Number",
"Definition:Fermat Number/Sequence"
] | [
"Divisor of Fermat Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Pri... |
proofwiki-12852 | Prime Decomposition of 5th Fermat Number | The prime decomposition of the $5$th Fermat number is given by:
{{begin-eqn}}
{{eqn | l = 2^{\paren {2^5} } + 1
| r = 4 \, 294 \, 967 \, 297
| c = Sequence of Fermat Numbers
}}
{{eqn | r = 641 \times 6 \, 700 \, 417
| c =
}}
{{eqn | r = \paren {5 \times 2^7 + 1} \times \paren {3 \times 17449 \times 2... | Note the remarkable coincidence that $2^4 + 5^4 = 2^7 \cdot 5 + 1 = 641$.
First we eliminate $y$ from $x^4 + y^4 = x^7 y + 1 = 0$:
{{begin-eqn}}
{{eqn | l = x^4 + y^4
| r = x^7 y + 1 = 0
| c =
}}
{{eqn | ll= \leadsto
| l = -x^4
| r = y^4
| c =
}}
{{eqn | ll= \leadsto
| l = x^{28} \... | The [[Definition:Prime Decomposition|prime decomposition]] of the $5$th [[Definition:Fermat Number|Fermat number]] is given by:
{{begin-eqn}}
{{eqn | l = 2^{\paren {2^5} } + 1
| r = 4 \, 294 \, 967 \, 297
| c = [[Definition:Fermat Number/Sequence|Sequence of Fermat Numbers]]
}}
{{eqn | r = 641 \times 6 \, ... | Note the remarkable coincidence that $2^4 + 5^4 = 2^7 \cdot 5 + 1 = 641$.
First we eliminate $y$ from $x^4 + y^4 = x^7 y + 1 = 0$:
{{begin-eqn}}
{{eqn | l = x^4 + y^4
| r = x^7 y + 1 = 0
| c =
}}
{{eqn | ll= \leadsto
| l = -x^4
| r = y^4
| c =
}}
{{eqn | ll= \leadsto
| l = x^{28... | Prime Decomposition of 5th Fermat Number/Proof 2 | https://proofwiki.org/wiki/Prime_Decomposition_of_5th_Fermat_Number | https://proofwiki.org/wiki/Prime_Decomposition_of_5th_Fermat_Number/Proof_2 | [
"4,294,967,297",
"641",
"6,700,417",
"Fermat Numbers",
"Prime Decomposition of 5th Fermat Number"
] | [
"Definition:Prime Decomposition",
"Definition:Fermat Number",
"Definition:Fermat Number/Sequence"
] | [
"Definition:Congruence (Number Theory)",
"Definition:Prime Number"
] |
proofwiki-12853 | Mapping Assigning to Element Its Compact Closure Preserves Infima and Directed Suprema | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice.
Let $C = \struct {\map K L, \preceq'}$ be an ordered subset of $L$
where $\map K L$ denotes the compact subset of $L$.
Let $P = \struct {\powerset {\map K L}, \precsim}$ be an inclusion ordered set of power set of $\map K L$.
Then there ... | By definitions of compact subset, compact closure, and subset:
:$\forall x \in S: x^{\mathrm{compact} } \subseteq \map K L$
By definition of power set:
:$\forall x \in S: x^{\mathrm{compact} } \in \powerset {\map K L}$
Define a mapping $f:S \to \powerset {\map K L}$ such that:
:$\forall x \in S: \map f x = x^{\mathrm{c... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Lattice (Order Theory)|lattice]].
Let $C = \struct {\map K L, \preceq'}$ be an [[Definition:Ordered Subset|ordered subset]] of $L$
where $\map K L$ denotes the [... | By definitions of [[Definition:Compact Subset of Lattice|compact subset]], [[Definition:Compact Closure|compact closure]], and [[Definition:Subset|subset]]:
:$\forall x \in S: x^{\mathrm{compact} } \subseteq \map K L$
By definition of [[Definition:Power Set|power set]]:
:$\forall x \in S: x^{\mathrm{compact} } \in \po... | Mapping Assigning to Element Its Compact Closure Preserves Infima and Directed Suprema | https://proofwiki.org/wiki/Mapping_Assigning_to_Element_Its_Compact_Closure_Preserves_Infima_and_Directed_Suprema | https://proofwiki.org/wiki/Mapping_Assigning_to_Element_Its_Compact_Closure_Preserves_Infima_and_Directed_Suprema | [
"Continuous Lattices",
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Ordered Subset",
"Definition:Compact Subset of Lattice",
"Definition:Inclusion Ordered Set",
"Definition:Power Set",
"Definition:Mapping Preserves Infimum/All",
"Definition:Mapping Pr... | [
"Definition:Compact Subset of Lattice",
"Definition:Compact Closure",
"Definition:Subset",
"Definition:Power Set",
"Definition:Mapping",
"Compact Closure is Directed",
"Definition:Directed Subset",
"Definition:Ordered Subset",
"Definition:Directed Subset",
"Definition:Lower Section",
"Definition... |
proofwiki-12854 | Power of 2 is Difference between Two Powers | Let $n \in \Z_{>0}$ be a power of $2$.
Then $n$ is the difference between powers of two positive integers greater than or equal to $2$.
{{questionable|This is so trivial I wonder whether something got lost in translation.}} | $2^k = 2^{k+1} - 2^k$
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Integer Power|power]] of $2$.
Then $n$ is the [[Definition:Integer Subtraction|difference]] between [[Definition:Integer Power|powers]] of two [[Definition:Positive Integer|positive integers]] greater than or equal to $2$.
{{questionable|This is so trivial I wonder whether somet... | $2^k = 2^{k+1} - 2^k$
{{qed}} | Power of 2 is Difference between Two Powers | https://proofwiki.org/wiki/Power_of_2_is_Difference_between_Two_Powers | https://proofwiki.org/wiki/Power_of_2_is_Difference_between_Two_Powers | [
"Power of 2 is Difference between Two Powers",
"Powers of 2"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Subtraction/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Positive/Integer"
] | [] |
proofwiki-12855 | Smallest Sequence of Three Consecutive Semiprimes | The smallest triple of consecutive semiprimes is:
:$33, 34, 35$ | We have:
{{begin-eqn}}
{{eqn | l = 33
| r = 3 \times 11
}}
{{eqn | l = 34
| r = 2 \times 17
}}
{{eqn | l = 35
| r = 5 \times 7
}}
{{end-eqn}}
It can be seen from the sequence of semiprimes that there exist no smaller such triples.
{{qed}} | The smallest [[Definition:Ordered Triple|triple]] of consecutive [[Definition:Semiprime Number|semiprimes]] is:
:$33, 34, 35$ | We have:
{{begin-eqn}}
{{eqn | l = 33
| r = 3 \times 11
}}
{{eqn | l = 34
| r = 2 \times 17
}}
{{eqn | l = 35
| r = 5 \times 7
}}
{{end-eqn}}
It can be seen from the [[Definition:Semiprime Number/Sequence|sequence of semiprimes]] that there exist no smaller such [[Definition:Ordered Triple|triples]]... | Smallest Sequence of Three Consecutive Semiprimes | https://proofwiki.org/wiki/Smallest_Sequence_of_Three_Consecutive_Semiprimes | https://proofwiki.org/wiki/Smallest_Sequence_of_Three_Consecutive_Semiprimes | [
"Semiprimes"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Semiprime Number"
] | [
"Definition:Semiprime Number/Sequence",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple"
] |
proofwiki-12856 | Integers not Sum of Distinct Triangular Numbers | The sequence of integers which cannot be expressed as the sum of distinct triangular numbers is:
:$2, 5, 8, 12, 23, 33$
{{OEIS|A053614}} | It will be proved that the largest integer which cannot be expressed as the sum of distinct triangular numbers is $33$.
The remaining integers in the sequence can be identified by inspection.
We prove this using a variant of Second Principle of Mathematical Induction.
Let $\map P n$ be the proposition:
:$n$ can be expr... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Triangular Number|triangular numbers]] is:
:$2, 5, 8, 12, 23, 33$
{{OEIS|A053614}} | It will be proved that the largest [[Definition:Integer|integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Triangular Number|triangular numbers]] is $33$.
The remaining [[Definition:Integer|integers]] in the [[Definition:Integer Sequ... | Integers not Sum of Distinct Triangular Numbers | https://proofwiki.org/wiki/Integers_not_Sum_of_Distinct_Triangular_Numbers | https://proofwiki.org/wiki/Integers_not_Sum_of_Distinct_Triangular_Numbers | [
"Triangular Numbers",
"33"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Distinct/Plural",
"Definition:Triangular Number"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Distinct/Plural",
"Definition:Triangular Number",
"Definition:Integer",
"Definition:Integer Sequence",
"Second Principle of Mathematical Induction",
"Definition:Addition/Integers",
"Definition:Distinct/Plural",
"Definition:Triangula... |
proofwiki-12857 | Powers of 2 and 5 without Zeroes | The following $n \in \Z$ are such that both $2^n$ and $5^n$ have no zeroes in their decimal representation:
:$0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 33$
{{OEIS|A007496}} | :{| border="1"
|-
! align="center" style = "padding: 2px 10px" | $n$
! align="center" style = "padding: 2px 10px" | $2^n$
! align="center" style = "padding: 2px 10px" | $5^n$
|-
| align="right" style = "padding: 2px 10px" | $0$
| align="right" style = "padding: 2px 10px" | $1$
| align="right" style = "padding: 2px 1... | The following $n \in \Z$ are such that both $2^n$ and $5^n$ have no [[Definition:Zero (Number)|zeroes]] in their [[Definition:Decimal Notation|decimal representation]]:
:$0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 33$
{{OEIS|A007496}} | :{| border="1"
|-
! align="center" style = "padding: 2px 10px" | $n$
! align="center" style = "padding: 2px 10px" | $2^n$
! align="center" style = "padding: 2px 10px" | $5^n$
|-
| align="right" style = "padding: 2px 10px" | $0$
| align="right" style = "padding: 2px 10px" | $1$
| align="right" style = "padding: 2px 1... | Powers of 2 and 5 without Zeroes | https://proofwiki.org/wiki/Powers_of_2_and_5_without_Zeroes | https://proofwiki.org/wiki/Powers_of_2_and_5_without_Zeroes | [
"Powers of 2",
"Powers of 5"
] | [
"Definition:Zero (Number)",
"Definition:Decimal Notation"
] | [] |
proofwiki-12858 | Vector Cross Product Distributes over Addition | The vector cross product is distributive over addition.
