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-13500 | Lower and Upper Bounds for Sequences/Warning | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $x_n \to l$ as $n \to \infty$.
Then it is '''not''' necessarily the case that:
:$(1): \quad \forall n \in \N: x_n > a \implies l > a$
:$(2): \quad \forall n \in \N: x_n < b \implies l < b$ | Take the examples:
:$(1): \quad \sequence {x_n} = \dfrac 1 n$
:$(2): \quad \sequence {y_n} = -\dfrac 1 n$
Then :
:$\forall n \in \N_{>0}: \dfrac 1 n > 0, -\dfrac 1 n < 0$
From Sequence of Reciprocals is Null Sequence, we have
:$x_n \to 0$
:$y_n \to 0$
as $n \to \infty$.
However, it is clearly '''false''' that $0 > 0$ a... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $x_n \to l$ as $n \to \infty$.
Then it is '''not''' necessarily the case that:
:$(1): \quad \forall n \in \N: x_n > a \implies l > a$
:$(2): \quad \forall n \in \N: x_n < b \implies l < b$ | Take the examples:
:$(1): \quad \sequence {x_n} = \dfrac 1 n$
:$(2): \quad \sequence {y_n} = -\dfrac 1 n$
Then :
:$\forall n \in \N_{>0}: \dfrac 1 n > 0, -\dfrac 1 n < 0$
From [[Sequence of Reciprocals is Null Sequence]], we have
:$x_n \to 0$
:$y_n \to 0$
as $n \to \infty$.
However, it is clearly '''false''' that ... | Lower and Upper Bounds for Sequences/Warning | https://proofwiki.org/wiki/Lower_and_Upper_Bounds_for_Sequences/Warning | https://proofwiki.org/wiki/Lower_and_Upper_Bounds_for_Sequences/Warning | [
"Limits of Sequences"
] | [
"Definition:Real Sequence"
] | [
"Sequence of Powers of Reciprocals is Null Sequence/Corollary"
] |
proofwiki-13501 | Squares of 23...3 | The following pattern holds:
{{begin-eqn}}
{{eqn | l = 3^2
| r = 9
}}
{{eqn | l = 23^2
| r = 529
}}
{{eqn | l = 233^2
| r = 54 \, 289
}}
{{eqn | l = 2333^2
| r = 5 \, 442 \, 889
}}
{{eqn | l = 23333^2
| r = 544 \, 428 \, 889
}}
{{end-eqn}}
and so on. | {{ProofWanted|Simple but tedious.}} | The following pattern holds:
{{begin-eqn}}
{{eqn | l = 3^2
| r = 9
}}
{{eqn | l = 23^2
| r = 529
}}
{{eqn | l = 233^2
| r = 54 \, 289
}}
{{eqn | l = 2333^2
| r = 5 \, 442 \, 889
}}
{{eqn | l = 23333^2
| r = 544 \, 428 \, 889
}}
{{end-eqn}}
and so on. | {{ProofWanted|Simple but tedious.}} | Squares of 23...3 | https://proofwiki.org/wiki/Squares_of_23...3 | https://proofwiki.org/wiki/Squares_of_23...3 | [
"Recreational Mathematics"
] | [] | [] |
proofwiki-13502 | Smallest Fourth Power as Sum and Difference of Fourth Powers | The smallest $4$th power that can be expressed as the sum of $2$ $4$th powers minus a $3$rd is:
:$2401 = 7^4 = 227^4 + 157^4 - 239^4$
with all numbers less than $10^4$. | {{begin-eqn}}
{{eqn | o =
| r = 227^4 + 157^4 - 239^4
| c =
}}
{{eqn | r = 2 \, 655 \, 237 \, 841 + 607 \, 573 \, 201 - 3 \, 262 \, 808 \, 641
| c =
}}
{{eqn | r = 2401
| c =
}}
{{eqn | r = 7^4
| c =
}}
{{end-eqn}}
{{ProofWanted|It remains to be shown that this is the smallest such.}... | The smallest [[Definition:Fourth Power|$4$th power]] that can be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Fourth Power|$4$th powers]] [[Definition:Integer Subtraction|minus]] a $3$rd is:
:$2401 = 7^4 = 227^4 + 157^4 - 239^4$
with all numbers less than $10^4$. | {{begin-eqn}}
{{eqn | o =
| r = 227^4 + 157^4 - 239^4
| c =
}}
{{eqn | r = 2 \, 655 \, 237 \, 841 + 607 \, 573 \, 201 - 3 \, 262 \, 808 \, 641
| c =
}}
{{eqn | r = 2401
| c =
}}
{{eqn | r = 7^4
| c =
}}
{{end-eqn}}
{{ProofWanted|It remains to be shown that this is the smallest such.... | Smallest Fourth Power as Sum and Difference of Fourth Powers | https://proofwiki.org/wiki/Smallest_Fourth_Power_as_Sum_and_Difference_of_Fourth_Powers | https://proofwiki.org/wiki/Smallest_Fourth_Power_as_Sum_and_Difference_of_Fourth_Powers | [
"2401",
"Fourth Powers"
] | [
"Definition:Fourth Power",
"Definition:Addition/Integers",
"Definition:Fourth Power",
"Definition:Subtraction/Integers"
] | [] |
proofwiki-13503 | Zsigmondy's Theorem for Sums | Let $a > b > 0$ be coprime positive integers.
Let $n \ge 1$ be a (strictly) positive integer.
Then there is a prime number $p$ such that
:$p$ divides $a^n + b^n$
:$p$ does not divide $a^k + b^k$ for all $k < n$
with the following exception:
:$n = 3$, $a = 2$, $b = 1$ | By Zsigmondy's Theorem, there exists a prime divisor $p$ of $a^{2 n} - b^{2 n}$ which does not divide $a^k - b^k$ for all $k < 2 n$ unless:
:$n = 1$ and $a + b$ is a power of $2$
:$n = 3$, $a = 2$, $b = 1$
In particular, $p$ does not divide $a^{2 k} - b^{2 k} = \paren {a^k - b^k} \paren {a^k + b^k}$ for $k < n$.
It rem... | Let $a > b > 0$ be [[Definition:Coprime Integers|coprime]] [[Definition:Positive Integer|positive integers]].
Let $n \ge 1$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then there is a [[Definition:Prime Number|prime number]] $p$ such that
:$p$ [[Definition:Divisor of Integer|divides]] ... | By [[Zsigmondy's Theorem]], there exists a [[Definition:Prime Divisor|prime divisor]] $p$ of $a^{2 n} - b^{2 n}$ which does not [[Definition:Divisor of Integer|divide]] $a^k - b^k$ for all $k < 2 n$ unless:
:$n = 1$ and $a + b$ is a [[Definition:Integer Power|power]] of $2$
:$n = 3$, $a = 2$, $b = 1$
In particular, $p... | Zsigmondy's Theorem for Sums | https://proofwiki.org/wiki/Zsigmondy's_Theorem_for_Sums | https://proofwiki.org/wiki/Zsigmondy's_Theorem_for_Sums | [
"Number Theory"
] | [
"Definition:Coprime/Integers",
"Definition:Positive/Integer",
"Definition:Strictly Positive/Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Zsigmondy's Theorem",
"Definition:Prime Factor",
"Definition:Divisor (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Odd Integer",
"Definition:Prime Factor",
"Definition:Coprime/Integers",
"Definit... |
proofwiki-13504 | Cyclotomic Polynomial of Index times Prime Power | Let $n, k \ge 1$ be natural numbers.
Let $p$ be a prime number.
Let $\Phi_n$ denote the $n$th cyclotomic polynomial.
Then $\map {\Phi_{p^k n}} x = \begin{cases}
\map {\Phi_n} {x^{p^k}} & \text{if } p \divides n\\
\dfrac {\map {\Phi_n} {x^{p^k}}} {\map {\Phi_n} {x^{p^{k - 1}}}} & \text{if } p \nmid n \end{cases}$ | Suppose $p \divides n$.
Then for all $m \in \Z$:
{{begin-eqn}}
{{eqn | l = m \perp n
| o = \implies
| r = m \perp n \land m \perp p
| c = Law of Identity; Divisor of One of Coprime Numbers is Coprime to Other
}}
{{eqn | o = \implies
| r = m \perp p^k n
| c = Integer Coprime to all Factors... | Let $n, k \ge 1$ be [[Definition:Natural Number|natural numbers]].
Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\Phi_n$ denote the $n$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]].
Then $\map {\Phi_{p^k n}} x = \begin{cases}
\map {\Phi_n} {x^{p^k}} & \text{if } p \divides n\\
\dfrac {... | Suppose $p \divides n$.
Then for all $m \in \Z$:
{{begin-eqn}}
{{eqn | l = m \perp n
| o = \implies
| r = m \perp n \land m \perp p
| c = [[Law of Identity]]; [[Divisor of One of Coprime Numbers is Coprime to Other]]
}}
{{eqn | o = \implies
| r = m \perp p^k n
| c = [[Integer Coprime to ... | Cyclotomic Polynomial of Index times Prime Power | https://proofwiki.org/wiki/Cyclotomic_Polynomial_of_Index_times_Prime_Power | https://proofwiki.org/wiki/Cyclotomic_Polynomial_of_Index_times_Prime_Power | [
"Cyclotomic Polynomials"
] | [
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Cyclotomic Polynomial"
] | [
"Law of Identity",
"Divisor of One of Coprime Numbers is Coprime to Other",
"Integer Coprime to all Factors is Coprime to Whole",
"Divisor of One of Coprime Numbers is Coprime to Other",
"Definition:Root of Unity/Complex/Primitive",
"Condition for Complex Root of Unity to be Primitive",
"Division Theore... |
proofwiki-13505 | Multiplicative Order of Roots of Cyclotomic Polynomial Modulo Prime | Let $n \ge 1$ be a natural number.
Let $\Phi_n$ be the $n$-th cyclotomic polynomial.
Let $p$ be a prime number.
Let $n = p^\alpha q$ where $\alpha = \map {\nu_p} n$ is the $p$-adic valuation of $n$.
Let $a \in \Z$ with $\map {\Phi_n} a \equiv 0 \pmod p$.
Then the order of $a$ modulo $p$ is $q$:
:$\map {\operatorname{or... | By Product of Cyclotomic Polynomials, $p \mid \map {\Phi_n} a \mid a^n-1$.
Thus $a$ is coprime to $p$.
By Fermat's Little Theorem, $1 \equiv a^n \equiv a^q \pmod p$.
Thus $\map {\operatorname{ord}_p} a \le q$.
{{AimForCont}} $\map {\operatorname{ord}_p} a = k < q$.
By Product of Cyclotomic Polynomials, $p \mid \map {\P... | Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
Let $\Phi_n$ be the $n$-th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]].
Let $p$ be a [[Definition:Prime Number|prime number]].
Let $n = p^\alpha q$ where $\alpha = \map {\nu_p} n$ is the [[Definition:P-adic Valuation|$p$-adic valuation]]... | By [[Product of Cyclotomic Polynomials]], $p \mid \map {\Phi_n} a \mid a^n-1$.
Thus $a$ is [[Definition:Coprime Integers|coprime]] to $p$.
By [[Fermat's Little Theorem]], $1 \equiv a^n \equiv a^q \pmod p$.
Thus $\map {\operatorname{ord}_p} a \le q$.
{{AimForCont}} $\map {\operatorname{ord}_p} a = k < q$.
By [[Prod... | Multiplicative Order of Roots of Cyclotomic Polynomial Modulo Prime | https://proofwiki.org/wiki/Multiplicative_Order_of_Roots_of_Cyclotomic_Polynomial_Modulo_Prime | https://proofwiki.org/wiki/Multiplicative_Order_of_Roots_of_Cyclotomic_Polynomial_Modulo_Prime | [
"Cyclotomic Polynomials"
] | [
"Definition:Natural Numbers",
"Definition:Cyclotomic Polynomial",
"Definition:Prime Number",
"Definition:P-adic Valuation",
"Definition:Multiplicative Order of Integer"
] | [
"Product of Cyclotomic Polynomials",
"Definition:Coprime/Integers",
"Fermat's Little Theorem",
"Product of Cyclotomic Polynomials",
"Definition:Multiplicity (Polynomial)",
"Double Root of Polynomial is Root of Derivative",
"Category:Cyclotomic Polynomials"
] |
proofwiki-13506 | Lower Topology is Unique | Let $T_1 = \left({S, \preceq, \tau_1}\right)$ and $T_2 = \left({S, \preceq, \tau_2}\right)$ be relational structures with lower topologies.
Then:
: $\tau_1 = \tau_2$ | Define:
: $B := \left\{ {\complement_S \left({x^\succeq}\right): x \in S}\right\}$
where $x^\succeq$ denotes the upper closure of $x$.
Thus:
{{begin-eqn}}
{{eqn | l = \tau_1
| r = \tau \left({B}\right)
| c = {{Defof|Topology Generated by Synthetic Sub-Basis}}
}}
{{eqn | r = \tau_2
| c = {{Defof|Topolo... | Let $T_1 = \left({S, \preceq, \tau_1}\right)$ and $T_2 = \left({S, \preceq, \tau_2}\right)$ be [[Definition:Relational Structure with Topology|relational structures]] with [[Definition:Lower Topology|lower topologies]].
Then:
: $\tau_1 = \tau_2$ | Define:
: $B := \left\{ {\complement_S \left({x^\succeq}\right): x \in S}\right\}$
where $x^\succeq$ denotes the [[Definition:Upper Closure of Element|upper closure]] of $x$.
Thus:
{{begin-eqn}}
{{eqn | l = \tau_1
| r = \tau \left({B}\right)
| c = {{Defof|Topology Generated by Synthetic Sub-Basis}}
}}
{{eq... | Lower Topology is Unique | https://proofwiki.org/wiki/Lower_Topology_is_Unique | https://proofwiki.org/wiki/Lower_Topology_is_Unique | [
"Topological Order Theory"
] | [
"Definition:Relational Structure with Topology",
"Definition:Lower Topology"
] | [
"Definition:Upper Closure/Element"
] |
proofwiki-13507 | Homogeneous Cyclotomic Polynomial is Symmetric | Let $n>1$ be a natural number.
Let $\Phi_n(x,y)$ be the $n$th homogeneous cyclotomic polynomial.
Then $\Phi_n(x,y) = \Phi_n(y,x)$, that is, $\Phi_n(x,y)$ is symmetric. | {{ProofWanted}}
Category:Cyclotomic Polynomials
1klkb4mge55s34ezlu50w1to0q0ktsa | Let $n>1$ be a [[Definition:Natural Number|natural number]].
Let $\Phi_n(x,y)$ be the $n$th [[Definition:Homogeneous Cyclotomic Polynomial|homogeneous cyclotomic polynomial]].
Then $\Phi_n(x,y) = \Phi_n(y,x)$, that is, $\Phi_n(x,y)$ is [[Definition:Symmetric Polynomial|symmetric]]. | {{ProofWanted}}
[[Category:Cyclotomic Polynomials]]
1klkb4mge55s34ezlu50w1to0q0ktsa | Homogeneous Cyclotomic Polynomial is Symmetric | https://proofwiki.org/wiki/Homogeneous_Cyclotomic_Polynomial_is_Symmetric | https://proofwiki.org/wiki/Homogeneous_Cyclotomic_Polynomial_is_Symmetric | [
"Cyclotomic Polynomials"
] | [
"Definition:Natural Numbers",
"Definition:Homogeneous Cyclotomic Polynomial",
"Definition:Symmetric Polynomial"
] | [
"Category:Cyclotomic Polynomials"
] |
proofwiki-13508 | Cyclotomic Polynomial of Index Power of Two | Let $n \ge 1$ be a natural number.
Then the $2^n$th cyclotomic polynomial is:
:$\map {\Phi_{2^n} } x = x^{2^{n - 1} } + 1$ | {{begin-eqn}}
{{eqn | l = \map {\Phi_{2^n} } x
| r = \prod_{\zeta} \paren {x - \zeta}
| c = where the product runs over all primitive complex $2^n$th roots of unity
}}
{{eqn | r = \prod_{\substack {1 \mathop \le k \mathop \le 2^n \\ \gcd \set {k, 2^n} = 1} } \paren {x - \map \exp {\frac {2 \pi i k} {2^n} } ... | Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
Then the $2^n$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]] is:
:$\map {\Phi_{2^n} } x = x^{2^{n - 1} } + 1$ | {{begin-eqn}}
{{eqn | l = \map {\Phi_{2^n} } x
| r = \prod_{\zeta} \paren {x - \zeta}
| c = where the product runs over all [[Definition:Primitive Complex Root of Unity|primitive complex $2^n$th roots of unity]]
}}
{{eqn | r = \prod_{\substack {1 \mathop \le k \mathop \le 2^n \\ \gcd \set {k, 2^n} = 1} } \p... | Cyclotomic Polynomial of Index Power of Two | https://proofwiki.org/wiki/Cyclotomic_Polynomial_of_Index_Power_of_Two | https://proofwiki.org/wiki/Cyclotomic_Polynomial_of_Index_Power_of_Two | [
"Cyclotomic Polynomials"
] | [
"Definition:Natural Numbers",
"Definition:Cyclotomic Polynomial"
] | [
"Definition:Root of Unity/Complex/Primitive",
"Condition for Complex Root of Unity to be Primitive",
"Definition:Odd Integer",
"Factorisation of z^n+1",
"Category:Cyclotomic Polynomials"
] |
proofwiki-13509 | Trivial Estimate for Cyclotomic Polynomials | Let $n \ge 1$ be a natural number.
Let $\Phi_n$ be the $n$th cyclotomic polynomial.
Let $\phi$ be the Euler totient function.
Let $z \in \C$ be a complex number.