That is, in general:
:$\mathbf a \times \paren {\mathbf b + \mathbf c} = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}$
for $\mathbf a, \mathbf b, \mathbf c \in \R^3$. | Let:
:$\mathbf a = \begin {bmatrix} a_x \\ a_y \\a_z \end {bmatrix}$, $\mathbf b = \begin {bmatrix} b_x \\ b_y \\ b_z \end {bmatrix}$, $\mathbf c = \begin {bmatrix} c_x \\ c_y \\ c_z \end {bmatrix}$
be vectors in $\R^3$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf a \times \paren {\mathbf b + \mathbf c}
| r = \begin ... | The [[Definition:Vector Cross Product|vector cross product]] is [[Definition:Distributive Operation|distributive]] over [[Definition:Vector Sum|addition]].
That is, in general:
:$\mathbf a \times \paren {\mathbf b + \mathbf c} = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}$
for $\mathbf ... | Let:
:$\mathbf a = \begin {bmatrix} a_x \\ a_y \\a_z \end {bmatrix}$, $\mathbf b = \begin {bmatrix} b_x \\ b_y \\ b_z \end {bmatrix}$, $\mathbf c = \begin {bmatrix} c_x \\ c_y \\ c_z \end {bmatrix}$
be [[Definition:Space Vector|vectors in $\R^3$]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf a \times \paren {\mathbf b + ... | Vector Cross Product Distributes over Addition/Proof 1 | https://proofwiki.org/wiki/Vector_Cross_Product_Distributes_over_Addition | https://proofwiki.org/wiki/Vector_Cross_Product_Distributes_over_Addition/Proof_1 | [
"Vector Cross Product",
"Vector Addition",
"Examples of Distributive Operations",
"Vector Cross Product Distributes over Addition"
] | [
"Definition:Vector Cross Product",
"Definition:Distributive Operation",
"Definition:Vector Sum"
] | [
"Definition:Vector/Real Euclidean Space/Space Vector",
"Real Multiplication Distributes over Addition",
"Real Addition is Commutative"
] |
proofwiki-12859 | Vector Cross Product Distributes over Addition | The vector cross product is distributive over addition.
That is, in general:
:$\mathbf a \times \paren {\mathbf b + \mathbf c} = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}$
for $\mathbf a, \mathbf b, \mathbf c \in \R^3$. | We draw a triangular prism whose parallel edges are in the direction of $\mathbf a$ and with its end faces as triangles with sides $\mathbf b$, $\mathbf c$ and $\mathbf b + \mathbf c$.
:600px
From Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors, the vector areas of these triangular e... | The [[Definition:Vector Cross Product|vector cross product]] is [[Definition:Distributive Operation|distributive]] over [[Definition:Vector Sum|addition]].
That is, in general:
:$\mathbf a \times \paren {\mathbf b + \mathbf c} = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}$
for $\mathbf ... | We draw a [[Definition:Triangular Prism|triangular prism]] whose [[Definition:Parallel Lines|parallel]] [[Definition:Edge of Polyhedron|edges]] are in the direction of $\mathbf a$ and with its end [[Definition:Face of Polyhedron|faces]] as [[Definition:Triangle (Geometry)|triangles]] with [[Definition:Side of Polygon|s... | Vector Cross Product Distributes over Addition/Proof 2 | https://proofwiki.org/wiki/Vector_Cross_Product_Distributes_over_Addition | https://proofwiki.org/wiki/Vector_Cross_Product_Distributes_over_Addition/Proof_2 | [
"Vector Cross Product",
"Vector Addition",
"Examples of Distributive Operations",
"Vector Cross Product Distributes over Addition"
] | [
"Definition:Vector Cross Product",
"Definition:Distributive Operation",
"Definition:Vector Sum"
] | [
"Definition:Prism/Triangular",
"Definition:Parallel (Geometry)/Lines",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"File:Cross-product-distributes-over-addition.png",
"Magnitude of Vector Cross Product equals Area of Parallel... |
proofwiki-12860 | Vector Cross Product Distributes over Addition | The vector cross product is distributive over addition.
That is, in general:
:$\mathbf a \times \paren {\mathbf b + \mathbf c} = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}$
for $\mathbf a, \mathbf b, \mathbf c \in \R^3$. | Let $\mathbf b'$ and $\mathbf c'$ be the projections of $\mathbf b$ and $\mathbf c$ onto the plane perpendicular to $\mathbf a$.
Then $\mathbf b' + \mathbf c'$ is the projection of $\mathbf b + \mathbf c$ onto that plane.
We have:
{{begin-eqn}}
{{eqn | l = \mathbf a \times \mathbf b'
| r = \mathbf a \times \mathb... | The [[Definition:Vector Cross Product|vector cross product]] is [[Definition:Distributive Operation|distributive]] over [[Definition:Vector Sum|addition]].
That is, in general:
:$\mathbf a \times \paren {\mathbf b + \mathbf c} = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}$
for $\mathbf ... | Let $\mathbf b'$ and $\mathbf c'$ be the [[Definition:Vector Projection|projections]] of $\mathbf b$ and $\mathbf c$ onto the [[Definition:Plane|plane]] [[Definition:Line Perpendicular to Plane|perpendicular]] to $\mathbf a$.
Then $\mathbf b' + \mathbf c'$ is the [[Definition:Vector Projection|projection]] of $\mathbf... | Vector Cross Product Distributes over Addition/Proof 3 | https://proofwiki.org/wiki/Vector_Cross_Product_Distributes_over_Addition | https://proofwiki.org/wiki/Vector_Cross_Product_Distributes_over_Addition/Proof_3 | [
"Vector Cross Product",
"Vector Addition",
"Examples of Distributive Operations",
"Vector Cross Product Distributes over Addition"
] | [
"Definition:Vector Cross Product",
"Definition:Distributive Operation",
"Definition:Vector Sum"
] | [
"Definition:Vector Projection",
"Definition:Plane Surface",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Vector Projection",
"Definition:Plane Surface",
"Definition:Right Angle/Perpendicular",
"Definition:Plane Surface",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Vector L... |
proofwiki-12861 | Jacobi's Equation is Variational Equation of Euler's Equation | The Variational equation of Euler's equation is Jacobi's equation. | Let Euler's equation be
:$\map {F_y} {x, \hat y, \hat y'} - \dfrac \d {\d x} \map {F_{y'} } {x, \hat y, \hat y'} = 0$
which is derived from:
:$\ds \int_a^b \paren {\map {F_y} {x, \hat y, \hat y'} - \frac \d {\d x} \map {F_{y'} } {x, \hat y, \hat y'} } \rd x = 0$
Let $\map {\hat y} x = \map y x$ and $\map {\hat y} x = \... | The [[Definition:Variational Equation of Differential Equation|Variational equation]] of [[Definition:Euler's Equation for Vanishing Variation|Euler's equation]] is [[Definition:Jacobi's Equation of Functional|Jacobi's equation]]. | Let [[Definition:Euler's Equation for Vanishing Variation|Euler's equation]] be
:$\map {F_y} {x, \hat y, \hat y'} - \dfrac \d {\d x} \map {F_{y'} } {x, \hat y, \hat y'} = 0$
which is derived from:
:$\ds \int_a^b \paren {\map {F_y} {x, \hat y, \hat y'} - \frac \d {\d x} \map {F_{y'} } {x, \hat y, \hat y'} } \rd x = 0... | Jacobi's Equation is Variational Equation of Euler's Equation | https://proofwiki.org/wiki/Jacobi's_Equation_is_Variational_Equation_of_Euler's_Equation | https://proofwiki.org/wiki/Jacobi's_Equation_is_Variational_Equation_of_Euler's_Equation | [
"Calculus of Variations"
] | [
"Definition:Variational Equation of Differential Equation",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Jacobi's Equation of Functional"
] | [
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Euler's Equation for Vanishing Variation",
"Taylor's Theorem",
"Definition:Ordered Tuple",
"Definition:Variable",
"Definition:Differential Equation/Solution",
"Definition:Euler's Equation for Vanishing Variation",
"Integration by Parts... |
proofwiki-12862 | Palindromes in Base 10 and Base 2 | The following $n \in \Z$ are palindromic in both decimal and binary:
:$0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, 7447, 9009, 15 \, 351, 32 \, 223, 39 \, 993, \ldots$
{{OEIS|A007632}} | :{| border="1"
|-
! align="center" style = "padding: 2px 10px" | $n_{10}$
! align="center" style = "padding: 2px 10px" | $n_2$
|-
| align="right" style = "padding: 2px 10px" | $0$
| align="right" style = "padding: 2px 10px" | $0$
|-
| align="right" style = "padding: 2px 10px" | $1$
| align="right" style = "padding: 2... | The following $n \in \Z$ are [[Definition:Palindromic Number|palindromic]] in both [[Definition:Decimal Notation|decimal]] and [[Definition:Binary Notation|binary]]:
:$0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, 7447, 9009, 15 \, 351, 32 \, 223, 39 \, 993, \ldots$
{{OEIS|A007632}} | :{| border="1"
|-
! align="center" style = "padding: 2px 10px" | $n_{10}$
! align="center" style = "padding: 2px 10px" | $n_2$
|-
| align="right" style = "padding: 2px 10px" | $0$
| align="right" style = "padding: 2px 10px" | $0$
|-
| align="right" style = "padding: 2px 10px" | $1$
| align="right" style = "padding: 2... | Palindromes in Base 10 and Base 2 | https://proofwiki.org/wiki/Palindromes_in_Base_10_and_Base_2 | https://proofwiki.org/wiki/Palindromes_in_Base_10_and_Base_2 | [
"Palindromic Numbers",
"2",
"10"
] | [
"Definition:Palindromic Number",
"Definition:Decimal Notation",
"Definition:Binary Notation"
] | [] |
proofwiki-12863 | Integer as Sum of 5 Non-Zero Squares | Let $n \in \Z$ be an integer such that $n > 33$.
Then $n$ can be expressed as the sum of $5$ non-zero squares. | From Lagrange's Four Square Theorem, every positive integer can be expressed as the sum of $4$ squares, some of which may be zero.
The existence of positive integers which cannot be expressed as the sum of $4$ non-zero squares is noted by the trivial examples $1$, $2$ and $3$.
Thus Lagrange's Four Square Theorem can be... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n > 33$.
Then $n$ can be expressed as the [[Definition:Integer Addition|sum]] of $5$ non-[[Definition:Zero (Number)|zero]] [[Definition:Square Number|squares]]. | From [[Lagrange's Four Square Theorem]], every [[Definition:Positive Integer|positive integer]] can be expressed as the [[Definition:Integer Addition|sum]] of $4$ [[Definition:Square Number|squares]], some of which may be [[Definition:Zero (Number)|zero]].
The existence of [[Definition:Positive Integer|positive intege... | Integer as Sum of 5 Non-Zero Squares | https://proofwiki.org/wiki/Integer_as_Sum_of_5_Non-Zero_Squares | https://proofwiki.org/wiki/Integer_as_Sum_of_5_Non-Zero_Squares | [
"Sums of Squares"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Zero (Number)",
"Definition:Square Number"
] | [
"Lagrange's Four Square Theorem",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Zero (Number)",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Zero (Number)",
"Definition:Square Number",
"Lagrange's Four Square Th... |
proofwiki-12864 | Bottom in Compact Closure | Let $L = \struct {S, \preceq}$ be a bounded below ordered set.
Let $x \in S$.
Then:
:$\bot \in x^{\mathrm{compact} }$
where:
:$\bot$ denotes the smallest element in $L$
:$ x^{\mathrm{compact} }$ denotes the compact closure of $x$. | By Bottom is Compact:
:$\bot$ is a compact element.
By definition of the smallest element:
:$\bot \preceq x$
Thus by definition of compact closure:
:$\bot \in x^{\mathrm{compact} }$
{{qed}}
Category:Way Below Relation
2z4a1pcn8t5hdg1ggxblyw896p9vkg1 | Let $L = \struct {S, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Ordered Set|ordered set]].
Let $x \in S$.
Then:
:$\bot \in x^{\mathrm{compact} }$
where:
:$\bot$ denotes the [[Definition:Smallest Element|smallest element]] in $L$
:$ x^{\mathrm{compact} }$ denotes the [[Definition:Comp... | By [[Bottom is Compact]]:
:$\bot$ is a [[Definition:Compact Element|compact element]].
By definition of the [[Definition:Smallest Element|smallest element]]:
:$\bot \preceq x$
Thus by definition of [[Definition:Compact Closure|compact closure]]:
:$\bot \in x^{\mathrm{compact} }$
{{qed}}
[[Category:Way Below Relation... | Bottom in Compact Closure | https://proofwiki.org/wiki/Bottom_in_Compact_Closure | https://proofwiki.org/wiki/Bottom_in_Compact_Closure | [
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Ordered Set",
"Definition:Smallest Element",
"Definition:Compact Closure"
] | [
"Bottom is Compact",
"Definition:Compact Element",
"Definition:Smallest Element",
"Definition:Compact Closure",
"Category:Way Below Relation"
] |
proofwiki-12865 | Triplets of Products of Two Distinct Primes | The following triplets of consecutive positive integers are the smallest in which each number is the product of $2$ distinct prime numbers:
:$33, 34, 35$
:$85, 86, 87$
:$93, 94, 95$
:$141, 142, 143$
:$201, 202, 203$
:$213, 214, 215$
:$217, 218, 219$ | Taking each triplet in turn:
{{begin-eqn}}
{{eqn | l = 33
| r = 3 \times 11
}}
{{eqn | l = 34
| r = 2 \times 17
}}
{{eqn | l = 35
| r = 5 \times 7
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 85
| r = 5 \times 17
}}
{{eqn | l = 86
| r = 2 \times 43
}}
{{eqn | l = 87
| r = 3 \times 29
}}
... | The following [[Definition:Ordered Triple|triplets]] of consecutive [[Definition:Positive Integer|positive integers]] are the smallest in which each number is the [[Definition:Integer Multiplication|product]] of $2$ [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime numbers]]:
:$33, 34, 35$
:$85, 86, 87$... | Taking each [[Definition:Ordered Triple|triplet]] in turn:
{{begin-eqn}}
{{eqn | l = 33
| r = 3 \times 11
}}
{{eqn | l = 34
| r = 2 \times 17
}}
{{eqn | l = 35
| r = 5 \times 7
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 85
| r = 5 \times 17
}}
{{eqn | l = 86
| r = 2 \times 43
}}
{{eqn | ... | Triplets of Products of Two Distinct Primes | https://proofwiki.org/wiki/Triplets_of_Products_of_Two_Distinct_Primes | https://proofwiki.org/wiki/Triplets_of_Products_of_Two_Distinct_Primes | [
"Semiprimes"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Positive/Integer",
"Definition:Multiplication/Integers",
"Definition:Distinct",
"Definition:Prime Number"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Semiprime Number",
"Definition:Multiplication/Integers",
"Definition:Distinct",
"Definition:Prime Number"
] |
proofwiki-12866 | Product of Two Distinct Primes has 4 Positive Divisors | Let $n \in \Z_{>0}$ be a positive integer which is the product of $2$ distinct primes.
Then $n$ has exactly $4$ positive divisors. | Let $n = p \times q$ where $p$ and $q$ are primes.
We have by definition of divisor:
{{begin-eqn}}
{{eqn | l = 1
| o = \divides
| r = n
| c = One Divides all Integers
}}
{{eqn | l = p
| o = \divides
| r = n
| c = {{Defof|Divisor of Integer}}
}}
{{eqn | l = q
| o = \divides
... | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]] which is the [[Definition:Integer Multiplication|product]] of $2$ [[Definition:Distinct|distinct]] [[Definition:Prime Number|primes]].
Then $n$ has exactly $4$ [[Definition:Positive Integer|positive]] [[Definition:Divisor of Integer|divisors]]. | Let $n = p \times q$ where $p$ and $q$ are [[Definition:Prime Number|primes]].
We have by definition of [[Definition:Divisor of Integer|divisor]]:
{{begin-eqn}}
{{eqn | l = 1
| o = \divides
| r = n
| c = [[One Divides all Integers]]
}}
{{eqn | l = p
| o = \divides
| r = n
| c = {{... | Product of Two Distinct Primes has 4 Positive Divisors | https://proofwiki.org/wiki/Product_of_Two_Distinct_Primes_has_4_Positive_Divisors | https://proofwiki.org/wiki/Product_of_Two_Distinct_Primes_has_4_Positive_Divisors | [
"Semiprimes"
] | [
"Definition:Positive/Integer",
"Definition:Multiplication/Integers",
"Definition:Distinct",
"Definition:Prime Number",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Integer Divisor Results/One Divides all Integers",
"Integer Divisor Results/Integer Divides Itself",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Coprime/Integers",
"Euclid's Lemma",
"Definition:Pri... |
proofwiki-12867 | Product of Two Distinct Primes is Multiplicatively Perfect | Let $n \in \Z_{>0}$ be a positive integer which is the product of $2$ distinct primes.
Then $n$ is multiplicatively perfect. | Let $n = p \times q$ where $p$ and $q$ are primes.
From Product of Two Distinct Primes has 4 Positive Divisors, the positive divisors of $n$ are:
:$1, p, q, pq$
Thus the product of all the divisors of $n$ is:
:$1 \times p \times q \times p q = p^2 q^2 = n^2$
Hence the result, by definition of multiplicatively perfect.
... | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]] which is the [[Definition:Integer Multiplication|product]] of $2$ [[Definition:Distinct|distinct]] [[Definition:Prime Number|primes]].
Then $n$ is [[Definition:Multiplicatively Perfect Number|multiplicatively perfect]]. | Let $n = p \times q$ where $p$ and $q$ are [[Definition:Prime Number|primes]].
From [[Product of Two Distinct Primes has 4 Positive Divisors]], the [[Definition:Positive Integer|positive]] [[Definition:Divisor of Integer|divisors]] of $n$ are:
:$1, p, q, pq$
Thus the [[Definition:Integer Multiplication|product]] of a... | Product of Two Distinct Primes is Multiplicatively Perfect | https://proofwiki.org/wiki/Product_of_Two_Distinct_Primes_is_Multiplicatively_Perfect | https://proofwiki.org/wiki/Product_of_Two_Distinct_Primes_is_Multiplicatively_Perfect | [
"Semiprimes",
"Multiplicatively Perfect Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Multiplication/Integers",
"Definition:Distinct",
"Definition:Prime Number",
"Definition:Multiplicatively Perfect Number"
] | [
"Definition:Prime Number",
"Product of Two Distinct Primes has 4 Positive Divisors",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Multiplication/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Multiplicatively Perfect Number"
] |
proofwiki-12868 | Cube of Prime has 4 Positive Divisors | Let $n \in \Z_{>0}$ be a positive integer which is the cube of a prime number.
Then $n$ has exactly $4$ positive divisors. | Let $n = p^3$ where $p$ is prime.
The positive divisors of $n$ are:
:$1, p, p^2, p^3$
This result follows from Divisors of Power of Prime.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]] which is the [[Definition:Cube (Algebra)|cube]] of a [[Definition:Prime Number|prime number]].
Then $n$ has exactly $4$ [[Definition:Positive Integer|positive]] [[Definition:Divisor of Integer|divisors]]. | Let $n = p^3$ where $p$ is [[Definition:Prime Number|prime]].
The [[Definition:Positive Integer|positive]] [[Definition:Divisor of Integer|divisors]] of $n$ are:
:$1, p, p^2, p^3$
This result follows from [[Divisors of Power of Prime]].
{{qed}} | Cube of Prime has 4 Positive Divisors | https://proofwiki.org/wiki/Cube_of_Prime_has_4_Positive_Divisors | https://proofwiki.org/wiki/Cube_of_Prime_has_4_Positive_Divisors | [
"Cube Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Cube/Algebra",
"Definition:Prime Number",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Prime Number",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Divisors of Power of Prime"
] |
proofwiki-12869 | Cube of Prime is Multiplicatively Perfect | Let $n \in \Z_{>0}$ be a positive integer which is the cube of a prime number.
Then $n$ is multiplicatively perfect. | Let $n = p^3$ where $p$ is prime.
From Cube of Prime has 4 Positive Divisors, the positive divisors of $n$ are:
:$1, p, p^2, p^3$
Thus the product of all the divisors of $n$ is:
:$1 \times p \times p^2 \times p^3 = p^6 = n^2$
Hence the result, by definition of multiplicatively perfect.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]] which is the [[Definition:Cube (Algebra)|cube]] of a [[Definition:Prime Number|prime number]].
Then $n$ is [[Definition:Multiplicatively Perfect Number|multiplicatively perfect]]. | Let $n = p^3$ where $p$ is [[Definition:Prime Number|prime]].