Then:
:$\size {\size z - 1}^{\map \phi n} \le \size {\map {\Phi_n} z} \le \paren {\size z + 1}^{\map \phi n}$
where:
:the first inequality becomes an equality... | {{ProofWanted}}
Category:Cyclotomic Polynomials
8irnibmc93ig6ztlrhmeb6bb0y6sfpt | Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
Let $\Phi_n$ be the $n$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]].
Let $\phi$ be the [[Definition:Euler Totient Function|Euler totient function]].
Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\size {\si... | {{ProofWanted}}
[[Category:Cyclotomic Polynomials]]
8irnibmc93ig6ztlrhmeb6bb0y6sfpt | Trivial Estimate for Cyclotomic Polynomials | https://proofwiki.org/wiki/Trivial_Estimate_for_Cyclotomic_Polynomials | https://proofwiki.org/wiki/Trivial_Estimate_for_Cyclotomic_Polynomials | [
"Cyclotomic Polynomials"
] | [
"Definition:Natural Numbers",
"Definition:Cyclotomic Polynomial",
"Definition:Euler Phi Function",
"Definition:Complex Number"
] | [
"Category:Cyclotomic Polynomials"
] |
proofwiki-13510 | 2520 equals Sum of 4 Divisors in 6 Ways | The number $2520$ can be expressed as the sum of $4$ of its divisors in $6$ different ways:
{{begin-eqn}}
{{eqn | l = 2520
| r = 1260 + 630 + 504 + 126
}}
{{eqn | r = 1260 + 630 + 421 + 210
}}
{{eqn | r = 1260 + 840 + 360 + 60
}}
{{eqn | r = 1260 + 840 + 315 + 105
}}
{{eqn | r = 1260 + 840 + 280 + 140
}}
{{eqn | ... | We apply 1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways:
{{:1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways}}
Find the maximum powers of the primes in each equation, and choose the largest that appears:
{{begin-eqn}}
{{eqn | l = 42
| r = 2^1 \times 3^1 \times 7^1
}}
{{eqn | l = ... | The number $2520$ can be expressed as the [[Definition:Integer Addition|sum]] of $4$ of its [[Definition:Divisor of Integer|divisors]] in $6$ different ways:
{{begin-eqn}}
{{eqn | l = 2520
| r = 1260 + 630 + 504 + 126
}}
{{eqn | r = 1260 + 630 + 421 + 210
}}
{{eqn | r = 1260 + 840 + 360 + 60
}}
{{eqn | r = 1260 +... | We apply [[1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways]]:
{{:1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways}}
Find the maximum [[Definition:Integer Power|powers]] of the [[Definition:Prime Number|primes]] in each equation, and choose the largest that appears:
{{begin-eqn}}
{{e... | 2520 equals Sum of 4 Divisors in 6 Ways | https://proofwiki.org/wiki/2520_equals_Sum_of_4_Divisors_in_6_Ways | https://proofwiki.org/wiki/2520_equals_Sum_of_4_Divisors_in_6_Ways | [
"Recreational Mathematics",
"2520"
] | [
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways",
"Definition:Power (Algebra)/Integer",
"Definition:Prime Number"
] |
proofwiki-13511 | Complete List of Special Highly Composite Numbers | There are exactly $6$ special highly composite numbers:
:$1, 2, 6, 12, 60, 2520$
{{OEIS|A106037}} | We have the following:
:$1$ is a Special Highly Composite Number
:$2$ is a Special Highly Composite Number
:$6$ is a Special Highly Composite Number
:$12$ is a Special Highly Composite Number
:$60$ is a Special Highly Composite Number
:$2520$ is a Special Highly Composite Number
By inspection of the sequence of highly ... | There are exactly $6$ [[Definition:Special Highly Composite Number|special highly composite numbers]]:
:$1, 2, 6, 12, 60, 2520$
{{OEIS|A106037}} | We have the following:
:[[Special Highly Composite Number/Examples/1|$1$ is a Special Highly Composite Number]]
:[[Special Highly Composite Number/Examples/2|$2$ is a Special Highly Composite Number]]
:[[Special Highly Composite Number/Examples/6|$6$ is a Special Highly Composite Number]]
:[[Special Highly Composit... | Complete List of Special Highly Composite Numbers | https://proofwiki.org/wiki/Complete_List_of_Special_Highly_Composite_Numbers | https://proofwiki.org/wiki/Complete_List_of_Special_Highly_Composite_Numbers | [
"Special Highly Composite Numbers"
] | [
"Definition:Special Highly Composite Number"
] | [
"Special Highly Composite Number/Examples/1",
"Special Highly Composite Number/Examples/2",
"Special Highly Composite Number/Examples/6",
"Special Highly Composite Number/Examples/12",
"Special Highly Composite Number/Examples/60",
"Special Highly Composite Number/Examples/2520",
"Definition:Highly Comp... |
proofwiki-13512 | Lifting The Exponent Lemma for Sums for p=2 | Let $x, y \in \Z$ be integers with $x + y \ne 0$.
Let $n \ge 1$ be an odd natural number.
Let:
:$2 \divides x + y$
where $\divides$ denotes divisibility.
Then:
:$\map {\nu_2} {x^n + y^n} = \map {\nu_2} {x + y}$
where $\nu_2$ denotes $2$-adic valuation.
</onlyinclude> | This follows from the Lifting The Exponent Lemma for p=2 with $y$ replaced by $-y$.
{{qed}} | Let $x, y \in \Z$ be [[Definition:Integer|integers]] with $x + y \ne 0$.
Let $n \ge 1$ be an [[Definition:Odd Integer|odd]] [[Definition:Natural Number|natural number]].
Let:
:$2 \divides x + y$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Then:
:$\map {\nu_2} {x^n + y^n} = \map {\nu_2}... | This follows from the [[Lifting The Exponent Lemma for p=2]] with $y$ replaced by $-y$.
{{qed}} | Lifting The Exponent Lemma for Sums for p=2 | https://proofwiki.org/wiki/Lifting_The_Exponent_Lemma_for_Sums_for_p=2 | https://proofwiki.org/wiki/Lifting_The_Exponent_Lemma_for_Sums_for_p=2 | [
"Lifting The Exponent Lemma"
] | [
"Definition:Integer",
"Definition:Odd Integer",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:P-adic Valuation"
] | [
"Lifting The Exponent Lemma for p=2"
] |
proofwiki-13513 | Prime Decomposition of Highly Composite Number | Let $n$ be a highly composite number.
Let the prime decomposition of $n$ be expressed as:
:$n = \ds \prod_{k \mathop \in \N} {p_k}^{r_k}$
where $p_k$ denotes the $k$th prime.
Then the sequence $\sequence {r_k}$ is decreasing.
That is:
:$\forall k \in \N: r_k \ge r_{k + 1}$ | Let $n = \ds \prod_{k \mathop \in \N} {p_k}^{r_k}$ be highly composite.
By definition of divisor count function:
:$\map {\sigma_0} n = \ds \prod_{k \mathop \in \N} \paren {r_k + 1}$
{{AimForCont}} $r_{l + 1} > r_l$ for some $l \in \N$.
Consider $m \in \Z$ whose prime decomposition of $n$ is expressed as:
:$m = \ds \pro... | Let $n$ be a [[Definition:Highly Composite Number|highly composite number]].
Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be expressed as:
:$n = \ds \prod_{k \mathop \in \N} {p_k}^{r_k}$
where $p_k$ denotes the [[Definition:Prime Number|$k$th prime]].
Then the [[Definition:Integer Sequence|s... | Let $n = \ds \prod_{k \mathop \in \N} {p_k}^{r_k}$ be [[Definition:Highly Composite Number|highly composite]].
By definition of [[Definition:Divisor Count Function|divisor count function]]:
:$\map {\sigma_0} n = \ds \prod_{k \mathop \in \N} \paren {r_k + 1}$
{{AimForCont}} $r_{l + 1} > r_l$ for some $l \in \N$.
Cons... | Prime Decomposition of Highly Composite Number | https://proofwiki.org/wiki/Prime_Decomposition_of_Highly_Composite_Number | https://proofwiki.org/wiki/Prime_Decomposition_of_Highly_Composite_Number | [
"Highly Composite Numbers"
] | [
"Definition:Highly Composite Number",
"Definition:Prime Decomposition",
"Definition:Prime Number",
"Definition:Integer Sequence",
"Definition:Decreasing/Sequence"
] | [
"Definition:Highly Composite Number",
"Definition:Divisor Count Function",
"Definition:Prime Decomposition",
"Definition:Highly Composite Number",
"Definition:Contradiction",
"Proof by Contradiction"
] |
proofwiki-13514 | Complement of Upper Closure of Element is Open in Lower Topology | Let $T = \struct {S, \preceq, \tau}$ be a relational structure with lower topology.
Let $x \in S$.
Then $\relcomp S {x^\succeq}$ is open and $x^\succeq$ is closed. | Define $B := \set {\relcomp S {y^\succeq}: y \in S}$
By definition of lower topology:
:$B$ is sub-basis of $T$.
By definition of sub-basis:
:$B \subseteq \tau$
By definition of $B$:
:$\relcomp S {x^\succeq} \in B$
Thus by definition of subset:
:$\relcomp S {x^\succeq} \in \tau$
Thus by definition:
:$x^\succeq$ is close... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Relational Structure with Topology|relational structure]] with [[Definition:Lower Topology|lower topology]].
Let $x \in S$.
Then $\relcomp S {x^\succeq}$ is [[Definition:Open Set (Topology)|open]] and $x^\succeq$ is [[Definition:Closed Set (Topology)|closed]]. | Define $B := \set {\relcomp S {y^\succeq}: y \in S}$
By definition of [[Definition:Lower Topology|lower topology]]:
:$B$ is [[Definition:Analytic Sub-Basis|sub-basis]] of $T$.
By definition of [[Definition:Analytic Sub-Basis|sub-basis]]:
:$B \subseteq \tau$
By definition of $B$:
:$\relcomp S {x^\succeq} \in B$
Thus... | Complement of Upper Closure of Element is Open in Lower Topology | https://proofwiki.org/wiki/Complement_of_Upper_Closure_of_Element_is_Open_in_Lower_Topology | https://proofwiki.org/wiki/Complement_of_Upper_Closure_of_Element_is_Open_in_Lower_Topology | [
"Topological Order Theory"
] | [
"Definition:Relational Structure with Topology",
"Definition:Lower Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology"
] | [
"Definition:Lower Topology",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Subset",
"Definition:Closed Set/Topology"
] |
proofwiki-13515 | Open Subset is Lower Section in Lower Topology | Let $T = \struct {S, \preceq, \tau}$ be a transitive relational structure with lower topology.
Let $A \subseteq S$ such that
:$A$ is open.
Then $A$ is a lower section of $S$. | Define $B = \set {\relcomp S {x^\succeq}: x \in S}$
By definition of lower topology:
:$B$ is sub-basis of $T$.
By definitions of sub-basis and basis:
:$\ds \BB = \set {\bigcap \FF: \FF \subseteq B, \FF \text{ is finite} }$ is a basis.
By definition of basis:
:$\ds \tau \subseteq \set {\bigcup X: X \subseteq \BB}$
Let $... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Transitive Relation|transitive]] [[Definition:Relational Structure with Topology|relational structure]] with [[Definition:Lower Topology|lower topology]].
Let $A \subseteq S$ such that
:$A$ is [[Definition:Open Set (Topology)|open]].
Then $A$ is a [[Definition:L... | Define $B = \set {\relcomp S {x^\succeq}: x \in S}$
By definition of [[Definition:Lower Topology|lower topology]]:
:$B$ is [[Definition:Analytic Sub-Basis|sub-basis]] of $T$.
By definitions of [[Definition:Analytic Sub-Basis|sub-basis]] and [[Definition:Analytic Basis|basis]]:
:$\ds \BB = \set {\bigcap \FF: \FF \subs... | Open Subset is Lower Section in Lower Topology | https://proofwiki.org/wiki/Open_Subset_is_Lower_Section_in_Lower_Topology | https://proofwiki.org/wiki/Open_Subset_is_Lower_Section_in_Lower_Topology | [
"Topological Order Theory"
] | [
"Definition:Transitive Relation",
"Definition:Relational Structure with Topology",
"Definition:Lower Topology",
"Definition:Open Set/Topology",
"Definition:Lower Section"
] | [
"Definition:Lower Topology",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Open Set/Topology",
"Definition:Subse... |
proofwiki-13516 | Ratio between Consecutive Highly Composite Numbers Greater than 2520 is Less than 2 | The ratio between $2$ consecutive highly composite numbers both greater than $2520$ is less than $2$. | {{AimForCont}} $n$ and $m$ are consecutive highly composite numbers such that:
:$2520 < n < m$
:$m / n \ge 2$
By definition of highly composite:
:$\map \tau m > \map \tau n$
and, {{hypothesis}}, $m$ is the smallest such integer.
We have that:
:$\map \tau {2 n} > \map \tau n$
so it follows that $m \le 2 n$, otherwise $m... | The [[Definition:Ratio|ratio]] between $2$ consecutive [[Definition:Highly Composite Number|highly composite numbers]] both greater than $2520$ is less than $2$. | {{AimForCont}} $n$ and $m$ are consecutive [[Definition:Highly Composite Number|highly composite numbers]] such that:
:$2520 < n < m$
:$m / n \ge 2$
By definition of [[Definition:Highly Composite Number|highly composite]]:
:$\map \tau m > \map \tau n$
and, {{hypothesis}}, $m$ is the smallest such [[Definition:Integer... | Ratio between Consecutive Highly Composite Numbers Greater than 2520 is Less than 2 | https://proofwiki.org/wiki/Ratio_between_Consecutive_Highly_Composite_Numbers_Greater_than_2520_is_Less_than_2 | https://proofwiki.org/wiki/Ratio_between_Consecutive_Highly_Composite_Numbers_Greater_than_2520_is_Less_than_2 | [
"Highly Composite Numbers"
] | [
"Definition:Ratio",
"Definition:Highly Composite Number"
] | [
"Definition:Highly Composite Number",
"Definition:Highly Composite Number",
"Definition:Integer",
"Definition:Integer",
"Special Highly Composite Number/Examples/2520",
"Definition:Prime Decomposition",
"Definition:Highly Composite Number",
"Definition:Special Highly Composite Number",
"Definition:D... |
proofwiki-13517 | Modified Kaprekar Process on 4-Digit Number terminates in 2538 | Let $n$ be a $4$-digit number.
Let $n$ be operated on by the modified Kaprekar process.
The eventual result is always $2538$. | {{ProofWanted|I find these processes dull. Does anyone else want to go into a detailed analysis of this and other similar?}} | Let $n$ be a [[Definition:Digit|$4$-digit]] [[Definition:Natural Number|number]].
Let $n$ be operated on by the [[Definition:Modified Kaprekar Process|modified Kaprekar process]].
The eventual result is always $2538$. | {{ProofWanted|I find these processes dull. Does anyone else want to go into a detailed analysis of this and other similar?}} | Modified Kaprekar Process on 4-Digit Number terminates in 2538 | https://proofwiki.org/wiki/Modified_Kaprekar_Process_on_4-Digit_Number_terminates_in_2538 | https://proofwiki.org/wiki/Modified_Kaprekar_Process_on_4-Digit_Number_terminates_in_2538 | [
"Kaprekar's Process"
] | [
"Definition:Digit",
"Definition:Natural Numbers",
"Definition:Modified Kaprekar Process"
] | [] |
proofwiki-13518 | Fibonacci Number equal to Sum of Sequence of Cubes | The following Fibonacci number can be expressed as the sum of a sequence of cubes:
:$F_{18} = 2584 = 7^3 + 8^3 + 9^3 + 10^3$
{{expand|Any more?}} | {{begin-eqn}}
{{eqn | l = 2584
| r = 343 + 512 + 729 + 1000
| c =
}}
{{eqn | r = 7^3 + 8^3 + 9^3 + 10^3
| c =
}}
{{end-eqn}}
{{qed}} | The following [[Definition:Fibonacci Number|Fibonacci number]] can be expressed as the [[Definition:Integer Addition|sum]] of a [[Definition:Integer Sequence|sequence]] of [[Definition:Cube Number|cubes]]:
:$F_{18} = 2584 = 7^3 + 8^3 + 9^3 + 10^3$
{{expand|Any more?}} | {{begin-eqn}}
{{eqn | l = 2584
| r = 343 + 512 + 729 + 1000
| c =
}}
{{eqn | r = 7^3 + 8^3 + 9^3 + 10^3
| c =
}}
{{end-eqn}}
{{qed}} | Fibonacci Number equal to Sum of Sequence of Cubes | https://proofwiki.org/wiki/Fibonacci_Number_equal_to_Sum_of_Sequence_of_Cubes | https://proofwiki.org/wiki/Fibonacci_Number_equal_to_Sum_of_Sequence_of_Cubes | [
"Fibonacci Numbers",
"Cube Numbers",
"2584"
] | [
"Definition:Fibonacci Number",
"Definition:Addition/Integers",
"Definition:Integer Sequence",
"Definition:Cube Number"
] | [] |
proofwiki-13519 | Dudeney's Property of 2592 | :$2592 = 2^5 \times 9^2$
It is the only number $n$ that has the property that:
:$n = \sqbrk {abcd} = a^b \times c^d$
where $\sqbrk {abcd}$ denotes the decimal representation of $n$. | First we verify that $2592$ does indeed satisfy the given property.
{{begin-eqn}}
{{eqn | l = 2592
| r = 2^5 \times 3^4
| c = Prime Decomposition of $2592$
}}
{{eqn | r = 2^5 \times \paren {3^2}^2
| c =
}}
{{eqn | r = 2^5 \times 9^2
| c =
}}
{{end-eqn}}
{{qed|lemma}}
It remains to be shown tha... | :$2592 = 2^5 \times 9^2$
It is the only [[Definition:Natural Number|number]] $n$ that has the property that:
:$n = \sqbrk {abcd} = a^b \times c^d$
where $\sqbrk {abcd}$ denotes the [[Definition:Decimal Notation|decimal representation]] of $n$. | First we verify that $2592$ does indeed satisfy the given property.
{{begin-eqn}}
{{eqn | l = 2592
| r = 2^5 \times 3^4
| c = [[Definition:Prime Decomposition|Prime Decomposition]] of $2592$
}}
{{eqn | r = 2^5 \times \paren {3^2}^2
| c =
}}
{{eqn | r = 2^5 \times 9^2
| c =
}}
{{end-eqn}}
{{qe... | Dudeney's Property of 2592 | https://proofwiki.org/wiki/Dudeney's_Property_of_2592 | https://proofwiki.org/wiki/Dudeney's_Property_of_2592 | [
"Recreational Mathematics",
"2592"
] | [
"Definition:Natural Numbers",
"Definition:Decimal Notation"
] | [
"Definition:Prime Decomposition",
"Definition:Digit",
"Definition:Digit",
"Definition:Power (Algebra)/Integer",
"Definition:Zero Digit",
"Definition:Power (Algebra)/Integer",
"Definition:Digit",
"Definition:Digit",
"Definition:Power (Algebra)/Integer",
"Definition:Digit",
"Definition:Digit",
"... |
proofwiki-13520 | 2601 as Sum of 3 Squares in 12 Different Ways | $2601$ can be expressed as the sum of $3$ squares in $12$ different ways. | {{begin-eqn}}
{{eqn | l = 2601
| r = 51^2
| c =
}}
{{eqn | r = 1^2 + 10^2 + 50^2
| c =
}}
{{eqn | r = 2^2 + 14^2 + 49^2
| c =
}}
{{eqn | r = 10^2 + 10^2 + 49^2
| c =
}}
{{eqn | r = 14^2 + 14^2 + 47^2
| c =
}}
{{eqn | r = 1^2 + 22^2 + 46^2
| c =
}}
{{eqn | r = 14^2 + 17^2 ... | $2601$ can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Square Number|squares]] in $12$ different ways. | {{begin-eqn}}
{{eqn | l = 2601
| r = 51^2
| c =
}}
{{eqn | r = 1^2 + 10^2 + 50^2
| c =
}}
{{eqn | r = 2^2 + 14^2 + 49^2
| c =
}}
{{eqn | r = 10^2 + 10^2 + 49^2
| c =
}}
{{eqn | r = 14^2 + 14^2 + 47^2
| c =
}}
{{eqn | r = 1^2 + 22^2 + 46^2
| c =
}}
{{eqn | r = 14^2 + 17^2 ... | 2601 as Sum of 3 Squares in 12 Different Ways | https://proofwiki.org/wiki/2601_as_Sum_of_3_Squares_in_12_Different_Ways | https://proofwiki.org/wiki/2601_as_Sum_of_3_Squares_in_12_Different_Ways | [
"Sums of Squares",
"2601"
] | [
"Definition:Addition/Integers",
"Definition:Square Number"
] | [] |
proofwiki-13521 | Reversal of Number Multiplied by 11 | Let $n \in \N$ be a number for which, when written in decimal notation, no two adjacent digits total to more than $9$.
Let $n'$ denote the reversal of $n$.
Then $n \times 11$ is the reversal of $n' \times 11$. | By Basis Representation Theorem, there exists one and only one sequence $\sequence {r_j}_{0 \mathop \le j \mathop \le t}$ such that:
:$(1): \quad \ds n = \sum_{k \mathop = 0}^t r_k 10^k$
:$(2): \quad \ds \forall k \in \closedint 0 t: r_k \in \N_{10}$
:$(3): \quad r_t \ne 0$
Since no two adjacent digits of $n$ total to ... | Let $n \in \N$ be a [[Definition:Natural Number|number]] for which, when written in [[Definition:Decimal Notation|decimal notation]], no two adjacent [[Definition:Digit|digits]] [[Definition:Integer Addition|total]] to more than $9$.
Let $n'$ denote the [[Definition:Reversal|reversal]] of $n$.
Then $n \times 11$ is t... | By [[Basis Representation Theorem]], there exists [[Definition:Exactly One|one and only one]] [[Definition:Sequence|sequence]] $\sequence {r_j}_{0 \mathop \le j \mathop \le t}$ such that:
:$(1): \quad \ds n = \sum_{k \mathop = 0}^t r_k 10^k$
:$(2): \quad \ds \forall k \in \closedint 0 t: r_k \in \N_{10}$
:$(3): \quad ... | Reversal of Number Multiplied by 11 | https://proofwiki.org/wiki/Reversal_of_Number_Multiplied_by_11 | https://proofwiki.org/wiki/Reversal_of_Number_Multiplied_by_11 | [
"Reversals",
"11",
"Reversal of Number Multiplied by 11"
] | [
"Definition:Natural Numbers",
"Definition:Decimal Notation",
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Reversal",
"Definition:Reversal"
] | [
"Basis Representation Theorem",
"Definition:Unique",
"Definition:Sequence",
"Definition:Digit",
"Definition:Addition/Integers",
"Translation of Index Variable of Summation",
"Definition:Basis Representation",
"Definition:Reversal",
"Translation of Index Variable of Summation",
"Definition:Basis Re... |
proofwiki-13522 | Reduction Formula for Integral of Power of Tangent | For all $n \in \Z_{> 1}$:
Let:
:$I_n := \ds \int \tan^n x \rd x$
Then:
:$I_n = \dfrac {\tan^{n - 1} x} {n - 1} - I_{n - 2}$
is a reduction formula for $\ds \int \tan^n x \rd x$. | {{begin-eqn}}
{{eqn | l = I_n
| r = \int \tan^n x \rd x
| c = by definition
}}
{{eqn | r = \int \tan^{n - 2} x \tan^2 x \rd x
}}
{{eqn | r = \int \tan^{n - 2} x \paren {\sec^2 x - 1} \rd x
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = \int \tan^{n - 2} x \sec^2 x \rd x - \int \tan^{n - 2} x \... | For all $n \in \Z_{> 1}$:
Let:
:$I_n := \ds \int \tan^n x \rd x$
Then:
:$I_n = \dfrac {\tan^{n - 1} x} {n - 1} - I_{n - 2}$
is a [[Definition:Reduction Formula (Calculus)|reduction formula]] for $\ds \int \tan^n x \rd x$. | {{begin-eqn}}
{{eqn | l = I_n
| r = \int \tan^n x \rd x
| c = by definition
}}
{{eqn | r = \int \tan^{n - 2} x \tan^2 x \rd x
}}
{{eqn | r = \int \tan^{n - 2} x \paren {\sec^2 x - 1} \rd x
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | r = \int \tan^{n - 2} x \sec^2 x \rd x - \int \tan^{n - 2}... | Reduction Formula for Integral of Power of Tangent | https://proofwiki.org/wiki/Reduction_Formula_for_Integral_of_Power_of_Tangent | https://proofwiki.org/wiki/Reduction_Formula_for_Integral_of_Power_of_Tangent | [
"Primitives involving Tangent Function",
"Reduction Formulae (Calculus)"
] | [
"Definition:Reduction Formula (Calculus)"
] | [
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite",
"Derivative of Tangent Function",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-13523 | Numbers Appearing 8 Times in Pascal's Triangle | Excluding $1$, the number $3003$ is the smallest integer to appear $8$ times in Pascal's triangle.
No other number below $2^{23}$ appears as often. | {{begin-eqn}}
{{eqn | l = 3003
| m = \frac {3003!} {3002! \times 1!}
| mo= =
| r = \dbinom {3003} 1
| c =
}}
{{eqn | m = \frac {78!} {76! \times 2!}
| mo= =
| r = \dbinom {78} 2
| c =
}}
{{eqn | m = \frac {15!} {10! \times 5!}
| mo= =
| r = \dbinom {15} 5
| ... | Excluding $1$, the number $3003$ is the smallest [[Definition:Positive Integer|integer]] to appear $8$ times in [[Definition:Pascal's Triangle|Pascal's triangle]].
No other number below $2^{23}$ appears as often. | {{begin-eqn}}
{{eqn | l = 3003
| m = \frac {3003!} {3002! \times 1!}
| mo= =
| r = \dbinom {3003} 1
| c =
}}
{{eqn | m = \frac {78!} {76! \times 2!}
| mo= =
| r = \dbinom {78} 2
| c =
}}
{{eqn | m = \frac {15!} {10! \times 5!}
| mo= =
| r = \dbinom {15} 5
| ... | Numbers Appearing 8 Times in Pascal's Triangle | https://proofwiki.org/wiki/Numbers_Appearing_8_Times_in_Pascal's_Triangle | https://proofwiki.org/wiki/Numbers_Appearing_8_Times_in_Pascal's_Triangle | [
"Pascal's Triangle",
"3003"
] | [
"Definition:Positive/Integer",
"Definition:Pascal's Triangle"
] | [] |
proofwiki-13524 | Squares of 3...34 | The following pattern holds:
{{begin-eqn}}
{{eqn | l = 4^2
| r = 16
}}
{{eqn | l = 34^2
| r = 1156
}}
{{eqn | l = 334^2
| r = 111 \, 556
}}
{{eqn | l = 3334^2
| r = 11 \, 115 \, 556
}}
{{eqn | l = 33334^2
| r = 1 \, 111 \, 155\, 556
}}
{{end-eqn}}
and so on. | {{ProofWanted|Simple but tedious.}} | The following pattern holds:
{{begin-eqn}}
{{eqn | l = 4^2
| r = 16
}}
{{eqn | l = 34^2
| r = 1156
}}
{{eqn | l = 334^2
| r = 111 \, 556
}}
{{eqn | l = 3334^2
| r = 11 \, 115 \, 556
}}
{{eqn | l = 33334^2
| r = 1 \, 111 \, 155\, 556
}}
{{end-eqn}}
and so on. | {{ProofWanted|Simple but tedious.}} | Squares of 3...34 | https://proofwiki.org/wiki/Squares_of_3...34 | https://proofwiki.org/wiki/Squares_of_3...34 | [
"Recreational Mathematics"
] | [] | [] |
proofwiki-13525 | 3367 Multiplied by 2-Digit Number | In order to multiply $3367$ by a $2$-digit integer $\sqbrk {xy}$:
:divide the $6$-digit integer $\sqbrk {xyxyxy}$ by $3$. | We have that:
:$10101 = 3367 \times 3$
Then:
:$10101 \times \sqbrk {xy} = \sqbrk {xyxyxy}$
The result follows.
{{qed}} | In order to [[Definition:Integer Multiplication|multiply]] $3367$ by a $2$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] $\sqbrk {xy}$:
:[[Definition:Integer Division|divide]] the $6$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] $\sqbrk {xyxyxy}$ by $3$. | We have that:
:$10101 = 3367 \times 3$
Then:
:$10101 \times \sqbrk {xy} = \sqbrk {xyxyxy}$
The result follows.
{{qed}} | 3367 Multiplied by 2-Digit Number | https://proofwiki.org/wiki/3367_Multiplied_by_2-Digit_Number | https://proofwiki.org/wiki/3367_Multiplied_by_2-Digit_Number | [
"Recreational Mathematics",
"3367"
] | [
"Definition:Multiplication/Integers",
"Definition:Digit",
"Definition:Integer",
"Definition:Integer Division",
"Definition:Digit",
"Definition:Integer"
] | [] |
proofwiki-13526 | Integer as Sum of 2 Cubes in 3 Ways | $728$ is the smallest natural number which can be expressed as the sum of $2$ cubes in $3$ different ways:
{{begin-eqn}}
{{eqn | l = 728
| r = 6^3 + 8^3
| c =
}}
{{eqn | r = \paren {-1}^3 + 9^3
| c =
}}
{{eqn | r = \paren {-10}^3 + 12^3
| c =
}}
{{end-eqn}} | We have:
{{begin-eqn}}
{{eqn | l = 6^3 + 8^3
| r = 216 + 512
| c =
}}
{{eqn | r = 728
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {-1}^3 + 9^3
| r = -1 + 729
| c =
}}
{{eqn | r = 728
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {-10}^3 + 12^3
| r = -1000 ... | $728$ is the smallest [[Definition:Natural Number|natural number]] which can be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Cube Number|cubes]] in $3$ different ways:
{{begin-eqn}}
{{eqn | l = 728
| r = 6^3 + 8^3
| c =
}}
{{eqn | r = \paren {-1}^3 + 9^3
| c =
}}
{{eqn |... | We have:
{{begin-eqn}}
{{eqn | l = 6^3 + 8^3
| r = 216 + 512
| c =
}}
{{eqn | r = 728
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {-1}^3 + 9^3
| r = -1 + 729
| c =
}}
{{eqn | r = 728
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {-10}^3 + 12^3
| r = -... | Integer as Sum of 2 Cubes in 3 Ways | https://proofwiki.org/wiki/Integer_as_Sum_of_2_Cubes_in_3_Ways | https://proofwiki.org/wiki/Integer_as_Sum_of_2_Cubes_in_3_Ways | [
"Sums of Cubes"
] | [
"Definition:Natural Numbers",
"Definition:Addition/Integers",
"Definition:Cube Number"
] | [
"Category:Sums of Cubes"
] |
proofwiki-13527 | Composite Fibonacci Numbers with Prime Index | The sequence of composite Fibonacci numbers with a prime index begins:
:$4181, 1 \, 346 \, 269, 24 \, 157 \, 817, 165 \, 580 \, 141, \ldots$
{{OEIS|A050937}}
The corresponding sequence of prime indices begins:
:$19, 31, 37, 41, 53, 59, 61, 67, 71, 73, 79, \ldots$
{{OEIS|A038672}} | By observation:
{{begin-eqn}}
{{eqn | l = F_{19}
| r = 4181
| c =
}}
{{eqn | r = 37 \times 113
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = F_{31}
| r = 1 \, 346 \, 269
| c =
}}
{{eqn | r = 557 \times 2417
| c =
}}
{{end-eqn}}
{{finish}} | The [[Definition:Integer Sequence|sequence]] of [[Definition:Composite Number|composite]] [[Definition:Fibonacci Number|Fibonacci numbers]] with a [[Definition:Prime Number|prime index]] begins:
:$4181, 1 \, 346 \, 269, 24 \, 157 \, 817, 165 \, 580 \, 141, \ldots$
{{OEIS|A050937}}
The corresponding [[Definition:Intege... | By observation:
{{begin-eqn}}
{{eqn | l = F_{19}
| r = 4181
| c =
}}
{{eqn | r = 37 \times 113
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = F_{31}
| r = 1 \, 346 \, 269
| c =
}}
{{eqn | r = 557 \times 2417
| c =
}}
{{end-eqn}}
{{finish}} | Composite Fibonacci Numbers with Prime Index | https://proofwiki.org/wiki/Composite_Fibonacci_Numbers_with_Prime_Index | https://proofwiki.org/wiki/Composite_Fibonacci_Numbers_with_Prime_Index | [
"Fibonacci Numbers",
"Prime Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Composite Number",
"Definition:Fibonacci Number",
"Definition:Prime Number",
"Definition:Integer Sequence",
"Definition:Prime Number"
] | [] |
proofwiki-13528 | Closed Subset is Upper Section in Lower Topology | Let $T = \struct {S, \preceq, \tau}$ be a transitive relational structure with lower topology.
Let $A \subseteq S$ such that
:$A$ is closed.
Then $A$ is an upper section of $S$. | By definition of closed set:
:$S \setminus A$ is open.
By Open Subset is Lower Section in Lower Topology:
:$S \setminus A$ is a lower section.
Thus by Complement of Lower Section is Upper Section and Relative Complement of Relative Complement:
:$A$ is an upper section of $S$.
{{qed}} | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Transitive Relation|transitive]] [[Definition:Relational Structure with Topology|relational structure]] with [[Definition:Lower Topology|lower topology]].
Let $A \subseteq S$ such that
:$A$ is [[Definition:Closed Set (Topology)|closed]].
Then $A$ is an [[Definit... | By definition of [[Definition:Closed Set (Topology)|closed set]]:
:$S \setminus A$ is [[Definition:Open Set (Topology)|open]].
By [[Open Subset is Lower Section in Lower Topology]]:
:$S \setminus A$ is a [[Definition:Lower Section|lower section]].
Thus by [[Complement of Lower Section is Upper Section]] and [[Relativ... | Closed Subset is Upper Section in Lower Topology | https://proofwiki.org/wiki/Closed_Subset_is_Upper_Section_in_Lower_Topology | https://proofwiki.org/wiki/Closed_Subset_is_Upper_Section_in_Lower_Topology | [
"Topological Order Theory"
] | [
"Definition:Transitive Relation",
"Definition:Relational Structure with Topology",
"Definition:Lower Topology",
"Definition:Closed Set/Topology",
"Definition:Upper Section"
] | [
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Open Subset is Lower Section in Lower Topology",
"Definition:Lower Section",
"Complement of Lower Section is Upper Section",
"Relative Complement of Relative Complement",
"Definition:Upper Section"
] |
proofwiki-13529 | Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice | Let $T = \struct {S, \preceq, \tau}$ and $Q = \struct {X, \preceq', \tau'}$ be complete topological lattices with lower topologies.
Let $f: S \to X$ be a mapping such that
:for all non-empty subsets $Y$ of $S$: $f$ preserves the infimum of $Y$.