From [[Cube of Prime has 4 Positive Divisors]], the [[Definition:Positive Integer|positive]] [[Definition:Divisor of Integer|divisors]] of $n$ are:
:$1, p, p^2, p^3$
Thus the [[Definition:Integer Multiplication|product]] of all the [[Definition:Divisor of ... | Cube of Prime is Multiplicatively Perfect | https://proofwiki.org/wiki/Cube_of_Prime_is_Multiplicatively_Perfect | https://proofwiki.org/wiki/Cube_of_Prime_is_Multiplicatively_Perfect | [
"Cube Numbers",
"Multiplicatively Perfect Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Cube/Algebra",
"Definition:Prime Number",
"Definition:Multiplicatively Perfect Number"
] | [
"Definition:Prime Number",
"Cube of Prime has 4 Positive Divisors",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Multiplication/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Multiplicatively Perfect Number"
] |
proofwiki-12870 | Compact Closure is Directed | Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.
Let $x \in S$.
Then $x^{\mathrm{compact} }$ is directed
where $x^{\mathrm{compact} }$ denotes the compact closure of $x$. | By Bottom in Compact Closure:
:$\bot \in x^{\mathrm{compact} }$
where $\bot$ denotes the smallest element in $L$.
Thus by definition:
:$x^{\mathrm{compact} }$ is non-empty.
Let $y, z \in x^{\mathrm{compact} }$
By definition of compact closure:
:$y$ and $z$ are compact elements and $y \preceq x$ and $z \preceq x$
By def... | Let $L = \left({S, \vee, \preceq}\right)$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Let $x \in S$.
Then $x^{\mathrm{compact} }$ is [[Definition:Directed Subset|directed]]
where $x^{\mathrm{compact} }$ denotes the [[Definition:Compact Closure|compact closur... | By [[Bottom in Compact Closure]]:
:$\bot \in x^{\mathrm{compact} }$
where $\bot$ denotes the [[Definition:Smallest Element|smallest element]] in $L$.
Thus by definition:
:$x^{\mathrm{compact} }$ is [[Definition:Non-Empty Set|non-empty]].
Let $y, z \in x^{\mathrm{compact} }$
By definition of [[Definition:Compact Clos... | Compact Closure is Directed | https://proofwiki.org/wiki/Compact_Closure_is_Directed | https://proofwiki.org/wiki/Compact_Closure_is_Directed | [
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Directed Subset",
"Definition:Compact Closure"
] | [
"Bottom in Compact Closure",
"Definition:Smallest Element",
"Definition:Non-Empty Set",
"Definition:Compact Closure",
"Definition:Compact Element",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Compact Element",
"Definition:Element is Way Below",
"Join Succeeds Operand... |
proofwiki-12871 | Sum of 2 Lucky Numbers in 4 Ways | The number $34$ is the smallest positive integer to be the sum of $2$ lucky numbers in $4$ different ways. | The sequence of lucky numbers begins:
:$1, 3, 7, 9, 13, 15, 21, 25, 31, 33, \ldots$
Thus we have:
{{begin-eqn}}
{{eqn | l = 34
| r = 1 + 33
| c =
}}
{{eqn | r = 3 + 31
| c =
}}
{{eqn | r = 9 + 25
| c =
}}
{{eqn | r = 13 + 21
| c =
}}
{{end-eqn}}
{{qed}} | The number $34$ is the smallest [[Definition:Positive Integer|positive integer]] to be the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Lucky Number|lucky numbers]] in $4$ different ways. | The [[Definition:Lucky Number/Sequence|sequence of lucky numbers]] begins:
:$1, 3, 7, 9, 13, 15, 21, 25, 31, 33, \ldots$
Thus we have:
{{begin-eqn}}
{{eqn | l = 34
| r = 1 + 33
| c =
}}
{{eqn | r = 3 + 31
| c =
}}
{{eqn | r = 9 + 25
| c =
}}
{{eqn | r = 13 + 21
| c =
}}
{{end-eqn}}
{{... | Sum of 2 Lucky Numbers in 4 Ways | https://proofwiki.org/wiki/Sum_of_2_Lucky_Numbers_in_4_Ways | https://proofwiki.org/wiki/Sum_of_2_Lucky_Numbers_in_4_Ways | [
"Lucky Numbers",
"34"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Lucky Number"
] | [
"Definition:Lucky Number/Sequence"
] |
proofwiki-12872 | 35 Hexominoes | There exist $35$ distinct free hexominoes:
:600px | {{ProofWanted|Work to be done yet to establish method of creation}} | There exist $35$ [[Definition:Distinct|distinct]] [[Definition:Free Polyomino|free]] [[Definition:Hexomino|hexominoes]]:
:[[File:35Hexominoes.png|600px]] | {{ProofWanted|Work to be done yet to establish method of creation}} | 35 Hexominoes | https://proofwiki.org/wiki/35_Hexominoes | https://proofwiki.org/wiki/35_Hexominoes | [
"Hexominoes",
"35"
] | [
"Definition:Distinct",
"Definition:Polyomino/Free",
"Definition:Hexomino",
"File:35Hexominoes.png"
] | [] |
proofwiki-12873 | Fixed Point of Permutation is Fixed Point of Power | Let $S_n$ denote the symmetric group on $n$ letters.
Let $\sigma \in S_n$.
Let $i \in \Fix \sigma$, where $\Fix \sigma$ denotes the set of fixed elements of $\sigma$.
Then for all $m \in \Z$:
:$i \in \Fix {\sigma^m}$ | It follows from Integers form Ordered Integral Domain that for any integer $m$ either:
:$m = 0$
or
:$m > 0$
or:
:$m < 0$ | Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $\sigma \in S_n$.
Let $i \in \Fix \sigma$, where $\Fix \sigma$ denotes the [[Definition:Set of Fixed Elements|set of fixed elements]] of $\sigma$.
Then for all $m \in \Z$:
:$i \in \Fix {\sigma^m}$ | It follows from [[Integers form Ordered Integral Domain]] that for any integer $m$ either:
:$m = 0$
or
:$m > 0$
or:
:$m < 0$ | Fixed Point of Permutation is Fixed Point of Power | https://proofwiki.org/wiki/Fixed_Point_of_Permutation_is_Fixed_Point_of_Power | https://proofwiki.org/wiki/Fixed_Point_of_Permutation_is_Fixed_Point_of_Power | [
"Symmetric Groups",
"Permutations"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Fixed Element under Permutation/Set of Fixed Elements"
] | [
"Integers form Ordered Integral Domain"
] |
proofwiki-12874 | Real Symmetric Matrix is Hermitian | Every real symmetric matrix is Hermitian. | Let $\mathbf A$ be a real symmetric matrix.
Then we have:
{{begin-eqn}}
{{eqn | l = \sqbrk {\mathbf A}^\dagger_{i j}
| r = \overline {\sqbrk {\mathbf A}_{ji} }
| c = {{Defof|Hermitian Conjugate}}
}}
{{eqn | r = \sqbrk {\mathbf A}_{ji}
| c = Complex Number equals Conjugate iff Wholly Real: $\mathbf A$ ... | Every [[Definition:Real Matrix|real]] [[Definition:Symmetric Matrix|symmetric matrix]] is [[Definition:Hermitian Matrix|Hermitian]]. | Let $\mathbf A$ be a [[Definition:Real Matrix|real]] [[Definition:Symmetric Matrix|symmetric matrix]].
Then we have:
{{begin-eqn}}
{{eqn | l = \sqbrk {\mathbf A}^\dagger_{i j}
| r = \overline {\sqbrk {\mathbf A}_{ji} }
| c = {{Defof|Hermitian Conjugate}}
}}
{{eqn | r = \sqbrk {\mathbf A}_{ji}
| c = ... | Real Symmetric Matrix is Hermitian | https://proofwiki.org/wiki/Real_Symmetric_Matrix_is_Hermitian | https://proofwiki.org/wiki/Real_Symmetric_Matrix_is_Hermitian | [
"Real Matrices",
"Symmetric Matrices",
"Hermitian Matrices"
] | [
"Definition:Real Matrix",
"Definition:Symmetric Matrix",
"Definition:Hermitian Matrix"
] | [
"Definition:Real Matrix",
"Definition:Symmetric Matrix",
"Complex Number equals Conjugate iff Wholly Real",
"Definition:Real Matrix",
"Definition:Symmetric Matrix",
"Definition:Hermitian Matrix"
] |
proofwiki-12875 | Hexominoes cannot form Rectangle | While there are a total of $210$ squares in a complete set of hexominoes, it is impossible to build them into a rectangle of side lengths $a$ and $b$ where $a \times b = 210$. | {{ProofWanted|I'll have to check this, but I think the proof is along the lines that if you coloured them in a checkerboard pattern, you can't get an equal number of black and white squares.}} | While there are a total of $210$ [[Definition:Square (Geometry)|squares]] in a complete set of [[Definition:Hexomino|hexominoes]], it is impossible to build them into a [[Definition:Rectangle|rectangle]] of [[Definition:Side of Polygon|side]] [[Definition:Length of Line|lengths]] $a$ and $b$ where $a \times b = 210$. | {{ProofWanted|I'll have to check this, but I think the proof is along the lines that if you coloured them in a checkerboard pattern, you can't get an equal number of black and white squares.}} | Hexominoes cannot form Rectangle | https://proofwiki.org/wiki/Hexominoes_cannot_form_Rectangle | https://proofwiki.org/wiki/Hexominoes_cannot_form_Rectangle | [
"Hexominoes"
] | [
"Definition:Quadrilateral/Square",
"Definition:Hexomino",
"Definition:Quadrilateral/Rectangle",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length"
] | [] |
proofwiki-12876 | Number of Heptominoes | There exist $108$ distinct free heptominoes, one of which has a hole:
:600px | {{ProofWanted|Work to be done yet to establish method of creation}} | There exist $108$ [[Definition:Distinct|distinct]] [[Definition:Free Polyomino|free]] [[Definition:Heptomino|heptominoes]], one of which has a hole:
:[[File:108Heptominoes.png|600px]] | {{ProofWanted|Work to be done yet to establish method of creation}} | Number of Heptominoes | https://proofwiki.org/wiki/Number_of_Heptominoes | https://proofwiki.org/wiki/Number_of_Heptominoes | [
"Heptominoes",
"108"
] | [
"Definition:Distinct",
"Definition:Polyomino/Free",
"Definition:Heptomino",
"File:108Heptominoes.png"
] | [] |
proofwiki-12877 | 369 Octominoes | There exist $369$ distinct free octominoes, $6$ of which have a hole:
:600px | {{ProofWanted|Work to be done yet to establish method of creation}} | There exist $369$ [[Definition:Distinct|distinct]] [[Definition:Free Polyomino|free]] [[Definition:Octomino|octominoes]], $6$ of which have a hole:
:[[File:369Octominoes.png|600px]] | {{ProofWanted|Work to be done yet to establish method of creation}} | 369 Octominoes | https://proofwiki.org/wiki/369_Octominoes | https://proofwiki.org/wiki/369_Octominoes | [
"Polyominoes",
"369"
] | [
"Definition:Distinct",
"Definition:Polyomino/Free",
"Definition:Octomino",
"File:369Octominoes.png"
] | [] |
proofwiki-12878 | 1285 9-ominoes | There exist $1285$ distinct free $9$-ominoes, $37$ of which have a hole:
:600px | {{ProofWanted|Work to be done yet to establish method of creation}} | There exist $1285$ [[Definition:Distinct|distinct]] [[Definition:Free Polyomino|free]] [[Definition:Polyomino|$9$-ominoes]], $37$ of which have a hole:
:[[File:1285-9ominoes.png|600px]] | {{ProofWanted|Work to be done yet to establish method of creation}} | 1285 9-ominoes | https://proofwiki.org/wiki/1285_9-ominoes | https://proofwiki.org/wiki/1285_9-ominoes | [
"Polyominoes",
"1285"
] | [
"Definition:Distinct",
"Definition:Polyomino/Free",
"Definition:Polyomino",
"File:1285-9ominoes.png"
] | [] |
proofwiki-12879 | Mapping Assigning to Element Its Compact Closure is Order Isomorphism | Let $L = \struct {S, \vee, \preceq}$ be a bounded below algebraic join semilattice.
Let $C = \struct {\map K L, \preceq'}$ be an ordered subset of $L$
where $\map K L$ denotes the compact subset of $L$.
Let $I = \struct {\map {\mathit {Ids} } C, \precsim}$ be an inclusion ordered set
where $\map {\mathit {Ids} } C$ den... | We will prove that
:$f$ is an order embedding.
Let $x, y \in S$.
'''Sufficient condition'''
Assume that
:$x \preceq y$
By Compact Closure is Increasing:
:$x^{\mathrm {compact} } \subseteq y^{\mathrm {compact} }$
By definition of $f$:
:$\map f x \subseteq \map f y$
Thus by definition of inclusion ordered set:
:$\map f x... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Join Semilattice|join semilattice]].
Let $C = \struct {\map K L, \preceq'}$ be an [[Definition:Ordered Subset|ordered subset]] of $L$
where $\map K L$ denotes the [[Defi... | We will prove that
:$f$ is an [[Definition:Order Embedding|order embedding]].
Let $x, y \in S$.
'''Sufficient condition'''
Assume that
:$x \preceq y$
By [[Compact Closure is Increasing]]:
:$x^{\mathrm {compact} } \subseteq y^{\mathrm {compact} }$
By definition of $f$:
:$\map f x \subseteq \map f y$
Thus by defin... | Mapping Assigning to Element Its Compact Closure is Order Isomorphism | https://proofwiki.org/wiki/Mapping_Assigning_to_Element_Its_Compact_Closure_is_Order_Isomorphism | https://proofwiki.org/wiki/Mapping_Assigning_to_Element_Its_Compact_Closure_is_Order_Isomorphism | [
"Order Isomorphisms"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Join Semilattice",
"Definition:Ordered Subset",
"Definition:Compact Subset of Lattice",
"Definition:Inclusion Ordered Set",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)",
"Definition:Mapping",
"Definition... | [
"Definition:Order Embedding",
"Compact Closure is Increasing",
"Definition:Inclusion Ordered Set",
"Definition:Inclusion Ordered Set",
"Supremum of Subset",
"Definition:Algebraic Ordered Set",
"Axiom:Axiom of K-Approximation",
"Axiom:Axiom of K-Approximation",
"Definition:Surjection",
"Definition:... |
proofwiki-12880 | Maximum Length of Non-Crossing Knight's Tour | The maximum length of a non-crossing knight's tour on a standard chessboard is $35$ moves. | {{ProofWanted|Lots of background material needed first.}} | The maximum length of a non-crossing [[Definition:Knight's Tour|knight's tour]] on a standard [[Definition:Chessboard|chessboard]] is $35$ moves. | {{ProofWanted|Lots of background material needed first.}} | Maximum Length of Non-Crossing Knight's Tour | https://proofwiki.org/wiki/Maximum_Length_of_Non-Crossing_Knight's_Tour | https://proofwiki.org/wiki/Maximum_Length_of_Non-Crossing_Knight's_Tour | [
"Recreational Mathematics",
"35",
"Recreational Chess",
"Knight's Tours"
] | [
"Definition:Knight's Tour",
"Definition:Chess/Chessboard"
] | [] |
proofwiki-12881 | Prime Factors of 35, 36, 4734 and 4735 | The integers:
:$35, 4374$
have the same prime factors between them as the integers:
:$36, 4375$ | We have:
{{begin-eqn}}
{{eqn | l = 35
| r = 5 \times 7
| c =
}}
{{eqn | l = 4374
| r = 2 \times 3^7
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 36
| r = 2^2 \times 3^2
| c =
}}
{{eqn | l = 4375
| r = 5^4 \times 7
| c =
}}
{{end-eqn}}
Thus both pairs of integers can... | The [[Definition:Integer|integers]]:
:$35, 4374$
have the same [[Definition:Prime Factor|prime factors]] between them as the [[Definition:Integer|integers]]:
:$36, 4375$ | We have:
{{begin-eqn}}
{{eqn | l = 35
| r = 5 \times 7
| c =
}}
{{eqn | l = 4374
| r = 2 \times 3^7
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 36
| r = 2^2 \times 3^2
| c =
}}
{{eqn | l = 4375
| r = 5^4 \times 7
| c =
}}
{{end-eqn}}
Thus both pairs of integer... | Prime Factors of 35, 36, 4734 and 4735 | https://proofwiki.org/wiki/Prime_Factors_of_35,_36,_4734_and_4735 | https://proofwiki.org/wiki/Prime_Factors_of_35,_36,_4734_and_4735 | [
"Factorization",
"35",
"36",
"4374",
"4375"
] | [
"Definition:Integer",
"Definition:Prime Factor",
"Definition:Integer"
] | [
"Definition:Prime Factor"
] |
proofwiki-12882 | Equivalence Class of Fixed Element | Let $S_n$ denote the symmetric group on $n$ letters.
Let $\sigma \in S_n$.
Let $\RR_\sigma$ denote the equivalence defined in Permutation Induces Equivalence Relation.
Let $i \in \N^*_{\le n}$.
Then:
:$i \in \Fix \sigma$ {{iff}} $\eqclass i {\RR_\sigma} = \set i$
where:
:$\eqclass i {\RR_\sigma}$ denotes the equivalenc... | By the definition of an equivalence relation it is easily seen that $\set i \subseteq \eqclass i {\RR_\sigma}$.
Suppose that $i \in \Fix \sigma$.
Let $j \in \eqclass i {\RR_\sigma}$.
Then by Condition for Membership of Equivalence Class and Permutation Induces Equivalence Relation:
:$j \in \eqclass i {\RR_\sigma} \iff ... | Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $\sigma \in S_n$.
Let $\RR_\sigma$ denote the [[Definition:Equivalence Relation Induced by Mapping|equivalence]] defined in [[Permutation Induces Equivalence Relation]].
Let $i \in \N^*_{\le n}$.
Then:
:$i \in \Fix... | By the definition of an [[Definition:Equivalence Relation|equivalence relation]] it is easily seen that $\set i \subseteq \eqclass i {\RR_\sigma}$.
Suppose that $i \in \Fix \sigma$.
Let $j \in \eqclass i {\RR_\sigma}$.
Then by [[Condition for Membership of Equivalence Class]] and [[Permutation Induces Equivalence ... | Equivalence Class of Fixed Element | https://proofwiki.org/wiki/Equivalence_Class_of_Fixed_Element | https://proofwiki.org/wiki/Equivalence_Class_of_Fixed_Element | [
"Symmetric Groups",
"Equivalence Classes"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Equivalence Relation Induced by Mapping",
"Permutation Induces Equivalence Relation",
"Definition:Equivalence Class",
"Definition:Fixed Element under Permutation/Set of Fixed Elements"
] | [
"Definition:Equivalence Relation",
"Condition for Membership of Equivalence Class",
"Permutation Induces Equivalence Relation",
"Fixed Point of Permutation is Fixed Point of Power",
"Category:Symmetric Groups",
"Category:Equivalence Classes"
] |
proofwiki-12883 | Number of Odd Entries in Row of Pascal's Triangle is Power of 2 | The number of odd entries in a row of Pascal's triangle is a power of $2$. | Let $n, k \in \Z$.
Let the representations of $n$ and $k$ to the base $p$ be given by:
:$n = 2^r a_r + \cdots + 2 a_1 + a_0$
:$k = 2^r b_r + \cdots + 2 b_1 + b_0$
By {{Corollary|Lucas' Theorem}}:
:$\ds \dbinom n k \equiv \prod_{j \mathop = 0}^r \dbinom {a_j} {b_j} \pmod 2$
By definition, $a_j$ and $b_j$ are either $0$ ... | The number of [[Definition:Odd Integer|odd]] entries in a [[Definition:Row of Pascal's Triangle|row]] of [[Definition:Pascal's Triangle|Pascal's triangle]] is a [[Definition:Integer Power|power]] of $2$. | Let $n, k \in \Z$.
Let the [[Definition:Number Base|representations of $n$ and $k$ to the base $p$]] be given by:
:$n = 2^r a_r + \cdots + 2 a_1 + a_0$
:$k = 2^r b_r + \cdots + 2 b_1 + b_0$
By {{Corollary|Lucas' Theorem}}:
:$\ds \dbinom n k \equiv \prod_{j \mathop = 0}^r \dbinom {a_j} {b_j} \pmod 2$
By definition,... | Number of Odd Entries in Row of Pascal's Triangle is Power of 2 | https://proofwiki.org/wiki/Number_of_Odd_Entries_in_Row_of_Pascal's_Triangle_is_Power_of_2 | https://proofwiki.org/wiki/Number_of_Odd_Entries_in_Row_of_Pascal's_Triangle_is_Power_of_2 | [
"Pascal's Triangle"
] | [
"Definition:Odd Integer",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle",
"Definition:Power (Algebra)/Integer"
] | [
"Definition:Number Base",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Product Rule for Counting",
"Definition:Odd Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Odd Integer",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle",
"Defi... |
proofwiki-12884 | Compact Closure is Increasing | Let $\left({S, \preceq}\right)$ be an ordered set.