Then $f$ is continuous mapping. | Define $B = \set {\relcomp X {x^{\succeq'} }: x \in X}$
We will prove that
:$\forall A \in B: f^{-1} \sqbrk {\relcomp X A}$ is closed.
Let $A \in B$.
By definition of $B$:
:$\exists x \in X: A = \relcomp X {x^{\succeq'} }$
By Relative Complement of Relative Complement:
:$\relcomp X A = x^{\succeq'}$
By Infimum of Upper... | Let $T = \struct {S, \preceq, \tau}$ and $Q = \struct {X, \preceq', \tau'}$ be [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Lower Topology|lower topologies]].
Let $f: S \to X$ be a [[Definition:Mapping|mapping]] such that
:for all [[Definition:Non-E... | Define $B = \set {\relcomp X {x^{\succeq'} }: x \in X}$
We will prove that
:$\forall A \in B: f^{-1} \sqbrk {\relcomp X A}$ is [[Definition:Closed Set (Topology)|closed]].
Let $A \in B$.
By definition of $B$:
:$\exists x \in X: A = \relcomp X {x^{\succeq'} }$
By [[Relative Complement of Relative Complement]]:
:$\re... | Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice | https://proofwiki.org/wiki/Mapping_Preserves_Non-Empty_Infima_implies_Mapping_is_Continuous_in_Lower_Topological_Lattice | https://proofwiki.org/wiki/Mapping_Preserves_Non-Empty_Infima_implies_Mapping_is_Continuous_in_Lower_Topological_Lattice | [
"Topological Order Theory"
] | [
"Definition:Complete Lattice",
"Definition:Topological Lattice",
"Definition:Lower Topology",
"Definition:Mapping",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Mapping Preserves Infimum/Subset",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Closed Set/Topology",
"Relative Complement of Relative Complement",
"Infimum of Upper Closure of Element",
"Empty Set is Closed/Topological Space",
"Definition:Closed Set/Topology",
"Definition:Mapping Preserves Infimum/Subset",
"Definition:Mapping Preserves Infimum/Subset",
"Definition:Co... |
proofwiki-13530 | 5040 is Product of Consecutive Numbers in Two Ways | :$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$ | Follows from Factorial as Product of Two Factorials:
:$10! = 6! \times 7!$
and so:
:$\dfrac {10!} {6!} = 10 \times 9 \times 8 \times 7 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Hence the result.
{{qed}} | :$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$ | Follows from [[Factorial as Product of Two Factorials]]:
:$10! = 6! \times 7!$
and so:
:$\dfrac {10!} {6!} = 10 \times 9 \times 8 \times 7 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Hence the result.
{{qed}} | 5040 is Product of Consecutive Numbers in Two Ways | https://proofwiki.org/wiki/5040_is_Product_of_Consecutive_Numbers_in_Two_Ways | https://proofwiki.org/wiki/5040_is_Product_of_Consecutive_Numbers_in_Two_Ways | [
"Factorials",
"5040"
] | [] | [
"Factorial as Product of Two Factorials"
] |
proofwiki-13531 | Products of Consecutive Integers in 2 Ways | The following integers are the product of consecutive integers in $2$ ways:
:$-720, 720, 5040$ | From 720 is Product of Consecutive Numbers in Two Ways:
:$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$
From 5040 is Product of Consecutive Numbers in Two Ways:
:$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$
Then:
:$-720 = \left({-6}\right) \left({-5}\righ... | The following [[Definition:Integer|integers]] are the [[Definition:Integer Multiplication|product]] of consecutive [[Definition:Integer|integers]] in $2$ ways:
:$-720, 720, 5040$ | From [[720 is Product of Consecutive Numbers in Two Ways]]:
:$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$
From [[5040 is Product of Consecutive Numbers in Two Ways]]:
:$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$
Then:
:$-720 = \left({-6}\right) \lef... | Products of Consecutive Integers in 2 Ways | https://proofwiki.org/wiki/Products_of_Consecutive_Integers_in_2_Ways | https://proofwiki.org/wiki/Products_of_Consecutive_Integers_in_2_Ways | [
"Factorials"
] | [
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Integer"
] | [
"720 is Product of Consecutive Numbers in Two Ways",
"5040 is Product of Consecutive Numbers in Two Ways",
"Definition:Divisor (Algebra)/Integer",
"Definition:Multiplication/Integers",
"Definition:Positive/Integer",
"Category:Factorials"
] |
proofwiki-13532 | 8 Mutually Non-Attacking Rooks on Chessboard | On a standard chessboard, it is possible to arrange a maximum of $8$ rooks so that no rook is attacking any other rook.
There are $5282$ such arrangements, up to rotation and reflection. | {{ProofWanted|No doubt we will eventually progress to chess problems of various styles.}} | On a standard [[Definition:Chessboard|chessboard]], it is possible to arrange a maximum of $8$ [[Definition:Chess Rook|rooks]] so that no [[Definition:Chess Rook|rook]] is attacking any other [[Definition:Chess Rook|rook]].
There are $5282$ such arrangements, up to rotation and reflection. | {{ProofWanted|No doubt we will eventually progress to chess problems of various styles.}} | 8 Mutually Non-Attacking Rooks on Chessboard | https://proofwiki.org/wiki/8_Mutually_Non-Attacking_Rooks_on_Chessboard | https://proofwiki.org/wiki/8_Mutually_Non-Attacking_Rooks_on_Chessboard | [
"5282",
"Recreational Chess"
] | [
"Definition:Chess/Chessboard",
"Definition:Chess/Piece/Rook",
"Definition:Chess/Piece/Rook",
"Definition:Chess/Piece/Rook"
] | [] |
proofwiki-13533 | Triangular Lucas Numbers | The only Lucas numbers which are also triangular are:
:$1, 3, 5778$
{{OEIS|A248506}} | {{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \times 2} 2
}}
{{eqn | l = 3
| r = \dfrac {2 \times 3} 2
| rr= = 2 + 1
}}
{{eqn | l = 5778
| r = \dfrac {107 \times 108} 2
| rr= = 2207 + 3571
}}
{{end-eqn}}
{{ProofWanted|It remains to be shown that these are the only ones.}} | The only [[Definition:Lucas Number|Lucas numbers]] which are also [[Definition:Triangular Number|triangular]] are:
:$1, 3, 5778$
{{OEIS|A248506}} | {{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \times 2} 2
}}
{{eqn | l = 3
| r = \dfrac {2 \times 3} 2
| rr= = 2 + 1
}}
{{eqn | l = 5778
| r = \dfrac {107 \times 108} 2
| rr= = 2207 + 3571
}}
{{end-eqn}}
{{ProofWanted|It remains to be shown that these are the only ones.}} | Triangular Lucas Numbers | https://proofwiki.org/wiki/Triangular_Lucas_Numbers | https://proofwiki.org/wiki/Triangular_Lucas_Numbers | [
"Lucas Numbers",
"Triangular Numbers"
] | [
"Definition:Lucas Number",
"Definition:Triangular Number"
] | [] |
proofwiki-13534 | Sum of Two Rational 4th Powers but not Two Integer 4th Powers | $5906$ is the smallest integer which can be expressed as the sum of two rational $4$th powers, but not two integer $4$th powers. | :$5906 = \paren {\dfrac {149} {17} }^4 + \paren {\dfrac {25} {17} }^4$
Suppose $5906$ is a sum of two integer $4$th powers.
We have:
:$9^4 = 6561 > 5906$
which shows that no $4$th power greater than $8^4$ is in the sum.
:$7^4 + 7^4 = 4802 < 5906$
which shows that some $4$th power greater than $7^4$ is in the sum.
So th... | $5906$ is the smallest [[Definition:Integer|integer]] which can be expressed as the [[Definition:Rational Addition|sum]] of two [[Definition:Rational Power|rational $4$th powers]], but not two [[Definition:Integer Power|integer $4$th powers]]. | :$5906 = \paren {\dfrac {149} {17} }^4 + \paren {\dfrac {25} {17} }^4$
Suppose $5906$ is a [[Definition:Integer Addition|sum]] of two [[Definition:Integer Power|integer $4$th powers]].
We have:
:$9^4 = 6561 > 5906$
which shows that no [[Definition:Integer Power|$4$th power]] greater than $8^4$ is in the [[Definiti... | Sum of Two Rational 4th Powers but not Two Integer 4th Powers | https://proofwiki.org/wiki/Sum_of_Two_Rational_4th_Powers_but_not_Two_Integer_4th_Powers | https://proofwiki.org/wiki/Sum_of_Two_Rational_4th_Powers_but_not_Two_Integer_4th_Powers | [
"Fourth Powers",
"5906"
] | [
"Definition:Integer",
"Definition:Addition/Rational Numbers",
"Definition:Power (Algebra)/Rational Number",
"Definition:Power (Algebra)/Integer"
] | [
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Addition... |
proofwiki-13535 | Subspace of Subspace is Subspace | Let $T = \struct{S, \tau}$ be a topological space.
Let $H \subseteq S$ be a non-empty subset of $S$ and $\tau_H$ be the subspace topology on $H$.
Let $K \subseteq S$ be a non-empty subset of $S$.
Then the subspace topology on $K$ induced by $\tau$ equals the subspace topology on $K$ induced by $\tau_H$.
{{explain|Why i... | Let $\tau_K$ be the subspace topology on $K$ induced by $\tau$.
Let $\tau'_K$ be the subspace topology on $K$ induced by $\tau_H$.
Then
{{begin-eqn}}
{{eqn | l = V \in \tau'_K
| o = \leadstoandfrom
| r = \exists U' \in \tau_H : V = U' \cap K
| c = {{Defof|Subspace Topology}} $\tau'_K$
}}
{{eqn | o = ... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$ and $\tau_H$ be the [[Definition:Subspace Topology|subspace topology]] on $H$.
Let $K \subseteq S$ be a [[Definition:Non-Empty Set|non... | Let $\tau_K$ be the [[Definition:Subspace Topology|subspace topology]] on $K$ induced by $\tau$.
Let $\tau'_K$ be the [[Definition:Subspace Topology|subspace topology]] on $K$ induced by $\tau_H$.
Then
{{begin-eqn}}
{{eqn | l = V \in \tau'_K
| o = \leadstoandfrom
| r = \exists U' \in \tau_H : V = U' \cap ... | Subspace of Subspace is Subspace | https://proofwiki.org/wiki/Subspace_of_Subspace_is_Subspace | https://proofwiki.org/wiki/Subspace_of_Subspace_is_Subspace | [
"Topological Subspaces"
] | [
"Definition:Topological Space",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Topological Subspace"
] | [
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Intersection is Associative",
"Intersection with Subset is Subset"
] |
proofwiki-13536 | Positive Integer Sum of 3 Fourth Powers in 2 Ways | The smallest positive integer which can be expressed as the sum of $3$ fourth powers in $2$ different ways is $6578$:
{{begin-eqn}}
{{eqn | l = 1^4 + 2^4 + 9^4
| r = 1 + 16 + 6561
| c =
}}
{{eqn | r = 6578
| c =
}}
{{eqn | r = 81 + 2401 + 4096
| c =
}}
{{eqn | r = 3^4 + 7^4 + 8^4
| c = ... | The fact that this is the smallest can be demonstrated by calculation.
{{qed}} | The smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Fourth Power|fourth powers]] in $2$ different ways is $6578$:
{{begin-eqn}}
{{eqn | l = 1^4 + 2^4 + 9^4
| r = 1 + 16 + 6561
| c =
}}
{{eqn | r = 6578
| ... | The fact that this is the smallest can be demonstrated by calculation.
{{qed}} | Positive Integer Sum of 3 Fourth Powers in 2 Ways | https://proofwiki.org/wiki/Positive_Integer_Sum_of_3_Fourth_Powers_in_2_Ways | https://proofwiki.org/wiki/Positive_Integer_Sum_of_3_Fourth_Powers_in_2_Ways | [
"Fourth Powers"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Fourth Power"
] | [] |
proofwiki-13537 | Mapping Preserves Infima implies Mapping is Continuous in Lower Topological Lattice | Let $T = \struct {S, \preceq, \tau}$ and $Q = \struct {X, \preceq', \tau'}$ be complete topological lattices with lower topologies.
Let $f: S \to X$ be a mapping such that
:$f$ preserves all infima.
Then $f$ is continuous mapping. | By assumption:
:for all non-empty subsets $Y$ of $S$: $f$ preserves the infimum of $Y$.
Thus by Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice:
:$f$ is continuous mapping.
{{qed}} | Let $T = \struct {S, \preceq, \tau}$ and $Q = \struct {X, \preceq', \tau'}$ be [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Lower Topology|lower topologies]].
Let $f: S \to X$ be a [[Definition:Mapping|mapping]] such that
:$f$ [[Definition:Mapping P... | By assumption:
:for all [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subsets]] $Y$ of $S$: $f$ [[Definition:Mapping Preserves Infimum/Subset|preserves the infimum]] of $Y$.
Thus by [[Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice]]:
:$f$ is [[Definition:Cont... | Mapping Preserves Infima implies Mapping is Continuous in Lower Topological Lattice | https://proofwiki.org/wiki/Mapping_Preserves_Infima_implies_Mapping_is_Continuous_in_Lower_Topological_Lattice | https://proofwiki.org/wiki/Mapping_Preserves_Infima_implies_Mapping_is_Continuous_in_Lower_Topological_Lattice | [
"Topological Order Theory"
] | [
"Definition:Complete Lattice",
"Definition:Topological Lattice",
"Definition:Lower Topology",
"Definition:Mapping",
"Definition:Mapping Preserves Infimum/All",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Mapping Preserves Infimum/Subset",
"Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice",
"Definition:Continuous Mapping (Topology)"
] |
proofwiki-13538 | Compact in Subspace is Compact in Topological Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $K \subseteq S$ be a subset.
Let $\tau_K$ be the subspace topology on $K$.
Let $T' = \struct {K, \tau_K}$ be the topological subspace of $T$ determined by $K$.
Let $H \subseteq K$ be compact in $T'$.
Then $H$ is compact in $T$. | Suppose that $H$ is compact in $T'$.
Let $\set {W_i}_{i \mathop \in J}$ be an open cover of $H$ in $T$.
Then $\ds H \subseteq \bigcup_{i \mathop \in J} W_i$.
Then:
{{begin-eqn}}
{{eqn | l = H
| r = H \cap K
| c = Intersection with Subset is Subset: from $H \subseteq K$
}}
{{eqn | o = \subseteq
| r = ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $K \subseteq S$ be a [[Definition:Subset|subset]].
Let $\tau_K$ be the [[Definition:Subspace Topology|subspace topology]] on $K$.
Let $T' = \struct {K, \tau_K}$ be the [[Definition:Topological Subspace|topological subspace]] of ... | Suppose that $H$ is [[Definition:Compact Topological Subspace|compact]] in $T'$.
Let $\set {W_i}_{i \mathop \in J}$ be an [[Definition:Open Cover|open cover]] of $H$ in $T$.
Then $\ds H \subseteq \bigcup_{i \mathop \in J} W_i$.
Then:
{{begin-eqn}}
{{eqn | l = H
| r = H \cap K
| c = [[Intersection with S... | Compact in Subspace is Compact in Topological Space | https://proofwiki.org/wiki/Compact_in_Subspace_is_Compact_in_Topological_Space | https://proofwiki.org/wiki/Compact_in_Subspace_is_Compact_in_Topological_Space | [
"Compact Topological Spaces",
"Topological Subspaces"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Open Cover",
"Intersection with Subset is Subset",
"Set Intersection Preserves Subsets/Corollary",
"Intersection Distributes over Union/Family of Sets",
"Definition:Open Cover",
"Definition:Compact Topological Space/Subspace",
"Definition:Su... |
proofwiki-13539 | If Infimum of Filtered Subset belongs to Element of Sub-Basis then Subset and Element Intersect implies Infimum of Subset belongs to Closure of Subset | Let $T = \struct {S, \preceq, \tau}$ be a complete topological lattice with lower topology.
Let $B$ be an analytic sub-basis of $T$.
Let $F$ be a filtered subset of $S$ such that
:$\forall A \in B: \inf F \in A \implies F \cap A \ne \O$
Then $\inf F \in F^-$
where $F^-$ denotes the topological closure of $F$. | We will prove that
:$\forall A \in B, x \in F \cap A, y \in F: y \preceq x \implies y \in A$
Let $A \in B$, $x \in F \cap A$, $y \in F$.
By definition of sub-basis:
:$A$ is open.
By Open Subset is Lower Section in Lower Topology:
:$A$ is a lower section.
By definition of intersection:
:$x \in A$.
Thus by definition of ... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Lower Topology|lower topology]].
Let $B$ be an [[Definition:Analytic Sub-Basis|analytic sub-basis]] of $T$.
Let $F$ be a [[Definition:Filtered Subset|filtered sub... | We will prove that
:$\forall A \in B, x \in F \cap A, y \in F: y \preceq x \implies y \in A$
Let $A \in B$, $x \in F \cap A$, $y \in F$.
By definition of [[Definition:Analytic Sub-Basis|sub-basis]]:
:$A$ is [[Definition:Open Set (Topology)|open]].