Let $x, y \in S$ such that
:$x \preceq y$
Then $x^{\mathrm{compact} } \subseteq y^{\mathrm{compact} }$
where $x^{\mathrm{compact} }$ denotes the compact closure of $x$. | Let $z \in x^{\mathrm{compact} }$
By definition of compact closure:
:$z$ is a compact element and $z \preceq x$
By definition of transitivity:
:$z \preceq y$
Thus by definition of compact closure:
:$z \in y^{\mathrm{compact} }$
{{qed}} | Let $\left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y \in S$ such that
:$x \preceq y$
Then $x^{\mathrm{compact} } \subseteq y^{\mathrm{compact} }$
where $x^{\mathrm{compact} }$ denotes the [[Definition:Compact Closure|compact closure]] of $x$. | Let $z \in x^{\mathrm{compact} }$
By definition of [[Definition:Compact Closure|compact closure]]:
:$z$ is a [[Definition:Compact Element|compact element]] and $z \preceq x$
By definition of [[Definition:Transitivity|transitivity]]:
:$z \preceq y$
Thus by definition of [[Definition:Compact Closure|compact closure]]:... | Compact Closure is Increasing | https://proofwiki.org/wiki/Compact_Closure_is_Increasing | https://proofwiki.org/wiki/Compact_Closure_is_Increasing | [
"Way Below Relation"
] | [
"Definition:Ordered Set",
"Definition:Compact Closure"
] | [
"Definition:Compact Closure",
"Definition:Compact Element",
"Definition:Transitive",
"Definition:Compact Closure"
] |
proofwiki-12885 | Sufficient Conditions for Weak Extremum | Let $J$ be a functional such that:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
:$\map y a = A$
:$\map y b = B$
Let $y = \map y x$ be an extremum.
Let the strengthened Legendre's Condition hold.
Let the strengthened Jacobi's Necessary Condition hold.
{{explain|specific links to those strengthened versions}}
The... | By the continuity of function $\map P x$ and the solution of Jacobi's equation:
:$\exists \epsilon > 0: \paren {\forall x \in \closedint a {b + \epsilon}:\map P x > 0} \land \paren {\tilde a \notin \closedint a {b + \epsilon} }$
Consider the quadratic functional:
:$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x - \alpha^2 ... | Let $J$ be a [[Definition:Real Functional|functional]] such that:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
:$\map y a = A$
:$\map y b = B$
Let $y = \map y x$ be an [[Definition:Extremum of Functional|extremum]].
Let the strengthened [[Legendre's Condition]] hold.
Let the strengthened [[Jacobi's Necessar... | By the [[Definition:Continuous Real Function on Interval|continuity]] of [[Definition:Function|function]] $\map P x$ and the [[Definition:Solution to Differential Equation|solution]] of [[Definition:Jacobi's Equation of Functional|Jacobi's equation]]:
:$\exists \epsilon > 0: \paren {\forall x \in \closedint a {b + \ep... | Sufficient Conditions for Weak Extremum | https://proofwiki.org/wiki/Sufficient_Conditions_for_Weak_Extremum | https://proofwiki.org/wiki/Sufficient_Conditions_for_Weak_Extremum | [
"Calculus of Variations"
] | [
"Definition:Functional/Real",
"Definition:Extremum/Functional",
"Legendre's Condition",
"Jacobi's Necessary Condition",
"Definition:Functional/Real",
"Definition:Weak Minimum"
] | [
"Definition:Continuous Real Function/Interval",
"Definition:Function",
"Definition:Differential Equation/Solution",
"Definition:Jacobi's Equation of Functional",
"Definition:Quadratic Functional",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Euler's Equation for Vanishing Variation... |
proofwiki-12886 | Sum of Entries in Row of Pascal's Triangle | The sum of all the entries in the $n$th row of Pascal's triangle is equal to $2^n$. | By definition, the entries in $n$th row of Pascal's triangle are exactly the binomial coefficients:
:$\dbinom n 0, \dbinom n 1, \ldots, \dbinom n n$
The result follows from Sum of Binomial Coefficients over Lower Index.
{{qed}} | The [[Definition:Integer Addition|sum]] of all the entries in the [[Definition:Row of Pascal's Triangle|$n$th row]] of [[Definition:Pascal's Triangle|Pascal's triangle]] is equal to $2^n$. | By definition, the entries in [[Definition:Row of Pascal's Triangle|$n$th row]] of [[Definition:Pascal's Triangle|Pascal's triangle]] are exactly the [[Definition:Binomial Coefficient|binomial coefficients]]:
:$\dbinom n 0, \dbinom n 1, \ldots, \dbinom n n$
The result follows from [[Sum of Binomial Coefficients over L... | Sum of Entries in Row of Pascal's Triangle/Proof 1 | https://proofwiki.org/wiki/Sum_of_Entries_in_Row_of_Pascal's_Triangle | https://proofwiki.org/wiki/Sum_of_Entries_in_Row_of_Pascal's_Triangle/Proof_1 | [
"Sum of Entries in Row of Pascal's Triangle",
"Pascal's Triangle"
] | [
"Definition:Addition/Integers",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle"
] | [
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle",
"Definition:Binomial Coefficient",
"Sum of Binomial Coefficients over Lower Index"
] |
proofwiki-12887 | Sum of Entries in Row of Pascal's Triangle | The sum of all the entries in the $n$th row of Pascal's triangle is equal to $2^n$. | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:The sum of all the entries in the $n$th row of Pascal's triangle is equal to $2^n$.
=== Basis for the Induction ===
$\map P 0$ is the case:
:The sum of all the entries in the row $0$ of Pascal's triangle is equal to $2^0 = ... | The [[Definition:Integer Addition|sum]] of all the entries in the [[Definition:Row of Pascal's Triangle|$n$th row]] of [[Definition:Pascal's Triangle|Pascal's triangle]] is equal to $2^n$. | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:The [[Definition:Integer Addition|sum]] of all the entries in the [[Definition:Row of Pascal's Triangle|$n$th row]] of [[Definition:Pascal's Triangle|Pasca... | Sum of Entries in Row of Pascal's Triangle/Proof 2 | https://proofwiki.org/wiki/Sum_of_Entries_in_Row_of_Pascal's_Triangle | https://proofwiki.org/wiki/Sum_of_Entries_in_Row_of_Pascal's_Triangle/Proof_2 | [
"Sum of Entries in Row of Pascal's Triangle",
"Pascal's Triangle"
] | [
"Definition:Addition/Integers",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Addition/Integers",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle",
"Definition:Addition/Integers",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle",
"Definition:Zero (Number)",
"Defi... |
proofwiki-12888 | Columns of Pascal's Triangle contain Simplicial Polytopic Numbers | The columns of Pascal's triangle contain the simplicial polytopic numbers:
: Column $0$: repeated instances of number $1$
: Column $1$: the (strictly) positive integers
: Column $2$: the triangular numbers
: Column $3$: the tetrahedral numbers
: Column $4$: the pentatope numbers
and so on. | {{ProofWanted|Need to define the simplicial polytopic numbers and demonstrate that the $n$th simplicial polytopic number of dimension $m$ is $\dbinom m n$ or whatever the formula is.}} | The [[Definition:Column of Pascal's Triangle|columns of Pascal's triangle]] contain the [[Definition:Simplicial Polytopic Number|simplicial polytopic numbers]]:
: [[Definition:Column of Pascal's Triangle|Column $0$]]: repeated instances of [[1|number $1$]]
: [[Definition:Column of Pascal's Triangle|Column $1$]]: the ... | {{ProofWanted|Need to define the [[Definition:Simplicial Polytopic Number|simplicial polytopic numbers]] and demonstrate that the $n$th [[Definition:Simplicial Polytopic Number|simplicial polytopic number]] of dimension $m$ is $\dbinom m n$ or whatever the formula is.}} | Columns of Pascal's Triangle contain Simplicial Polytopic Numbers | https://proofwiki.org/wiki/Columns_of_Pascal's_Triangle_contain_Simplicial_Polytopic_Numbers | https://proofwiki.org/wiki/Columns_of_Pascal's_Triangle_contain_Simplicial_Polytopic_Numbers | [
"Pascal's Triangle",
"Simplicial Polytopic Numbers"
] | [
"Definition:Pascal's Triangle/Column",
"Definition:Simplicial Polytopic Number",
"Definition:Pascal's Triangle/Column",
"1",
"Definition:Pascal's Triangle/Column",
"Definition:Strictly Positive/Integer",
"Definition:Pascal's Triangle/Column",
"Definition:Triangular Number",
"Definition:Pascal's Tria... | [
"Definition:Simplicial Polytopic Number",
"Definition:Simplicial Polytopic Number"
] |
proofwiki-12889 | Equivalence Class of Fixed Element/Corollary | :$i \notin \Fix \sigma$ {{iff}} $\eqclass i {\RR_\sigma}$ contains more than one element | From Equivalence Class of Fixed Element and Biconditional Equivalent to Biconditional of Negations:
:$i \notin \Fix \sigma \iff \set i \ne \eqclass i {\RR_\sigma}$
Because the Biconditional is Transitive, it suffices to show that:
:$\set i \ne \eqclass i {\RR_\sigma}$
{{iff}}:
:$\eqclass i {\RR_\sigma}$ contains more t... | :$i \notin \Fix \sigma$ {{iff}} $\eqclass i {\RR_\sigma}$ contains more than one [[Definition:Element|element]] | From [[Equivalence Class of Fixed Element]] and [[Biconditional Equivalent to Biconditional of Negations]]:
:$i \notin \Fix \sigma \iff \set i \ne \eqclass i {\RR_\sigma}$
Because the [[Biconditional is Transitive]], it suffices to show that:
:$\set i \ne \eqclass i {\RR_\sigma}$
{{iff}}:
:$\eqclass i {\RR_\sigma}$ ... | Equivalence Class of Fixed Element/Corollary | https://proofwiki.org/wiki/Equivalence_Class_of_Fixed_Element/Corollary | https://proofwiki.org/wiki/Equivalence_Class_of_Fixed_Element/Corollary | [
"Symmetric Groups",
"Equivalence Classes"
] | [
"Definition:Element"
] | [
"Equivalence Class of Fixed Element",
"Biconditional Equivalent to Biconditional of Negations",
"Biconditional is Transitive",
"Definition:Element",
"Definition:Equivalence Relation",
"Definition:Element",
"Definition:Element",
"Category:Symmetric Groups",
"Category:Equivalence Classes"
] |
proofwiki-12890 | Sum of Entries in Lesser Diagonal of Pascal's Triangle equal Fibonacci Number | The sum of the entries in the $n$th lesser diagonal of Pascal's triangle equals the $n + 1$th Fibonacci number. | By definition, the entries in the $n$th lesser diagonal of Pascal's triangle are:
:$\dbinom n 0, \dbinom {n - 1} 1, \dbinom {n - 2} 2, \dbinom {n - 3} 3, \ldots$
and so the statement can be written:
:$F_{n + 1} = \ds \sum_{k \mathop \ge 0} \dbinom {n - k} k$
The proof proceeds by strong induction.