By [[Open Subset is Lower Section in Lower Topology]]:
:$A$ is a [[De... | If Infimum of Filtered Subset belongs to Element of Sub-Basis then Subset and Element Intersect implies Infimum of Subset belongs to Closure of Subset | https://proofwiki.org/wiki/If_Infimum_of_Filtered_Subset_belongs_to_Element_of_Sub-Basis_then_Subset_and_Element_Intersect_implies_Infimum_of_Subset_belongs_to_Closure_of_Subset | https://proofwiki.org/wiki/If_Infimum_of_Filtered_Subset_belongs_to_Element_of_Sub-Basis_then_Subset_and_Element_Intersect_implies_Infimum_of_Subset_belongs_to_Closure_of_Subset | [
"Topological Order Theory"
] | [
"Definition:Complete Lattice",
"Definition:Topological Lattice",
"Definition:Lower Topology",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Filtered Subset",
"Definition:Closure (Topology)"
] | [
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Open Set/Topology",
"Open Subset is Lower Section in Lower Topology",
"Definition:Lower Section",
"Definition:Set Intersection",
"Definition:Lower Section",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Basis (Topology)/Analytic Basis",
... |
proofwiki-13540 | Square of Repdigit Number consisting of Instances of 6 | The following pattern holds:
{{begin-eqn}}
{{eqn | l = 6^2
| r = 36
}}
{{eqn | l = 3 + 6
| r = 9
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 66^2
| r = 4356
}}
{{eqn | l = 43 + 56
| r = 99
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 666^2
| r = 443 \, 556
}}
{{eqn | l = 443 + 556
| r = 999... | {{ProofWanted|Simple but tedious.}} | The following pattern holds:
{{begin-eqn}}
{{eqn | l = 6^2
| r = 36
}}
{{eqn | l = 3 + 6
| r = 9
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 66^2
| r = 4356
}}
{{eqn | l = 43 + 56
| r = 99
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 666^2
| r = 443 \, 556
}}
{{eqn | l = 443 + 556
| r ... | {{ProofWanted|Simple but tedious.}} | Square of Repdigit Number consisting of Instances of 6 | https://proofwiki.org/wiki/Square_of_Repdigit_Number_consisting_of_Instances_of_6 | https://proofwiki.org/wiki/Square_of_Repdigit_Number_consisting_of_Instances_of_6 | [
"Square Numbers",
"Repdigit Numbers"
] | [] | [] |
proofwiki-13541 | Square of Repdigit Number consisting of Instances of 3 | The following pattern holds:
{{begin-eqn}}
{{eqn | l = 3^2
| r = 09
}}
{{eqn | l = 0 + 9
| r = 9
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 33^2
| r = 1089
}}
{{eqn | l = 10 + 89
| r = 99
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 333^2
| r = 110 \, 889
}}
{{eqn | l = 110 + 889
| r = 999... | {{ProofWanted|Simple but tedious.}} | The following pattern holds:
{{begin-eqn}}
{{eqn | l = 3^2
| r = 09
}}
{{eqn | l = 0 + 9
| r = 9
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 33^2
| r = 1089
}}
{{eqn | l = 10 + 89
| r = 99
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 333^2
| r = 110 \, 889
}}
{{eqn | l = 110 + 889
| r ... | {{ProofWanted|Simple but tedious.}} | Square of Repdigit Number consisting of Instances of 3 | https://proofwiki.org/wiki/Square_of_Repdigit_Number_consisting_of_Instances_of_3 | https://proofwiki.org/wiki/Square_of_Repdigit_Number_consisting_of_Instances_of_3 | [
"Square Numbers",
"Repdigit Numbers"
] | [] | [] |
proofwiki-13542 | Product with Repdigit can be Split into Parts which Add to Repdigit | Let $n$ be a positive integer with $d_1$ digits.
Let $m$ be a repdigit number with $d_2$ digits such that $d_2 > d_1$.
Let $r$ consist of the result when the rightmost $d_2$ digits of $m n$ is cut off and added to the remaining left hand portion.
Then $r$ is a repdigit number. | Let $b > 1$ be the base we are working on.
Let $m = \sqbrk {aaa \dots a}_b$.
Let $R = \dfrac m a = \sqbrk {111 \dots 1}_b = \dfrac {b^{d_2} - 1} {b - 1}$.
Let the rightmost $d_2$ digits of $m n$ be $y$ and the remaining left hand portion be $x$.
Then we have:
{{begin-eqn}}
{{eqn | l = 0
| o = \equiv
| r = m... | Let $n$ be a [[Definition:Positive Integer|positive integer]] with $d_1$ [[Definition:Digit|digits]].
Let $m$ be a [[Definition:Repdigit Number|repdigit number]] with $d_2$ [[Definition:Digit|digits]] such that $d_2 > d_1$.
Let $r$ consist of the result when the rightmost $d_2$ [[Definition:Digit|digits]] of $m n$ is... | Let $b > 1$ be the [[Definition:Number Base|base]] we are working on.
Let $m = \sqbrk {aaa \dots a}_b$.
Let $R = \dfrac m a = \sqbrk {111 \dots 1}_b = \dfrac {b^{d_2} - 1} {b - 1}$.
Let the rightmost $d_2$ [[Definition:Digit|digits]] of $m n$ be $y$ and the remaining left hand portion be $x$.
Then we have:
{{begin-... | Product with Repdigit can be Split into Parts which Add to Repdigit | https://proofwiki.org/wiki/Product_with_Repdigit_can_be_Split_into_Parts_which_Add_to_Repdigit | https://proofwiki.org/wiki/Product_with_Repdigit_can_be_Split_into_Parts_which_Add_to_Repdigit | [
"Repdigit Numbers",
"Product with Repdigit can be Split into Parts which Add to Repdigit"
] | [
"Definition:Positive/Integer",
"Definition:Digit",
"Definition:Repdigit Number",
"Definition:Digit",
"Definition:Digit",
"Definition:Repdigit Number"
] | [
"Definition:Number Base",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Digit",
"Definition:Repdigit Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit"
] |
proofwiki-13543 | Sum of Reciprocals of Squares of Odd Integers as Double Integral | :$\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2} = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y$ | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2}
| r = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1} \paren {2 n - 1} }
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \int_0^1 x^{2 n - 2} \rd x \int_0^1 y^{2 n - 2} \rd y
| c = Primitive of Power
}}
{{eqn | r = \sum_{n \... | :$\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2} = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y$ | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2}
| r = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1} \paren {2 n - 1} }
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \int_0^1 x^{2 n - 2} \rd x \int_0^1 y^{2 n - 2} \rd y
| c = [[Primitive of Power]]
}}
{{eqn | r = \sum_... | Sum of Reciprocals of Squares of Odd Integers as Double Integral/Proof 1 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Squares_of_Odd_Integers_as_Double_Integral | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Squares_of_Odd_Integers_as_Double_Integral/Proof_1 | [
"Riemann Zeta Function",
"Sum of Reciprocals of Squares of Odd Integers as Double Integral"
] | [] | [
"Primitive of Power",
"Exponent Combination Laws/Product of Powers",
"Exponent Combination Laws/Power of Product",
"Fubini's Theorem",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-13544 | Sum of Reciprocals of Squares of Odd Integers as Double Integral | :$\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2} = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y
| r = \int_0^1 \int_0^1 \paren {1 + x^2 y^2 + x^4 y^4 + x^6 y^6 + \cdots} \rd x \rd y
| c = Sum of Infinite Geometric Sequence
}}
{{eqn | r = \int_0^1 \intlimits {y + \frac {x^2y^3} 3 + \frac {x^4y^5} 5 + \frac {x^6y^7} 7 + \cdots... | :$\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2} = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y
| r = \int_0^1 \int_0^1 \paren {1 + x^2 y^2 + x^4 y^4 + x^6 y^6 + \cdots} \rd x \rd y
| c = [[Sum of Infinite Geometric Sequence]]
}}
{{eqn | r = \int_0^1 \intlimits {y + \frac {x^2y^3} 3 + \frac {x^4y^5} 5 + \frac {x^6y^7} 7 + \c... | Sum of Reciprocals of Squares of Odd Integers as Double Integral/Proof 2 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Squares_of_Odd_Integers_as_Double_Integral | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Squares_of_Odd_Integers_as_Double_Integral/Proof_2 | [
"Riemann Zeta Function",
"Sum of Reciprocals of Squares of Odd Integers as Double Integral"
] | [] | [
"Sum of Infinite Geometric Sequence"
] |
proofwiki-13545 | Three Tri-Automorphic Numbers for each Number of Digits | Let $d \in \Z_{>0}$ be a (strictly) positive integer.
Then there exist exactly $3$ tri-automorphic numbers with exactly $d$ digits.
These tri-automorphic numbers all end in $2$, $5$ or $7$. | Let $n$ be a tri-automorphic number with $d$ digits.
Let $n = 10 a + b$.
Then:
:$3 n^2 = 300a^2 + 60 a b + 3 b^2$
As $n$ is tri-automorphic, we have:
:$(1): \quad 300 a^2 + 60 a b + 3 b^2 = 1000 z + 100 y + 10 a + b$
and:
:$(2): \quad 3 b^2 - b = 10 x$
where $x$ is an integer.
This condition is only satisfied by $b = 2... | Let $d \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then there exist exactly $3$ [[Definition:Tri-Automorphic Number|tri-automorphic numbers]] with exactly $d$ [[Definition:Digit|digits]].
These [[Definition:Tri-Automorphic Number|tri-automorphic numbers]] all end in $2$, $... | Let $n$ be a [[Definition:Tri-Automorphic Number|tri-automorphic number]] with $d$ [[Definition:Digit|digits]].
Let $n = 10 a + b$.
Then:
:$3 n^2 = 300a^2 + 60 a b + 3 b^2$
As $n$ is [[Definition:Tri-Automorphic Number|tri-automorphic]], we have:
:$(1): \quad 300 a^2 + 60 a b + 3 b^2 = 1000 z + 100 y + 10 a + b$
an... | Three Tri-Automorphic Numbers for each Number of Digits | https://proofwiki.org/wiki/Three_Tri-Automorphic_Numbers_for_each_Number_of_Digits | https://proofwiki.org/wiki/Three_Tri-Automorphic_Numbers_for_each_Number_of_Digits | [
"Tri-Automorphic Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Tri-Automorphic Number",
"Definition:Digit",
"Definition:Tri-Automorphic Number"
] | [
"Definition:Tri-Automorphic Number",
"Definition:Digit",
"Definition:Tri-Automorphic Number",
"Definition:Integer"
] |
proofwiki-13546 | Fourier Series/x squared over Minus Pi to Pi | :$\ds x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$ | From Even Power is Even Function, $x^2$ is an even function.
By Fourier Series for Even Function over Symmetric Range, we have:
:$\ds x^2 \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \frac 2 \pi \int_0^\pi x^2 \map \cos {n x} \rd x
}}
{{eqn | ll= \leads... | :$\ds x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$ | From [[Even Power is Even Function]], $x^2$ is an [[Definition:Even Function|even function]].
By [[Fourier Series for Even Function over Symmetric Range]], we have:
:$\ds x^2 \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \frac 2 \pi \int_0^\pi x^2 \m... | Fourier Series/x squared over Minus Pi to Pi | https://proofwiki.org/wiki/Fourier_Series/x_squared_over_Minus_Pi_to_Pi | https://proofwiki.org/wiki/Fourier_Series/x_squared_over_Minus_Pi_to_Pi | [
"Examples of Fourier Series"
] | [] | [
"Even Power is Even Function",
"Definition:Even Function",
"Fourier Series for Even Function over Symmetric Range",
"Cosine of Zero is One",
"Primitive of Power",
"Fundamental Theorem of Calculus",
"Primitive of x squared by Cosine of a x",
"Fundamental Theorem of Calculus",
"Sine of Integer Multipl... |
proofwiki-13547 | Integer and its Double forming Pandigital Pair | $6729$ and its double contain all the digits from $1$ to $9$ between them. | $2 \times 6729 = 13 \, 458$
{{qed}} | $6729$ and its double contain all the [[Definition:Digit|digits]] from $1$ to $9$ between them. | $2 \times 6729 = 13 \, 458$
{{qed}} | Integer and its Double forming Pandigital Pair | https://proofwiki.org/wiki/Integer_and_its_Double_forming_Pandigital_Pair | https://proofwiki.org/wiki/Integer_and_its_Double_forming_Pandigital_Pair | [
"Recreational Mathematics"
] | [
"Definition:Digit"
] | [] |
proofwiki-13548 | 4-Digit Numbers forming Longest Reverse-and-Add Sequence | Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation.
Let $r \left({m}\right)$ be the reverse-and-add process on $m$.
Let $r$ be applied iteratively to $m$.
The $4$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are:
:$6999, 7998, 8997, 9996$
all o... | {{begin-eqn}}
{{eqn | n = 1
| o =
| m = 6999 + 9996
| mo= =
| r = 16995
}}
{{eqn | n = 2
| o =
| m = 16995 + 59961
| mo= =
| r = 76956
}}
{{eqn | n = 3
| o =
| m = 76956 + 65967
| mo= =
| r = 142923
}}
{{eqn | n = 4
| o =
| m = 1429... | Let $m \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]] expressed in [[Definition:Decimal Notation|decimal notation]].
Let $r \left({m}\right)$ be the [[Definition:Reverse-and-Add|reverse-and-add process]] on $m$.
Let $r$ be applied iteratively to $m$.
The $4$-[[Definition:Digit|digit]] [[Definiti... | {{begin-eqn}}
{{eqn | n = 1
| o =
| m = 6999 + 9996
| mo= =
| r = 16995
}}
{{eqn | n = 2
| o =
| m = 16995 + 59961
| mo= =
| r = 76956
}}
{{eqn | n = 3
| o =
| m = 76956 + 65967
| mo= =
| r = 142923
}}
{{eqn | n = 4
| o =
| m = 1429... | 4-Digit Numbers forming Longest Reverse-and-Add Sequence | https://proofwiki.org/wiki/4-Digit_Numbers_forming_Longest_Reverse-and-Add_Sequence | https://proofwiki.org/wiki/4-Digit_Numbers_forming_Longest_Reverse-and-Add_Sequence | [
"Reverse-and-Add"
] | [
"Definition:Positive/Integer",
"Definition:Decimal Notation",
"Definition:Reverse-and-Add",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Palindromic Number"
] | [
"Definition:Palindromic Number",
"Definition:Reversal"
] |
proofwiki-13549 | Fourier Series/Identity Function over Minus Pi to Pi | For $x \in \openint {-\pi} \pi$:
:$\ds x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin n x$ | From Odd Power is Odd Function, $x$ is an odd function.
By Fourier Series for Odd Function over Symmetric Range, we have:
:$\ds x \sim \sum_{n \mathop = 1}^\infty b_n \sin n x$
where:
{{begin-eqn}}
{{eqn | l = b_n
| r = \frac 2 \pi \int_0^\pi x \sin n x \rd x
| c =
}}
{{eqn | r = \frac 2 \pi \intlimits {\f... | For $x \in \openint {-\pi} \pi$:
:$\ds x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin n x$ | From [[Odd Power is Odd Function]], $x$ is an [[Definition:Odd Function|odd function]].
By [[Fourier Series for Odd Function over Symmetric Range]], we have:
:$\ds x \sim \sum_{n \mathop = 1}^\infty b_n \sin n x$
where:
{{begin-eqn}}
{{eqn | l = b_n
| r = \frac 2 \pi \int_0^\pi x \sin n x \rd x
| c =
}... | Fourier Series/Identity Function over Minus Pi to Pi/Proof 1 | https://proofwiki.org/wiki/Fourier_Series/Identity_Function_over_Minus_Pi_to_Pi | https://proofwiki.org/wiki/Fourier_Series/Identity_Function_over_Minus_Pi_to_Pi/Proof_1 | [
"Fourier Series for Identity Function"
] | [] | [
"Odd Power is Odd Function",
"Definition:Odd Function",
"Fourier Series for Odd Function over Symmetric Range",
"Primitive of x by Sine of a x",
"Fundamental Theorem of Calculus",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple of Pi"
] |
proofwiki-13550 | Fourier Series/Identity Function over Minus Pi to Pi | For $x \in \openint {-\pi} \pi$:
:$\ds x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin n x$ | By Fourier Series for Identity Function over Symmetric Range, the function $f: \openint {-\lambda} \lambda \to \R$ defined as:
:$\forall x \in \openint {-\lambda} \lambda: \map f x = x$
has a Fourier series:
:$\map f x \sim \dfrac {2 \lambda} \pi \ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin \frac... | For $x \in \openint {-\pi} \pi$:
:$\ds x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin n x$ | By [[Fourier Series for Identity Function over Symmetric Range]], the [[Definition:Real Function|function]] $f: \openint {-\lambda} \lambda \to \R$ defined as:
:$\forall x \in \openint {-\lambda} \lambda: \map f x = x$
has a [[Definition:Fourier Series|Fourier series]]:
:$\map f x \sim \dfrac {2 \lambda} \pi \ds \su... | Fourier Series/Identity Function over Minus Pi to Pi/Proof 2 | https://proofwiki.org/wiki/Fourier_Series/Identity_Function_over_Minus_Pi_to_Pi | https://proofwiki.org/wiki/Fourier_Series/Identity_Function_over_Minus_Pi_to_Pi/Proof_2 | [
"Fourier Series for Identity Function"
] | [] | [
"Fourier Series/Identity Function over Symmetric Range",
"Definition:Real Function",
"Definition:Fourier Series"
] |
proofwiki-13551 | Fourier Series/Fourth Power of x over Minus Pi to Pi | :$\ds x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \cos n \pi \cos n x$ | Since $x^4 = \paren {-x}^4$, $x^4$ is an even function.
By Fourier Series for Even Function over Symmetric Range, the Fourier series of $\map f x$ can be expressed as:
:$x^4 \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{> 0}$:
{{begin-eqn}}
{{eqn | l = a_n
| r = \df... | :$\ds x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \cos n \pi \cos n x$ | Since $x^4 = \paren {-x}^4$, $x^4$ is an [[Definition:Even Function|even function]].
By [[Fourier Series for Even Function over Symmetric Range]], the [[Definition:Fourier Series|Fourier series]] of $\map f x$ can be expressed as:
:$x^4 \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for a... | Fourier Series/Fourth Power of x over Minus Pi to Pi | https://proofwiki.org/wiki/Fourier_Series/Fourth_Power_of_x_over_Minus_Pi_to_Pi | https://proofwiki.org/wiki/Fourier_Series/Fourth_Power_of_x_over_Minus_Pi_to_Pi | [
"Examples of Fourier Series"
] | [] | [
"Definition:Even Function",
"Fourier Series for Even Function over Symmetric Range",
"Definition:Fourier Series",
"Primitive of x fourth by Cosine of a x",
"Primitive of Power",
"Category:Examples of Fourier Series"
] |
proofwiki-13552 | Fourier Series/Absolute Value of x over Minus Pi to Pi | For $x \in \openint {-\pi} \pi$:
:$\ds \size x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\map \cos {2 n - 1} x} {\paren {2 n - 1}^2}$ | By definition, the absolute value function is an even function:
:$\size {-x} = x = \size x$
Thus by Fourier Series for Even Function over Symmetric Range, $\size x$ can be expressed as:
:$\ds \size x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{\ge 0}$:
:$a_n = \ds \frac 2 \pi... | For $x \in \openint {-\pi} \pi$:
:$\ds \size x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\map \cos {2 n - 1} x} {\paren {2 n - 1}^2}$ | By definition, the [[Definition:Absolute Value|absolute value]] function is an [[Definition:Even Function|even function]]:
:$\size {-x} = x = \size x$
Thus by [[Fourier Series for Even Function over Symmetric Range]], $\size x$ can be expressed as:
:$\ds \size x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \... | Fourier Series/Absolute Value of x over Minus Pi to Pi/Proof 1 | https://proofwiki.org/wiki/Fourier_Series/Absolute_Value_of_x_over_Minus_Pi_to_Pi | https://proofwiki.org/wiki/Fourier_Series/Absolute_Value_of_x_over_Minus_Pi_to_Pi/Proof_1 | [
"Fourier Series for Absolute Value Function",
"Fourier Series/Absolute Value of x over Minus Pi to Pi"
] | [] | [
"Definition:Absolute Value",
"Definition:Even Function",
"Fourier Series for Even Function over Symmetric Range",
"Definition:Real Interval",
"Half-Range Fourier Cosine Series/Identity Function/0 to Pi"
] |
proofwiki-13553 | Fourier Series/Absolute Value of x over Minus Pi to Pi | For $x \in \openint {-\pi} \pi$:
:$\ds \size x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\map \cos {2 n - 1} x} {\paren {2 n - 1}^2}$ | By Fourier Series for Absolute Value Function over Symmetric Range, the function $f: \openint {-\lambda} \lambda \to \R$ defined as:
:$\forall x \in \openint {-\lambda} \lambda: \map f x = \size x$
has a Fourier series:
:$\map f x \sim \dfrac \lambda 2 - \ds \dfrac {4 \lambda} {\pi^2} \sum_{n \mathop = 0}^\infty \frac ... | For $x \in \openint {-\pi} \pi$:
:$\ds \size x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\map \cos {2 n - 1} x} {\paren {2 n - 1}^2}$ | By [[Fourier Series for Absolute Value Function over Symmetric Range]], the [[Definition:Real Function|function]] $f: \openint {-\lambda} \lambda \to \R$ defined as:
:$\forall x \in \openint {-\lambda} \lambda: \map f x = \size x$
has a [[Definition:Fourier Series|Fourier series]]:
:$\map f x \sim \dfrac \lambda 2 -... | Fourier Series/Absolute Value of x over Minus Pi to Pi/Proof 2 | https://proofwiki.org/wiki/Fourier_Series/Absolute_Value_of_x_over_Minus_Pi_to_Pi | https://proofwiki.org/wiki/Fourier_Series/Absolute_Value_of_x_over_Minus_Pi_to_Pi/Proof_2 | [
"Fourier Series for Absolute Value Function",
"Fourier Series/Absolute Value of x over Minus Pi to Pi"
] | [] | [
"Fourier Series/Absolute Value Function over Symmetric Range",
"Definition:Real Function",
"Definition:Fourier Series"
] |
proofwiki-13554 | Fourier Series/Pi minus x over 0 to 2 Pi | :$\ds \pi - x = 2 \sum_{n \mathop = 1}^\infty \frac {\sin n x} n$ | By definition of Fourier series:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \dfrac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x
}}
{{eqn | l = b_n
| r = \dfrac 1 \pi \int_0^{2 \pi} \map f x \sin n x \rd x
}... | :$\ds \pi - x = 2 \sum_{n \mathop = 1}^\infty \frac {\sin n x} n$ | By definition of [[Definition:Fourier Series over Range 2 Pi|Fourier series]]:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \dfrac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x
}}
{{eqn | l = b_n
| r = \dfra... | Fourier Series/Pi minus x over 0 to 2 Pi | https://proofwiki.org/wiki/Fourier_Series/Pi_minus_x_over_0_to_2_Pi | https://proofwiki.org/wiki/Fourier_Series/Pi_minus_x_over_0_to_2_Pi | [
"Examples of Fourier Series"
] | [] | [
"Definition:Fourier Series/Range 2 Pi",
"Cosine of Zero is One",
"Primitive of Power",
"Integral over 2 pi of Cosine of n x",
"Primitive of x by Cosine of a x",
"Sine of Integer Multiple of Pi",
"Sine and Cosine are Periodic on Reals",
"Integral over 2 pi of Sine of n x",
"Primitive of x by Sine of ... |
proofwiki-13555 | Cube of 20 is Sum of Sequence of 4 Consecutive Cubes | :$20^3 = \ds \sum_{k \mathop = 11}^{14} k^3$
That is:
:$20^3 = 11^3 + 12^3 + 13^3 + 14^3$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{14} k^3
| r = \paren {\dfrac {14 \paren {14 + 1} } 2}^2
| c = Sum of Sequence of Cubes
}}
{{eqn | r = 11 \, 025
| c =
}}
{{eqn | l = \sum_{k \mathop = 1}^{10} k^3
| r = \paren {\dfrac {10 \paren {10 + 1} } 2}^2
| c = Sum of Sequence of Cubes... | :$20^3 = \ds \sum_{k \mathop = 11}^{14} k^3$
That is:
:$20^3 = 11^3 + 12^3 + 13^3 + 14^3$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{14} k^3
| r = \paren {\dfrac {14 \paren {14 + 1} } 2}^2
| c = [[Sum of Sequence of Cubes]]
}}
{{eqn | r = 11 \, 025
| c =
}}
{{eqn | l = \sum_{k \mathop = 1}^{10} k^3
| r = \paren {\dfrac {10 \paren {10 + 1} } 2}^2
| c = [[Sum of Sequence of... | Cube of 20 is Sum of Sequence of 4 Consecutive Cubes | https://proofwiki.org/wiki/Cube_of_20_is_Sum_of_Sequence_of_4_Consecutive_Cubes | https://proofwiki.org/wiki/Cube_of_20_is_Sum_of_Sequence_of_4_Consecutive_Cubes | [
"Cube Numbers",
"20",
"8000"
] | [] | [
"Sum of Sequence of Cubes",
"Sum of Sequence of Cubes"
] |
proofwiki-13556 | Mapping is Continuous implies Mapping Preserves Filtered Infima in Lower Topological Lattice | Let $T = \struct {S, \preceq, \tau}$ and $Q = \struct {X, \preceq', \tau'}$ be complete topological lattices with lower topologies.
Let $f: S \to X$ be a mapping such that:
:$f$ is a continuous mapping.
Then $f$ preserves filtered infima. | Define $B := \set {\relcomp S {x^\succeq}: x \in S}$
By definition of lower topology:
:$B$ is an analytic sub-basis.
Let $F$ be a filtered subset of $S$ such that:
:$F$ admits an infimum in $T$.
Thus by definition of complete lattice:
:$f \sqbrk F$ admits an infimum in $Q$.
We will prove that:
:$\forall A \in B: \inf F... | Let $T = \struct {S, \preceq, \tau}$ and $Q = \struct {X, \preceq', \tau'}$ be [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Lower Topology|lower topologies]].
Let $f: S \to X$ be a [[Definition:Mapping|mapping]] such that:
:$f$ is a [[Definition:Con... | Define $B := \set {\relcomp S {x^\succeq}: x \in S}$
By definition of [[Definition:Lower Topology|lower topology]]:
:$B$ is an [[Definition:Analytic Sub-Basis|analytic sub-basis]].
Let $F$ be a [[Definition:Filtered Subset|filtered subset]] of $S$ such that:
:$F$ admits an [[Definition:Infimum of Set|infimum]] in $T$... | Mapping is Continuous implies Mapping Preserves Filtered Infima in Lower Topological Lattice | https://proofwiki.org/wiki/Mapping_is_Continuous_implies_Mapping_Preserves_Filtered_Infima_in_Lower_Topological_Lattice | https://proofwiki.org/wiki/Mapping_is_Continuous_implies_Mapping_Preserves_Filtered_Infima_in_Lower_Topological_Lattice | [
"Topological Order Theory"
] | [
"Definition:Complete Lattice",
"Definition:Topological Lattice",
"Definition:Lower Topology",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Mapping Preserves Infimum/Filtered"
] | [
"Definition:Lower Topology",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Filtered Subset",
"Definition:Infimum of Set",
"Definition:Complete Lattice",
"Definition:Infimum of Set",
"Definition:Relative Complement",
"Definition:Upper Closure/Element",
"Definition:Infimum of Set",
"Definit... |
proofwiki-13557 | 3 Numbers in A.P. whose 4th Powers are Sum of Four 4th Powers | The following triplets of integers in arithmetic sequence with common difference of $60$ can all be expressed as the sum of four $4$th powers:
:$\tuple {8373, 8433, 8493}, \tuple {8517, 8577, 8637}, \ldots$ | {{begin-eqn}}
{{eqn | l = 8373^4
| r = 4450^4 + 5500^4 + 5670^4 + 7123^4
}}
{{eqn | l = 8433^4
| r = 4730^4 + 4806^4 + 5230^4 + 7565^4
}}
{{eqn | l = 8493^4
| r = 524^4 + 4910^4 + 5925^4 + 7630^4
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 8517^4
| r = 1642^4 + 3440^4 + 6100^4 + 7815^4
}}
{{eqn | l... | The following [[Definition:Ordered Triple|triplets]] of [[Definition:Integer|integers]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] with [[Definition:Common Difference|common difference]] of $60$ can all be expressed as the [[Definition:Integer Addition|sum]] of four [[Definition:Fourth Power|$4$th powers... | {{begin-eqn}}
{{eqn | l = 8373^4
| r = 4450^4 + 5500^4 + 5670^4 + 7123^4
}}
{{eqn | l = 8433^4
| r = 4730^4 + 4806^4 + 5230^4 + 7565^4
}}
{{eqn | l = 8493^4
| r = 524^4 + 4910^4 + 5925^4 + 7630^4
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 8517^4
| r = 1642^4 + 3440^4 + 6100^4 + 7815^4
}}
{{eqn |... | 3 Numbers in A.P. whose 4th Powers are Sum of Four 4th Powers | https://proofwiki.org/wiki/3_Numbers_in_A.P._whose_4th_Powers_are_Sum_of_Four_4th_Powers | https://proofwiki.org/wiki/3_Numbers_in_A.P._whose_4th_Powers_are_Sum_of_Four_4th_Powers | [
"Fourth Powers"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Integer",
"Definition:Arithmetic Sequence",
"Definition:Arithmetic Sequence/Common Difference",
"Definition:Addition/Integers",
"Definition:Fourth Power"
] | [] |
proofwiki-13558 | Intersection of Chain of Prime Ideals of Commutative Ring is Prime Ideal | Let $R$ be a commutative ring.
Let $\Spec R$ be the spectrum of $R$, ordered by the subset relation.
Let $\set {P_\alpha}_{\alpha \mathop \in A}$ be a non-empty chain of prime ideals of $\Spec R$.
Let $\ds P = \bigcap_{\alpha \mathop \in A} P_\alpha$ be their intersection.
Then $P$ is a prime ideal of $R$. | By Intersection of Ring Ideals is Ideal, $P$ is an ideal of $R$.
We show that $P$ is prime.
Let $a, b\in R$ with $a, b \notin P$.
We show that $a b \notin P$.
Because $a \notin P$, there exists $\alpha \in A$ with $a \notin P_\alpha$.
Because $b \notin P$, there exists $\beta \in A$ with $b \notin P_\beta$.
Because $\s... | Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\Spec R$ be the [[Definition:Spectrum of Ring|spectrum]] of $R$, [[Definition:Ordered Set|ordered]] by the [[Definition:Subset Relation|subset relation]].
Let $\set {P_\alpha}_{\alpha \mathop \in A}$ be a [[Definition:Non-Empty Set|non-empty]] [[Defi... | By [[Intersection of Ring Ideals is Ideal]], $P$ is an [[Definition:Ideal of Ring|ideal]] of $R$.
We show that $P$ is [[Definition:Prime Ideal of Ring|prime]].
Let $a, b\in R$ with $a, b \notin P$.
We show that $a b \notin P$.
Because $a \notin P$, there exists $\alpha \in A$ with $a \notin P_\alpha$.
Because $b \... | Intersection of Chain of Prime Ideals of Commutative Ring is Prime Ideal | https://proofwiki.org/wiki/Intersection_of_Chain_of_Prime_Ideals_of_Commutative_Ring_is_Prime_Ideal | https://proofwiki.org/wiki/Intersection_of_Chain_of_Prime_Ideals_of_Commutative_Ring_is_Prime_Ideal | [
"Commutative Rings",
"Prime Ideals of Rings"
] | [
"Definition:Commutative Ring",
"Definition:Prime Spectrum of Ring",
"Definition:Ordered Set",
"Definition:Subset Relation",
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)/Subset Relation",
"Definition:Prime Ideal of Ring",
"Definition:Set Intersection",
"Definition:Prime Ideal of Ring"... | [
"Intersection of Ring Ideals is Ideal",
"Definition:Ideal of Ring",
"Definition:Prime Ideal of Ring",
"Definition:Total Ordering",
"Definition:Prime Ideal of Ring"
] |
proofwiki-13559 | Number of Different Ways to play First n Moves in Chess | The sequence formed from the number of ways to play the first $n$ moves in chess begins:
:$20, 400, 8902, 197 \, 742, \ldots$
{{OEIS|A007545}}
The count for the fourth move is already ambiguous, as it depends on whether only legal moves count, or whether all moves, legal or illegal, are included.
The count as given her... | There are $20$ ways to make the $1$st move by White:
:Each of the $8$ pawns may be moved either $1$ or $2$ squares forward, making $16$ moves
:Each of the $2$ knights may be moved to either of $2$ squares before it, making $4$ moves.
For each of those $20$ first moves by White, Black has the same $20$ options.
Thus the... | The [[Definition:Integer Sequence|sequence]] formed from the number of ways to play the first $n$ moves in [[Definition:Chess|chess]] begins:
:$20, 400, 8902, 197 \, 742, \ldots$
{{OEIS|A007545}}
The count for the fourth move is already ambiguous, as it depends on whether only legal moves count, or whether all moves, ... | There are $20$ ways to make the $1$st move by White:
:Each of the $8$ [[Definition:Chess Pawn|pawns]] may be moved either $1$ or $2$ squares forward, making $16$ moves
:Each of the $2$ [[Definition:Chess Knight|knights]] may be moved to either of $2$ squares before it, making $4$ moves.
For each of those $20$ first m... | Number of Different Ways to play First n Moves in Chess | https://proofwiki.org/wiki/Number_of_Different_Ways_to_play_First_n_Moves_in_Chess | https://proofwiki.org/wiki/Number_of_Different_Ways_to_play_First_n_Moves_in_Chess | [
"Chess"
] | [
"Definition:Integer Sequence",
"Definition:Chess"
] | [
"Definition:Chess/Pawn",
"Definition:Chess/Piece/Knight",
"Definition:Chess/Pawn",
"Definition:Chess/Piece/Knight",
"Definition:Chess/Pawn",
"Definition:Chess/Pawn",
"Definition:Chess/Pawn",
"Definition:Chess/Pawn",
"Definition:Chess/Piece/Knight",
"Definition:Chess/Pawn",
"Definition:Chess/Pawn... |
proofwiki-13560 | Largest Product of Pandigital Factors | The largest integer that can be obtained by multiplying $2$ integers which between them use all the digits from $1$ to $9$ is:
:$843 \, 973 \, 902 = 9642 \times 87531$ | {{ProofWanted|probably quite simple}} | The largest [[Definition:Integer|integer]] that can be obtained by [[Definition:Integer Multiplication|multiplying]] $2$ [[Definition:Integer|integers]] which between them use all the [[Definition:Digit|digits]] from $1$ to $9$ is:
:$843 \, 973 \, 902 = 9642 \times 87531$ | {{ProofWanted|probably quite simple}} | Largest Product of Pandigital Factors | https://proofwiki.org/wiki/Largest_Product_of_Pandigital_Factors | https://proofwiki.org/wiki/Largest_Product_of_Pandigital_Factors | [
"Recreational Mathematics"
] | [
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Integer",
"Definition:Digit"
] | [] |
proofwiki-13561 | Sequence of 9 Primes of form 4n+1 | The following sequence of $9$ consecutive prime numbers are all of the form $4 n + 1$:
:$11 \, 593, 11 \, 597, 11 \, 617, 11 \, 621, 11 \, 633, 11 \, 657, 11 \, 677, 11 \, 681, 11 \, 689$ | {{begin-eqn}}
{{eqn | l = 11 \, 593
| r = 4 \times 2898 + 1
}}
{{eqn | l = 11 \, 597
| r = 4 \times 2899 + 1
}}
{{eqn | l = 11 \, 617
| r = 4 \times 2904 + 1
}}
{{eqn | l = 11 \, 621
| r = 4 \times 2905 + 1
}}
{{eqn | l = 11 \, 633
| r = 4 \times 2908 + 1
}}
{{eqn | l = 11 \, 657
| r... | The following [[Definition:Integer Sequence|sequence]] of $9$ consecutive [[Definition:Prime Number|prime numbers]] are all of the form $4 n + 1$:
:$11 \, 593, 11 \, 597, 11 \, 617, 11 \, 621, 11 \, 633, 11 \, 657, 11 \, 677, 11 \, 681, 11 \, 689$ | {{begin-eqn}}
{{eqn | l = 11 \, 593
| r = 4 \times 2898 + 1
}}
{{eqn | l = 11 \, 597
| r = 4 \times 2899 + 1
}}
{{eqn | l = 11 \, 617
| r = 4 \times 2904 + 1
}}
{{eqn | l = 11 \, 621
| r = 4 \times 2905 + 1
}}
{{eqn | l = 11 \, 633
| r = 4 \times 2908 + 1
}}
{{eqn | l = 11 \, 657
| r... | Sequence of 9 Primes of form 4n+1 | https://proofwiki.org/wiki/Sequence_of_9_Primes_of_form_4n+1 | https://proofwiki.org/wiki/Sequence_of_9_Primes_of_form_4n+1 | [
"Prime Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-13562 | Smallest Penholodigital Square | The smallest penholodigital square number is:
:$11 \, 826^2 = 139 \, 854 \, 276$ | Let $n$ be the smallest positive integer whose square is penholodigital.