For all $n \in \Z_{>0... | The [[Definition:Integer Addition|sum]] of the entries in the $n$th [[Definition:Lesser Diagonal of Pascal's Triangle|lesser diagonal]] of [[Definition:Pascal's Triangle|Pascal's triangle]] equals the $n + 1$th [[Definition:Fibonacci Number|Fibonacci number]]. | By definition, the entries in the $n$th [[Definition:Lesser Diagonal of Pascal's Triangle|lesser diagonal]] of [[Definition:Pascal's Triangle|Pascal's triangle]] are:
:$\dbinom n 0, \dbinom {n - 1} 1, \dbinom {n - 2} 2, \dbinom {n - 3} 3, \ldots$
and so the statement can be written:
:$F_{n + 1} = \ds \sum_{k \mathop \... | Sum of Entries in Lesser Diagonal of Pascal's Triangle equal Fibonacci Number | https://proofwiki.org/wiki/Sum_of_Entries_in_Lesser_Diagonal_of_Pascal's_Triangle_equal_Fibonacci_Number | https://proofwiki.org/wiki/Sum_of_Entries_in_Lesser_Diagonal_of_Pascal's_Triangle_equal_Fibonacci_Number | [
"Pascal's Triangle",
"Fibonacci Numbers"
] | [
"Definition:Addition/Integers",
"Definition:Pascal's Triangle/Lesser Diagonal",
"Definition:Pascal's Triangle",
"Definition:Fibonacci Number"
] | [
"Definition:Pascal's Triangle/Lesser Diagonal",
"Definition:Pascal's Triangle",
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-12891 | Extension of Infima Preserving Mapping to Complete Lattice Preserves Infima | Let $L_1 = \struct {S_1, \preceq_1}$, $L_2 = \struct {S_2, \preceq_2}$ be ordered sets.
Let $L_3 = \struct {S_3, \preceq_3}$ be a complete lattice.
Suppose that.
:$L_2$ is an infima inheriting ordered subset of $L_3$.
Let $f: S_1 \to S_2$ be a mapping such that:
:$f$ preserves infima.
Then
$f: S_1 \to S_3$ preserves in... | By definition of ordered subset:
:$S_2 \subseteq S_3$
Then define $g = f:S_1 \to S_3$
Let $X$ be a subset of $S_1$ such that
:$X$ admits a infimum in $L_1$.
Thus by definition of complete lattice:
:$g \sqbrk X$ admits a infimum in $L_3$.
By definition of mapping preserves infima:
:$f \sqbrk X$ admits a infimum in $L_2$... | Let $L_1 = \struct {S_1, \preceq_1}$, $L_2 = \struct {S_2, \preceq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $L_3 = \struct {S_3, \preceq_3}$ be a [[Definition:Complete Lattice|complete lattice]].
Suppose that.
:$L_2$ is an [[Definition:Infima Inheriting|infima inheriting]] [[Definition:Ordered Subset|orde... | By definition of [[Definition:Ordered Subset|ordered subset]]:
:$S_2 \subseteq S_3$
Then define $g = f:S_1 \to S_3$
Let $X$ be a [[Definition:Subset|subset]] of $S_1$ such that
:$X$ admits a [[Definition:Infimum of Set|infimum]] in $L_1$.
Thus by definition of [[Definition:Complete Lattice|complete lattice]]:
:$g \s... | Extension of Infima Preserving Mapping to Complete Lattice Preserves Infima | https://proofwiki.org/wiki/Extension_of_Infima_Preserving_Mapping_to_Complete_Lattice_Preserves_Infima | https://proofwiki.org/wiki/Extension_of_Infima_Preserving_Mapping_to_Complete_Lattice_Preserves_Infima | [
"Order Theory",
"Complete Lattices"
] | [
"Definition:Ordered Set",
"Definition:Complete Lattice",
"Definition:Infima Inheriting",
"Definition:Ordered Subset",
"Definition:Mapping",
"Definition:Mapping Preserves Infimum/All",
"Definition:Mapping Preserves Infimum/All"
] | [
"Definition:Ordered Subset",
"Definition:Subset",
"Definition:Infimum of Set",
"Definition:Complete Lattice",
"Definition:Infimum of Set",
"Definition:Mapping Preserves Infimum/All",
"Definition:Infimum of Set",
"Definition:Infima Inheriting"
] |
proofwiki-12892 | Extension of Directed Suprema Preserving Mapping to Complete Lattice Preserves Directed Suprema | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be ordered sets.
Let $L_3 = \struct {S_3, \preceq_3}$ be a complete lattice.
Suppose that
:$L_2$ is directed suprema inheriting ordered subset of $L_3$.
Let $f:S_1 \to S_2$ be a mapping such that
:$f$ preserves directed suprema.
Then
$f:S_1 \to S... | By definition of ordered subset:
:$S_2 \subseteq S_3$
Then define $g = f: S_1 \to S_3$
Let $X$ be a directed subset of $S_1$ such that
:$X$ admits a supremum in $L_1$.
Thus by definition of complete lattice:
:$g \sqbrk X$ admits a supremum in $L_3$.
By definition of mapping preserves directed suprema:
:$f \sqbrk X$ adm... | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $L_3 = \struct {S_3, \preceq_3}$ be a [[Definition:Complete Lattice|complete lattice]].
Suppose that
:$L_2$ is [[Definition:Directed Suprema Inheriting|directed suprema inheriting]] [[Definition:O... | By definition of [[Definition:Ordered Subset|ordered subset]]:
:$S_2 \subseteq S_3$
Then define $g = f: S_1 \to S_3$
Let $X$ be a [[Definition:Directed Subset|directed subset]] of $S_1$ such that
:$X$ admits a [[Definition:Supremum of Set|supremum]] in $L_1$.
Thus by definition of [[Definition:Complete Lattice|compl... | Extension of Directed Suprema Preserving Mapping to Complete Lattice Preserves Directed Suprema | https://proofwiki.org/wiki/Extension_of_Directed_Suprema_Preserving_Mapping_to_Complete_Lattice_Preserves_Directed_Suprema | https://proofwiki.org/wiki/Extension_of_Directed_Suprema_Preserving_Mapping_to_Complete_Lattice_Preserves_Directed_Suprema | [
"Order Theory",
"Complete Lattices"
] | [
"Definition:Ordered Set",
"Definition:Complete Lattice",
"Definition:Directed Suprema Inheriting",
"Definition:Ordered Subset",
"Definition:Mapping",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Mapping Preserves Supremum/Directed"
] | [
"Definition:Ordered Subset",
"Definition:Directed Subset",
"Definition:Supremum of Set",
"Definition:Complete Lattice",
"Definition:Supremum of Set",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Supremum of Set",
"Directed Suprema Preserving Mapping is Increasing",
"Definition:Incre... |
proofwiki-12893 | Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence | There are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence. | Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in an arithmetic sequence.
Then:
{{begin-eqn}}
{{eqn | l = \dbinom n {k + 2} - \dbinom n {k + 1}
| r = \dbinom n {k + 1} - \dbinom n k
| c = {{Defof|Arithmetic Sequence}}
}}
{{eqn | l = \frac {n!} {\paren {n - k - 2}! \paren {k + 2}!} - ... | There are an [[Definition:Infinite Set|infinite number]] of [[Definition:Row of Pascal's Triangle|rows]] of [[Definition:Pascal's Triangle|Pascal's triangle]] which contain $3$ [[Definition:Integer|integers]] in [[Definition:Arithmetic Sequence|arithmetic sequence]]. | Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in an [[Definition:Arithmetic Sequence|arithmetic sequence]].
Then:
{{begin-eqn}}
{{eqn | l = \dbinom n {k + 2} - \dbinom n {k + 1}
| r = \dbinom n {k + 1} - \dbinom n k
| c = {{Defof|Arithmetic Sequence}}
}}
{{eqn | l = \frac {n!} {\p... | Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence | https://proofwiki.org/wiki/Rows_in_Pascal's_Triangle_containing_Numbers_in_Arithmetic_Sequence | https://proofwiki.org/wiki/Rows_in_Pascal's_Triangle_containing_Numbers_in_Arithmetic_Sequence | [
"Pascal's Triangle",
"Arithmetic Sequences",
"Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence"
] | [
"Definition:Infinite Set",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle",
"Definition:Integer",
"Definition:Arithmetic Sequence"
] | [
"Definition:Arithmetic Sequence",
"Solution to Quadratic Equation",
"Definition:Rational Number",
"Definition:Square Number",
"Definition:Odd Integer",
"Definition:Square Number",
"Definition:Odd Integer",
"Definition:Infinite Set",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle... |
proofwiki-12894 | Rows in Pascal's Triangle containing Numbers in Geometric Sequence | There exist no rows of Pascal's triangle which contain $3$ integers in geometric sequence. | Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in a geometric sequence.
Then:
{{begin-eqn}}
{{eqn | l = \dbinom n {k + 2} / \dbinom n {k + 1}
| r = \dbinom n {k + 1} / \dbinom n k
| c = {{Defof|Geometric Sequence}}
}}
{{eqn | l = \paren {\frac {n!} {\paren {n - k - 2}! \paren {k + 2}... | There exist no [[Definition:Row of Pascal's Triangle|rows]] of [[Definition:Pascal's Triangle|Pascal's triangle]] which contain $3$ [[Definition:Integer|integers]] in [[Definition:Geometric Sequence|geometric sequence]]. | Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in a [[Definition:Geometric Sequence|geometric sequence]].
Then:
{{begin-eqn}}
{{eqn | l = \dbinom n {k + 2} / \dbinom n {k + 1}
| r = \dbinom n {k + 1} / \dbinom n k
| c = {{Defof|Geometric Sequence}}
}}
{{eqn | l = \paren {\frac {n!}... | Rows in Pascal's Triangle containing Numbers in Geometric Sequence | https://proofwiki.org/wiki/Rows_in_Pascal's_Triangle_containing_Numbers_in_Geometric_Sequence | https://proofwiki.org/wiki/Rows_in_Pascal's_Triangle_containing_Numbers_in_Geometric_Sequence | [
"Pascal's Triangle",
"Geometric Sequences"
] | [
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle",
"Definition:Integer",
"Definition:Geometric Sequence"
] | [
"Definition:Geometric Sequence",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle",
"Definition:Integer",
"Definition:Geometric Sequence",
"Definition:Binomial Coefficient",
"Negated Upper Index of Binomial Coefficient",
"Definition:Geometric Sequence"
] |
proofwiki-12895 | Rows in Pascal's Triangle containing Numbers in Harmonic Sequence | There exist no rows of Pascal's triangle which contain $3$ integers in harmonic sequence. | Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in a harmonic sequence.