First it is noted that the smallest penholodigital number is $123 \, 456 \, 789$.
Hence any square penholodigital number must be at least as large as that.
Thus we can can say that:
:$n \ge \ceiling {\sqrt {123 \, 456 \, 789} } = 11 \, 112$
where ... | The smallest [[Definition:Penholodigital Number|penholodigital]] [[Definition:Square Number|square number]] is:
:$11 \, 826^2 = 139 \, 854 \, 276$ | Let $n$ be the smallest [[Definition:Positive Integer|positive integer]] whose [[Definition:Square (Algebra)|square]] is [[Definition:Penholodigital Number|penholodigital]].
First it is noted that the smallest [[Definition:Penholodigital Number|penholodigital number]] is $123 \, 456 \, 789$.
Hence any [[Definition:Sq... | Smallest Penholodigital Square | https://proofwiki.org/wiki/Smallest_Penholodigital_Square | https://proofwiki.org/wiki/Smallest_Penholodigital_Square | [
"Square Numbers",
"Penholodigital Integers",
"11,826",
"139,854,276"
] | [
"Definition:Pandigital Set/Penholodigital/Integer",
"Definition:Square Number"
] | [
"Definition:Positive/Integer",
"Definition:Square/Function",
"Definition:Pandigital Set/Penholodigital/Integer",
"Definition:Pandigital Set/Penholodigital/Integer",
"Definition:Square Number",
"Definition:Pandigital Set/Penholodigital/Integer",
"Definition:Ceiling Function",
"Definition:Positive/Integ... |
proofwiki-13563 | Locally Compact Space is Weakly Locally Compact | Let $T = \struct {S, \tau}$ be a locally compact topological space.
Then $T$ is weakly locally compact. | We are given $T$ is locally compact, so every point of $S$ has a neighborhood basis consisting of compact sets.
Let $x \in S$ be arbitrary.
From Topological Space is Neighborhood of all its Points, $S$ is a neighborhood of $x$ in $S$.
So, every point contained in $S$ has a compact neighborhood.
Hence, $T$ is weakly lo... | Let $T = \struct {S, \tau}$ be a [[Definition:Locally Compact Space|locally compact topological space]].
Then $T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]]. | We are [[Definition:Given|given]] $T$ is [[Definition:Locally Compact Space|locally compact]], so every point of $S$ has a [[Definition:Neighborhood Basis|neighborhood basis]] consisting of [[Definition:Compact Topological Subspace|compact sets]].
Let $x \in S$ be arbitrary.
From [[Topological Space is Neighborhood ... | Locally Compact Space is Weakly Locally Compact | https://proofwiki.org/wiki/Locally_Compact_Space_is_Weakly_Locally_Compact | https://proofwiki.org/wiki/Locally_Compact_Space_is_Weakly_Locally_Compact | [
"Locally Compact Spaces",
"Weakly Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Locally Compact Space",
"Definition:Weakly Locally Compact Space"
] | [
"Definition:Given",
"Definition:Locally Compact Space",
"Definition:Neighborhood Basis",
"Definition:Compact Topological Space/Subspace",
"Topological Space is Neighborhood of all its Points",
"Definition:Neighborhood (Topology)/Point",
"Definition:Compact Topological Space/Subspace",
"Definition:Neig... |
proofwiki-13564 | Topological Space is Neighborhood of all its Points | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Then $S$ is a neighborhood of $x$. | By the definition of the topology $\tau$, $S$ is an open set.
From Set is Open iff Neighborhood of all its Points, $S$ is a neighborhood of $x$.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Then $S$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$. | By the definition of the [[Definition:Topology|topology]] $\tau$, $S$ is an [[Definition:Open Set (Topology)|open set]].
From [[Set is Open iff Neighborhood of all its Points]], $S$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$.
{{qed}} | Topological Space is Neighborhood of all its Points | https://proofwiki.org/wiki/Topological_Space_is_Neighborhood_of_all_its_Points | https://proofwiki.org/wiki/Topological_Space_is_Neighborhood_of_all_its_Points | [
"Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Topology",
"Definition:Open Set/Topology",
"Set is Open iff Neighborhood of all its Points",
"Definition:Neighborhood (Topology)/Point"
] |
proofwiki-13565 | Equivalence of Definitions of Noetherian Module | Let $A$ be a commutative ring with unity.
Let $M$ be an $A$-module.
{{TFAE|def = Noetherian Module}} | {{tidy}} | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $M$ be an [[Definition:Module over Ring|$A$-module]].
{{TFAE|def = Noetherian Module}} | {{tidy}} | Equivalence of Definitions of Noetherian Module | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Noetherian_Module | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Noetherian_Module | [
"Noetherian Modules"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Module over Ring"
] | [] |
proofwiki-13566 | Particular Point Space is Locally Compact | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is locally compact. | {{Recall|Particular Point Topology|particular point space}}
{{:Definition:Particular Point Topology}}
{{Recall|Locally Compact Space|locally compact space}}
{{:Definition:Locally Compact Space}}
Let $x \in S$.
Consider the set $\set {p, x}$.
From the definition of particular point topology, $\set {p, x}$ is open in $T$... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is [[Definition:Locally Compact Space|locally compact]]. | {{Recall|Particular Point Topology|particular point space}}
{{:Definition:Particular Point Topology}}
{{Recall|Locally Compact Space|locally compact space}}
{{:Definition:Locally Compact Space}}
Let $x \in S$.
Consider the set $\set {p, x}$.
From the definition of [[Definition:Particular Point Topology|particular p... | Particular Point Space is Locally Compact | https://proofwiki.org/wiki/Particular_Point_Space_is_Locally_Compact | https://proofwiki.org/wiki/Particular_Point_Space_is_Locally_Compact | [
"Particular Point Topologies",
"Examples of Locally Compact Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Locally Compact Space"
] | [
"Definition:Particular Point Topology",
"Definition:Open Set/Topology",
"Finite Topological Space is Compact",
"Definition:Compact Topological Space",
"Definition:Neighborhood (Topology)",
"Definition:Particular Point Topology",
"Definition:Neighborhood Basis",
"Definition:Locally Compact Space"
] |
proofwiki-13567 | Product of Summations is Summation Over Cartesian Product of Products | This is a generalization of the distributive law:
:$\ds \prod_{a \mathop \in A} \sum_{b \mathop \in B_a} t_{a, b} = \sum_{c \mathop \in \prod \limits_{a \mathop \in A} B_a} \prod_{a \mathop \in A} t_{a, c_a}$
where the product of sets $\ds \prod_{a \mathop \in A} B_a$ is taken to be a cartesian product.
{{explain|In or... | For simplicity, let $A = \closedint 1 n$.
This reduces the complexity without loss of generality, as if we wanted to use an arbitrary set we could store the actual elements in an $n$-tuple and index them.
So we can think of $\closedint 1 n$ as representing the actual elements.
Use induction on $n$:
For $n = 1$:
:$\ds \... | This is a generalization of the distributive law:
:$\ds \prod_{a \mathop \in A} \sum_{b \mathop \in B_a} t_{a, b} = \sum_{c \mathop \in \prod \limits_{a \mathop \in A} B_a} \prod_{a \mathop \in A} t_{a, c_a}$
where the product of sets $\ds \prod_{a \mathop \in A} B_a$ is taken to be a [[Definition:Cartesian Product|c... | For simplicity, let $A = \closedint 1 n$.
This reduces the complexity without loss of generality, as if we wanted to use an arbitrary set we could store the actual elements in an $n$-tuple and index them.
So we can think of $\closedint 1 n$ as representing the actual elements.
Use induction on $n$:
For $n = 1$:
:$... | Product of Summations is Summation Over Cartesian Product of Products | https://proofwiki.org/wiki/Product_of_Summations_is_Summation_Over_Cartesian_Product_of_Products | https://proofwiki.org/wiki/Product_of_Summations_is_Summation_Over_Cartesian_Product_of_Products | [
"Algebra"
] | [
"Definition:Cartesian Product"
] | [
"Category:Algebra"
] |
proofwiki-13568 | Square and Tetrahedral Numbers | The only positive integers which are simultaneously tetrahedral and square are:
:$1, 4, 19 \, 600$ | First we confirm that these $3$ numbers have that property:
{{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6
| c = Closed Form for Tetrahedral Numbers
}}
{{eqn | r = 1^2
| c = {{Defof|Square Number}}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 4
| r = \dfrac {2 \paren {... | The only [[Definition:Positive Integer|positive integers]] which are simultaneously [[Definition:Tetrahedral Number|tetrahedral]] and [[Definition:Square Number|square]] are:
:$1, 4, 19 \, 600$ | First we confirm that these $3$ [[Definition:Natural Number|numbers]] have that property:
{{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6
| c = [[Closed Form for Tetrahedral Numbers]]
}}
{{eqn | r = 1^2
| c = {{Defof|Square Number}}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn |... | Square and Tetrahedral Numbers/Proof | https://proofwiki.org/wiki/Square_and_Tetrahedral_Numbers | https://proofwiki.org/wiki/Square_and_Tetrahedral_Numbers/Proof | [
"Square and Tetrahedral Numbers",
"Square Numbers",
"Tetrahedral Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Tetrahedral Number",
"Definition:Square Number"
] | [
"Definition:Natural Numbers",
"Closed Form for Tetrahedral Numbers",
"Closed Form for Tetrahedral Numbers",
"Closed Form for Tetrahedral Numbers",
"Definition:Tetrahedral Number",
"Definition:Square Number",
"Closed Form for Tetrahedral Numbers",
"Proof by Cases",
"Definition:Even Integer",
"Defin... |
proofwiki-13569 | Largest Integer not Sum of Two Abundant Numbers | The largest integer which is not the sum of $2$ abundant numbers is $20 \, 161$. | First we show that for $1 < k < 90$, $315 k$ is abundant.
If $k$ is divisible by $3, 5, 7$, note that:
:$945, 1575, 2205$
are all abundant, and $315 k$ is a multiple of at least one of them.
Hence $315 k$ is abundant by Multiple of Abundant Number is Abundant.
If $k$ is not divisible by $3, 5, 7$:
Let $p$ be a prime su... | The largest [[Definition:Integer|integer]] which is not the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Abundant Number|abundant numbers]] is $20 \, 161$. | First we show that for $1 < k < 90$, $315 k$ is [[Definition:Abundant Number|abundant]].
If $k$ is [[Definition:Divisor of Integer|divisible]] by $3, 5, 7$, note that:
:$945, 1575, 2205$
are all [[Definition:Abundant Number|abundant]], and $315 k$ is a multiple of at least one of them.
Hence $315 k$ is [[Definition:... | Largest Integer not Sum of Two Abundant Numbers | https://proofwiki.org/wiki/Largest_Integer_not_Sum_of_Two_Abundant_Numbers | https://proofwiki.org/wiki/Largest_Integer_not_Sum_of_Two_Abundant_Numbers | [
"Abundant Numbers"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Abundant Number"
] | [
"Definition:Abundant Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Abundant Number",
"Definition:Abundant Number",
"Multiple of Abundant Number is Abundant",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Abundant Number",
"Definition:Coprime/Integers",... |
proofwiki-13570 | Smallest Integer using Three Words in English Description | The smallest integer which uses exactly $3$ words in its standard (British) English description is:
:$21 \, 000$: '''twenty-one thousand'''
counting hyphenations as separate words. | All integers up to $100$ ('''one hundred''') use either $1$ or $2$ words:
:'''one'''
:'''sixty'''
:'''seventeen'''
:'''ninety-eight'''
All integers of the form $100 n$ for $n = 1, 2, \ldots 9$ use exactly $2$ words:
:'''one hundred'''
:'''seven hundred'''
:'''nine hundred'''
In British English, the technique for descri... | The smallest [[Definition:Integer|integer]] which uses exactly $3$ [[Definition:Word (Natural Language)|words]] in its standard (British) English description is:
:$21 \, 000$: '''twenty-one thousand'''
counting hyphenations as separate [[Definition:Word (Natural Language)|words]]. | All [[Definition:Integer|integers]] up to $100$ ('''one hundred''') use either $1$ or $2$ [[Definition:Word (Natural Language)|words]]:
:'''one'''
:'''sixty'''
:'''seventeen'''
:'''ninety-eight'''
All [[Definition:Integer|integers]] of the form $100 n$ for $n = 1, 2, \ldots 9$ use exactly $2$ [[Definition:Word (Natur... | Smallest Integer using Three Words in English Description | https://proofwiki.org/wiki/Smallest_Integer_using_Three_Words_in_English_Description | https://proofwiki.org/wiki/Smallest_Integer_using_Three_Words_in_English_Description | [
"Recreational Mathematics"
] | [
"Definition:Integer",
"Definition:Word (Natural Language)",
"Definition:Word (Natural Language)"
] | [
"Definition:Integer",
"Definition:Word (Natural Language)",
"Definition:Integer",
"Definition:Word (Natural Language)",
"Definition:Integer",
"Definition:Integer",
"Definition:Word (Natural Language)",
"Definition:Integer",
"Definition:Word (Natural Language)",
"Definition:Integer",
"Definition:... |
proofwiki-13571 | Smallest Pythagorean Quadrilateral with Integer Sides | The smallest Pythagorean quadrilateral in which the sides of the $4$ right triangles formed by its sides and perpendicular diagonals are all integers has an area of $21 \, 576$.
The sides of the right triangles in question are:
:$25, 60, 65$
:$91, 60, 109$
:$91, 312, 325$
:$25, 312, 313$ | :800px
The $4$ right triangles are inspected:
{{begin-eqn}}
{{eqn | l = 25^2 + 60^2
| r = 625 + 3600
}}
{{eqn | r = 4225
| c =
}}
{{eqn | r = 65^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 91^2 + 60^2
| r = 8281 + 3600
}}
{{eqn | r = 11 \, 881
| c =
}}
{{eqn | r = 109^2
| c =... | The smallest [[Definition:Pythagorean Quadrilateral|Pythagorean quadrilateral]] in which the [[Definition:Side of Polygon|sides]] of the $4$ [[Definition:Right Triangle|right triangles]] formed by its [[Definition:Side of Polygon|sides]] and [[Definition:Perpendicular|perpendicular]] [[Definition:Diagonal of Quadrilate... | :[[File:SmallestPythagoreanQuadrilateral.png|800px]]
The $4$ [[Definition:Right Triangle|right triangles]] are inspected:
{{begin-eqn}}
{{eqn | l = 25^2 + 60^2
| r = 625 + 3600
}}
{{eqn | r = 4225
| c =
}}
{{eqn | r = 65^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 91^2 + 60^2
| r = 82... | Smallest Pythagorean Quadrilateral with Integer Sides | https://proofwiki.org/wiki/Smallest_Pythagorean_Quadrilateral_with_Integer_Sides | https://proofwiki.org/wiki/Smallest_Pythagorean_Quadrilateral_with_Integer_Sides | [
"Pythagorean Triangles",
"Quadrilaterals"
] | [
"Definition:Pythagorean Quadrilateral",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Polygon/Side",
"Definition:Right Angle/Perpendicular",
"Definition:Diameter of Quadrilateral",
"Definition:Integer",
"Definition:Area",
"Definition:Polygon/Side",
"Definiti... | [
"File:SmallestPythagoreanQuadrilateral.png",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Area",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Area"
] |
proofwiki-13572 | Integers whose Divisor Count equals Cube Root | There are $3$ positive integers whose divisor count function equals its cube root:
{{begin-eqn}}
{{eqn | l = 1 = 1^3
| o = :
| r = \map {\sigma_0} 1 = 1
| c = {{DCFLink|1}}
}}
{{eqn | l = 21 \, 952 = 28^3
| o = :
| r = \map {\sigma_0} {21 \, 952} = 28
| c = {{DCFLink|21,952|21 \, 952... | Suppose $N = \map {\sigma_0} {N^3}$.
The case $N = 1$ is trivial.
Suppose $N$ is a prime power.
Write $N = p^n$.