Then:
{{begin-eqn}}
{{eqn | l = \dbinom n {k + 2}^{-1} - \dbinom n {k + 1}^{-1}
| r = \dbinom n {k + 1}^{-1} - \dbinom n k^{-1}
| c = {{Defof|Harmonic Sequence}}
}}
{{eqn | l = \frac {\paren {n - k - 2}! \paren {k... | There exist no [[Definition:Row of Pascal's Triangle|rows]] of [[Definition:Pascal's Triangle|Pascal's triangle]] which contain $3$ [[Definition:Integer|integers]] in [[Definition:Harmonic Sequence|harmonic sequence]]. | Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in a [[Definition:Harmonic Sequence|harmonic sequence]].
Then:
{{begin-eqn}}
{{eqn | l = \dbinom n {k + 2}^{-1} - \dbinom n {k + 1}^{-1}
| r = \dbinom n {k + 1}^{-1} - \dbinom n k^{-1}
| c = {{Defof|Harmonic Sequence}}
}}
{{eqn | l = \... | Rows in Pascal's Triangle containing Numbers in Harmonic Sequence | https://proofwiki.org/wiki/Rows_in_Pascal's_Triangle_containing_Numbers_in_Harmonic_Sequence | https://proofwiki.org/wiki/Rows_in_Pascal's_Triangle_containing_Numbers_in_Harmonic_Sequence | [
"Pascal's Triangle",
"Harmonic Sequences"
] | [
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle",
"Definition:Integer",
"Definition:Harmonic Sequence"
] | [
"Definition:Harmonic Sequence",
"Definition:Quadratic Equation",
"Definition:Discriminant of Polynomial/Quadratic Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle",
"Definition:Integer",
"Definition:Harmonic Sequence"
] |
proofwiki-12896 | Reciprocal as Summation of Binomial Coefficients of Reciprocals | :$\forall n \in \Z_{>0}: \dfrac 1 n = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \dbinom {n - 1} k \dfrac 1 {k + 1}$
where $\dbinom {n - 1} k$ denotes a binomial coefficient.
That is, for example:
{{begin-eqn}}
{{eqn | l = \dfrac 1 1
| r = 1
}}
{{eqn | l = \dfrac 1 2
| r = 1 - \dfrac 1 2
}}
{{eqn | l = ... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \dbinom {n - 1} k \dfrac 1 {k + 1}
| r = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \binom n {k + 1}
| c = Factors of Binomial Coefficient
}}
{{eqn | r = \frac 1 n \sum_{k \mathop = 1}^n \paren {-1}^{k + 1} \binom n k
| c =... | :$\forall n \in \Z_{>0}: \dfrac 1 n = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \dbinom {n - 1} k \dfrac 1 {k + 1}$
where $\dbinom {n - 1} k$ denotes a [[Definition:Binomial Coefficient|binomial coefficient]].
That is, for example:
{{begin-eqn}}
{{eqn | l = \dfrac 1 1
| r = 1
}}
{{eqn | l = \dfrac 1 2
... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \dbinom {n - 1} k \dfrac 1 {k + 1}
| r = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \binom n {k + 1}
| c = [[Factors of Binomial Coefficient]]
}}
{{eqn | r = \frac 1 n \sum_{k \mathop = 1}^n \paren {-1}^{k + 1} \binom n k
|... | Reciprocal as Summation of Binomial Coefficients of Reciprocals | https://proofwiki.org/wiki/Reciprocal_as_Summation_of_Binomial_Coefficients_of_Reciprocals | https://proofwiki.org/wiki/Reciprocal_as_Summation_of_Binomial_Coefficients_of_Reciprocals | [
"Binomial Coefficients",
"Reciprocals"
] | [
"Definition:Binomial Coefficient"
] | [
"Factors of Binomial Coefficient",
"Translation of Index Variable of Summation",
"Binomial Theorem"
] |
proofwiki-12897 | Element of Leibniz Harmonic Triangle is Sum of Numbers Below | The elements in the Leibniz harmonic triangle are the sum of the elements immediately below them.
{{refactor|Rework this as another definition of LHT, establishing that column and diagonal $0$ are defined as the reciprocals.|level = medium}} | By definition of Leibniz harmonic triangle, element $\tuple {n, m}$ is:
:$\dfrac 1 {\paren {n + 1} \binom n m}$
Thus we have:
{{begin-eqn}}
{{eqn | r = \dfrac 1 {\paren {n + 2} \binom {n + 1} m} + \dfrac 1 {\paren {n + 2} \binom {n + 1} {m + 1} }
| o =
| c = Elements of Leibniz Harmonic Triangle immediatel... | The elements in the [[Definition:Leibniz Harmonic Triangle|Leibniz harmonic triangle]] are the [[Definition:Rational Addition|sum]] of the elements immediately below them.
{{refactor|Rework this as another definition of LHT, establishing that column and diagonal $0$ are defined as the reciprocals.|level = medium}} | By definition of [[Definition:Leibniz Harmonic Triangle|Leibniz harmonic triangle]], element $\tuple {n, m}$ is:
:$\dfrac 1 {\paren {n + 1} \binom n m}$
Thus we have:
{{begin-eqn}}
{{eqn | r = \dfrac 1 {\paren {n + 2} \binom {n + 1} m} + \dfrac 1 {\paren {n + 2} \binom {n + 1} {m + 1} }
| o =
| c = Eleme... | Element of Leibniz Harmonic Triangle is Sum of Numbers Below | https://proofwiki.org/wiki/Element_of_Leibniz_Harmonic_Triangle_is_Sum_of_Numbers_Below | https://proofwiki.org/wiki/Element_of_Leibniz_Harmonic_Triangle_is_Sum_of_Numbers_Below | [
"Leibniz Harmonic Triangle"
] | [
"Definition:Leibniz Harmonic Triangle",
"Definition:Addition/Rational Numbers"
] | [
"Definition:Leibniz Harmonic Triangle",
"Definition:Leibniz Harmonic Triangle",
"Pascal's Rule"
] |
proofwiki-12898 | Complement of Irreducible Topological Subset is Prime Element | Let $T = \struct {S, \tau}$ be a topological space.
Let $X$ be an irreducible subset of $S$ such that:
:$\relcomp S X \in \tau$
Let $L = \struct {\tau, \preceq}$ be an inclusion ordered set of topology $\tau$.
Then $\relcomp S X$ is prime element in $L$. | {{tidy|This proof lends itself to be rendered neatly by means of the {{TL|eqn}} template}}
Let $Y, Z \in \tau$ such that
:$Y \wedge Z \preceq \relcomp S X$
By definition of topological space:
:$Y \cap Z \in \tau$
By Meet in Inclusion Ordered Set:
:$Y \cap Z = Y \wedge Z$
By definition of inclusion ordered set:
:$Y \cap... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $X$ be an [[Definition:Irreducible Space|irreducible]] [[Definition:Subset|subset]] of $S$ such that:
:$\relcomp S X \in \tau$
Let $L = \struct {\tau, \preceq}$ be an [[Definition:Inclusion Ordered Set|inclusion ordered set]] of ... | {{tidy|This proof lends itself to be rendered neatly by means of the {{TL|eqn}} template}}
Let $Y, Z \in \tau$ such that
:$Y \wedge Z \preceq \relcomp S X$
By definition of [[Definition:Topological Space|topological space]]:
:$Y \cap Z \in \tau$
By [[Meet in Inclusion Ordered Set]]:
:$Y \cap Z = Y \wedge Z$
By defi... | Complement of Irreducible Topological Subset is Prime Element | https://proofwiki.org/wiki/Complement_of_Irreducible_Topological_Subset_is_Prime_Element | https://proofwiki.org/wiki/Complement_of_Irreducible_Topological_Subset_is_Prime_Element | [
"Prime Elements",
"Irreducible Spaces"
] | [
"Definition:Topological Space",
"Definition:Irreducible Space",
"Definition:Subset",
"Definition:Inclusion Ordered Set",
"Definition:Prime Element (Order Theory)"
] | [
"Definition:Topological Space",
"Meet in Inclusion Ordered Set",
"Definition:Inclusion Ordered Set",
"Relative Complement inverts Subsets",
"Relative Complement of Relative Complement",
"De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection",
"Intersection with Subset is Subset",
... |
proofwiki-12899 | Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below | Consider the Leibniz harmonic triangle:
{{:Definition:Leibniz Harmonic Triangle}}
Let $\tuple {n, m}$ be the element in the $n$th row and $m$th column.
Then:
:$\tuple {n, m} = \ds \sum_{k \mathop \ge 0} \tuple {n + 1 + k, m + k}$ | Taking $r \to \infty$ in Lemma 2:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \tuple {n + 1 + k, m + k}
| r = \sum_{k \mathop \ge 1} \tuple {n + k, m + k - 1}
| c = Translation of Index Variable of Summation
}}
{{eqn | r = \lim_{r \to \infty} \paren {\tuple {n, m} - \tuple {n + r, m + r} }
| c = ... | Consider the [[Definition:Leibniz Harmonic Triangle|Leibniz harmonic triangle]]:
{{:Definition:Leibniz Harmonic Triangle}}
Let $\tuple {n, m}$ be the element in the $n$th [[Definition:Row of Leibniz Harmonic Triangle|row]] and $m$th [[Definition:Column of Leibniz Harmonic Triangle|column]].
Then:
:$\tuple {n, m} = \d... | Taking $r \to \infty$ in [[Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below/Lemma 2|Lemma 2]]:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \tuple {n + 1 + k, m + k}
| r = \sum_{k \mathop \ge 1} \tuple {n + k, m + k - 1}
| c = [[Translation of Index Variable of Summation]]
... | Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below | https://proofwiki.org/wiki/Element_of_Leibniz_Harmonic_Triangle_as_Sum_of_Elements_on_Diagonal_from_Below | https://proofwiki.org/wiki/Element_of_Leibniz_Harmonic_Triangle_as_Sum_of_Elements_on_Diagonal_from_Below | [
"Leibniz Harmonic Triangle",
"Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below"
] | [
"Definition:Leibniz Harmonic Triangle",
"Definition:Leibniz Harmonic Triangle/Row",
"Definition:Leibniz Harmonic Triangle/Column"
] | [
"Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below/Lemma 2",
"Translation of Index Variable of Summation",
"Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below/Lemma 2"
] |
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