By Divisor Count Function of Power of Prime:
:$N = \map {\sigma_0} {p^{3 n} } = 3 n + 1$
By Bernoulli's Inequality:
:$N = p^n \ge 1 + n \paren {p - 1}$
This gives us the inequality:
:$3 n + 1 \ge 1 + n \pare... | There are $3$ [[Definition:Positive Integer|positive integers]] whose [[Definition:Divisor Count Function|divisor count function]] equals its [[Definition:Cube Root|cube root]]:
{{begin-eqn}}
{{eqn | l = 1 = 1^3
| o = :
| r = \map {\sigma_0} 1 = 1
| c = {{DCFLink|1}}
}}
{{eqn | l = 21 \, 952 = 28^3
... | Suppose $N = \map {\sigma_0} {N^3}$.
The case $N = 1$ is trivial.
Suppose $N$ is a [[Definition:Prime Power|prime power]].
Write $N = p^n$.
By [[Divisor Count Function of Power of Prime]]:
:$N = \map {\sigma_0} {p^{3 n} } = 3 n + 1$
By [[Bernoulli's Inequality]]:
:$N = p^n \ge 1 + n \paren {p - 1}$
This gives us... | Integers whose Divisor Count equals Cube Root | https://proofwiki.org/wiki/Integers_whose_Divisor_Count_equals_Cube_Root | https://proofwiki.org/wiki/Integers_whose_Divisor_Count_equals_Cube_Root | [
"Divisor Count Function",
"Cube Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Divisor Count Function",
"Definition:Cube Root"
] | [
"Definition:Prime Power",
"Divisor Count Function of Power of Prime",
"Bernoulli's Inequality",
"Definition:Prime Number",
"Definition:Prime Power",
"Divisor Count Function is Multiplicative",
"Definition:Integer",
"Definition:Prime Power"
] |
proofwiki-13573 | Smallest Integer which is Product of 4 Triples all with Same Sum | The smallest integer which can be expressed as the product of $4$ different triplets of integers each of which has the same sum is:
{{begin-eqn}}
{{eqn | l = 25 \, 200
| r = 6 \times 56 \times 75
}}
{{eqn | r = 7 \times 40 \times 90
}}
{{eqn | r = 9 \times 28 \times 100
}}
{{eqn | r = 12 \times 20 \times 105
}}
{... | We have:
{{begin-eqn}}
{{eqn | l = 6 \times 56 \times 75
| r = \paren {2 \times 3} \times \paren {2^3 \times 7} \times \paren {3 \times 5^2}
| c =
}}
{{eqn | r = 2^4 \times 3^2 \times 5^2 \times 7
| c =
}}
{{eqn | l = 6 + 56 + 75
| r = 137
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = ... | The smallest [[Definition:Integer|integer]] which can be expressed as the [[Definition:Integer Multiplication|product]] of $4$ different [[Definition:Ordered Triple|triplets]] of [[Definition:Integer|integers]] each of which has the same [[Definition:Integer Addition|sum]] is:
{{begin-eqn}}
{{eqn | l = 25 \, 200
... | We have:
{{begin-eqn}}
{{eqn | l = 6 \times 56 \times 75
| r = \paren {2 \times 3} \times \paren {2^3 \times 7} \times \paren {3 \times 5^2}
| c =
}}
{{eqn | r = 2^4 \times 3^2 \times 5^2 \times 7
| c =
}}
{{eqn | l = 6 + 56 + 75
| r = 137
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l... | Smallest Integer which is Product of 4 Triples all with Same Sum | https://proofwiki.org/wiki/Smallest_Integer_which_is_Product_of_4_Triples_all_with_Same_Sum | https://proofwiki.org/wiki/Smallest_Integer_which_is_Product_of_4_Triples_all_with_Same_Sum | [
"Recreational Mathematics",
"25,200"
] | [
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Integer",
"Definition:Addition/Integers"
] | [] |
proofwiki-13574 | 4 Integers whose Euler Phi Value is 10,368 | :$\map \phi {25 \, 930} = \map \phi {25 \, 935} = \map \phi {25 \, 940} = \map \phi {25 \, 942} = 10 \, 368 = 2^7 \times 3^4$
where $\phi$ denotes the Euler $\phi$ function. | {{begin-eqn}}
{{eqn | l = \map \phi {25 \, 930}
| r = 10 \, 368
| c = {{EulerPhiLink|25,930|25 \, 930}}
}}
{{eqn | l = \map \phi {25 \, 935}
| r = 10 \, 368
| c = {{EulerPhiLink|25,935|25 \, 935}}
}}
{{eqn | l = \map \phi {25 \, 940}
| r = 10 \, 368
| c = {{EulerPhiLink|25,940|25 \, ... | :$\map \phi {25 \, 930} = \map \phi {25 \, 935} = \map \phi {25 \, 940} = \map \phi {25 \, 942} = 10 \, 368 = 2^7 \times 3^4$
where $\phi$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]]. | {{begin-eqn}}
{{eqn | l = \map \phi {25 \, 930}
| r = 10 \, 368
| c = {{EulerPhiLink|25,930|25 \, 930}}
}}
{{eqn | l = \map \phi {25 \, 935}
| r = 10 \, 368
| c = {{EulerPhiLink|25,935|25 \, 935}}
}}
{{eqn | l = \map \phi {25 \, 940}
| r = 10 \, 368
| c = {{EulerPhiLink|25,940|25 \, ... | 4 Integers whose Euler Phi Value is 10,368 | https://proofwiki.org/wiki/4_Integers_whose_Euler_Phi_Value_is_10,368 | https://proofwiki.org/wiki/4_Integers_whose_Euler_Phi_Value_is_10,368 | [
"Euler Phi Function",
"10,368"
] | [
"Definition:Euler Phi Function"
] | [] |
proofwiki-13575 | Fourth Power expressible as Sum of 6 Fourth Powers | $28 \, 561$ can be expressed as the sum of $6$ fourth powers:
:$28 \, 561 = 13^4 = 12^4 + 8^4 + 7^4 + 6^4 + 2^4 + 2^4$ | {{begin-eqn}}
{{eqn | o =
| r = 12^4 + 8^4 + 7^4 + 6^4 + 2^4 + 2^4
| c =
}}
{{eqn | r = 20 \, 736 + 4096 + 2401 + 1296 + 16 + 16
| c =
}}
{{eqn | r = 28 \, 561
| c =
}}
{{eqn | r = 13^4
| c =
}}
{{end-eqn}}
{{qed}} | $28 \, 561$ can be expressed as the [[Definition:Integer Addition|sum]] of $6$ [[Definition:Fourth Power|fourth powers]]:
:$28 \, 561 = 13^4 = 12^4 + 8^4 + 7^4 + 6^4 + 2^4 + 2^4$ | {{begin-eqn}}
{{eqn | o =
| r = 12^4 + 8^4 + 7^4 + 6^4 + 2^4 + 2^4
| c =
}}
{{eqn | r = 20 \, 736 + 4096 + 2401 + 1296 + 16 + 16
| c =
}}
{{eqn | r = 28 \, 561
| c =
}}
{{eqn | r = 13^4
| c =
}}
{{end-eqn}}
{{qed}} | Fourth Power expressible as Sum of 6 Fourth Powers | https://proofwiki.org/wiki/Fourth_Power_expressible_as_Sum_of_6_Fourth_Powers | https://proofwiki.org/wiki/Fourth_Power_expressible_as_Sum_of_6_Fourth_Powers | [
"Fourth Powers",
"28,561"
] | [
"Definition:Addition/Integers",
"Definition:Fourth Power"
] | [] |
proofwiki-13576 | Smallest Fermat Pseudoprime to Bases 2, 3, 5 and 7 | The smallest Fermat pseudoprime to bases $2$, $3$, $5$ and $7$ is $29 \, 341$. | {{ProofWanted|We have the list of Poulet numbers and Fermat pseudoprimes base $3$, but not of bases $5$ and $7$. Once we get those lists, we can find the numbers on the lists for both.}} | The smallest [[Definition:Fermat Pseudoprime|Fermat pseudoprime]] to bases $2$, $3$, $5$ and $7$ is $29 \, 341$. | {{ProofWanted|We have the list of [[Definition:Poulet Number|Poulet numbers]] and [[Definition:Fermat Pseudoprime/Base 3|Fermat pseudoprimes base $3$]], but not of bases $5$ and $7$. Once we get those lists, we can find the numbers on the lists for both.}} | Smallest Fermat Pseudoprime to Bases 2, 3, 5 and 7 | https://proofwiki.org/wiki/Smallest_Fermat_Pseudoprime_to_Bases_2,_3,_5_and_7 | https://proofwiki.org/wiki/Smallest_Fermat_Pseudoprime_to_Bases_2,_3,_5_and_7 | [
"Fermat Pseudoprimes",
"29,341"
] | [
"Definition:Fermat Pseudoprime"
] | [
"Definition:Poulet Number",
"Definition:Fermat Pseudoprime/Base 3"
] |
proofwiki-13577 | Smallest Differences between Fractional Parts of Square and Cube Roots | Apart from $6$th powers, the value of $n$ less than $50 \, 000$ for which the difference between the fractional parts of $\sqrt n$ and $\sqrt [3] n$ is smallest is $30 \, 739$.
The next integer to produce a smaller difference above that is $62 \, 324$. | {{begin-eqn}}
{{eqn | l = \sqrt {30 \, 739}
| o = \approx
| r = 175 \cdotp 32541 \, 17349
| c =
}}
{{eqn | l = \sqrt [3] {30 \, 739}
| o = \approx
| r = 31 \cdotp 32539 \, 66116
| c =
}}
{{eqn | ll= \leadsto
| l = \sqrt {30 \, 739} - \sqrt [3] {30 \, 739}
| o = \approx
... | Apart from [[Definition:Sixth Power|$6$th powers]], the value of $n$ less than $50 \, 000$ for which the [[Definition:Real Subtraction|difference]] between the [[Definition:Fractional Part|fractional parts]] of $\sqrt n$ and $\sqrt [3] n$ is smallest is $30 \, 739$.
The next integer to produce a smaller difference ab... | {{begin-eqn}}
{{eqn | l = \sqrt {30 \, 739}
| o = \approx
| r = 175 \cdotp 32541 \, 17349
| c =
}}
{{eqn | l = \sqrt [3] {30 \, 739}
| o = \approx
| r = 31 \cdotp 32539 \, 66116
| c =
}}
{{eqn | ll= \leadsto
| l = \sqrt {30 \, 739} - \sqrt [3] {30 \, 739}
| o = \approx
... | Smallest Differences between Fractional Parts of Square and Cube Roots | https://proofwiki.org/wiki/Smallest_Differences_between_Fractional_Parts_of_Square_and_Cube_Roots | https://proofwiki.org/wiki/Smallest_Differences_between_Fractional_Parts_of_Square_and_Cube_Roots | [
"Square Roots",
"Cube Roots"
] | [
"Definition:Sixth Power",
"Definition:Subtraction/Real Numbers",
"Definition:Fractional Part"
] | [] |
proofwiki-13578 | Smallest Triplet of Consecutive Integers each Divisible by Fourth Power | This triplet of consecutive integers has the property that each of them is divisible by a fourth power:
:$33 \, 614, 33 \, 615, 33 \, 616$
This is the smallest such triplet. | {{begin-eqn}}
{{eqn | l = 33 \, 614
| r = 14 \times 7^4
| c =
}}
{{eqn | l = 33 \, 615
| r = 415 \times 3^4
| c =
}}
{{eqn | l = 33 \, 616
| r = 2101 \times 2^4
| c =
}}
{{end-eqn}}
Each number in such triplets of consecutive integers is divisible by a fourth power of some prime n... | This [[Definition:Ordered Triple|triplet]] of consecutive [[Definition:Positive Integer|integers]] has the property that each of them is [[Definition:Divisor of Integer|divisible]] by a [[Definition:Fourth Power|fourth power]]:
:$33 \, 614, 33 \, 615, 33 \, 616$
This is the smallest such [[Definition:Ordered Triple|tr... | {{begin-eqn}}
{{eqn | l = 33 \, 614
| r = 14 \times 7^4
| c =
}}
{{eqn | l = 33 \, 615
| r = 415 \times 3^4
| c =
}}
{{eqn | l = 33 \, 616
| r = 2101 \times 2^4
| c =
}}
{{end-eqn}}
Each number in such [[Definition:Ordered Triple|triplets]] of consecutive [[Definition:Integer|in... | Smallest Triplet of Consecutive Integers each Divisible by Fourth Power | https://proofwiki.org/wiki/Smallest_Triplet_of_Consecutive_Integers_each_Divisible_by_Fourth_Power | https://proofwiki.org/wiki/Smallest_Triplet_of_Consecutive_Integers_each_Divisible_by_Fourth_Power | [
"Fourth Powers"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Fourth Power",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Fourth Power",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Fourth Power",
"Definition:Prime Number",
"Definition:Ordered Tuple as Ord... |
proofwiki-13579 | Abundancy of Integers in form 945 + 630n | A large number of integers of the form $945 + 630 n$, for $n \in \Z_{\ge 0}$, are abundant.
The first counterexample is for $n = 52$.
{{WIP|Add this sequence to the number pages}} | {{begin-eqn}}
{{eqn | n = n = 0
| l = \map {\sigma_1} {945} - 945
| r = 1920 - 945
| c = {{DSFLink|945}}
}}
{{eqn | r = 975
| c =
}}
{{eqn | n = n = 1
| l = \map {\sigma_1} {1575} - 1575
| r = 3224 - 1575
| c = {{DSFLink|1575}}
}}
{{eqn | r = 1649
| c =
}}
{{eqn | n = n... | A large number of [[Definition:Positive Integer|integers]] of the form $945 + 630 n$, for $n \in \Z_{\ge 0}$, are [[Definition:Abundant Number|abundant]].
The first counterexample is for $n = 52$.
{{WIP|Add this sequence to the number pages}} | {{begin-eqn}}
{{eqn | n = n = 0
| l = \map {\sigma_1} {945} - 945
| r = 1920 - 945
| c = {{DSFLink|945}}
}}
{{eqn | r = 975
| c =
}}
{{eqn | n = n = 1
| l = \map {\sigma_1} {1575} - 1575
| r = 3224 - 1575
| c = {{DSFLink|1575}}
}}
{{eqn | r = 1649
| c =
}}
{{eqn | n = n... | Abundancy of Integers in form 945 + 630n | https://proofwiki.org/wiki/Abundancy_of_Integers_in_form_945_+_630n | https://proofwiki.org/wiki/Abundancy_of_Integers_in_form_945_+_630n | [
"Abundancy",
"Abundant Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Abundant Number"
] | [
"Definition:Abundant Number"
] |
proofwiki-13580 | Affine Group of One Dimension as Semidirect Product | Let $\map {\operatorname{Af}_1} \R$ be the $1$-dimensional affine group on $\R$.
Let $\R^+$ be the additive group of real numbers.
Let $\R^\times$ be the multiplicative group of real numbers.
Let $\phi: \R^\times \to \Aut {\R^+}$ be defined as:
:$\forall b \in \R^\times: \map \phi b = \paren {a \mapsto a b}$
Let $\R^+ ... | By definition, a (group) isomorphism is a (group) homomorphism which is a bijection.
Recall the definition of underlying set of $1$-dimensional affine group on $\R$:
:$S = \set {f_{a b}: x \mapsto a x + b : a \in \R_{\ne 0}, b \in \R}$
So the bijection $\psi: \map {\operatorname {Af}_1} \R \to \R^+ \rtimes_\phi \R^\tim... | Let $\map {\operatorname{Af}_1} \R$ be the [[Definition:Affine Group of One Dimension|$1$-dimensional affine group on $\R$]].
Let $\R^+$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $\R^\times$ be the [[Definition:Multiplicative Group of Real Numbers|multiplicative group of... | By definition, a [[Definition:Group Isomorphism|(group) isomorphism]] is a [[Definition:Group Homomorphism|(group) homomorphism]] which is a [[Definition:Bijection|bijection]].
Recall the definition of [[Definition:Underlying Set of Structure|underlying set]] of [[Definition:Affine Group of One Dimension|$1$-dimensio... | Affine Group of One Dimension as Semidirect Product | https://proofwiki.org/wiki/Affine_Group_of_One_Dimension_as_Semidirect_Product | https://proofwiki.org/wiki/Affine_Group_of_One_Dimension_as_Semidirect_Product | [
"Affine Groups"
] | [
"Definition:Affine Group of One Dimension",
"Definition:Additive Group of Real Numbers",
"Definition:Multiplicative Group of Real Numbers",
"Definition:Semidirect Product",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Group Homomorphism",
"Definition:Bijection",
"Definition:Underlying Set/Abstract Algebra",
"Definition:Affine Group of One Dimension",
"Definition:Bijection",
"Definition:Group Homomorphism",
"Real Addition is Commutative",
"... |
proofwiki-13581 | Sequence of Consecutive Integers with Same Number of Divisors | The following sequence of consecutive integers all have the same number of divisors, that is, $8$:
:$40 \, 311, 40 \, 312, 40 \, 313, 40 \, 314, 40 \, 315$
This is the longest such sequence known. | In the below, $\sigma_0$ denotes the divisor count function.
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {40 \, 311}
| r = 8
| c = {{DCFLink|40,311|40 \, 311}}
}}
{{eqn | l = \map {\sigma_0} {40 \, 312}
| r = 8
| c = {{DCFLink|40,312|40 \, 312}}
}}
{{eqn | l = \map {\sigma_0} {40 \, 313}
| r... | The following [[Definition:Integer Sequence|sequence]] of consecutive [[Definition:Integer|integers]] all have the same number of [[Definition:Divisor of Integer|divisors]], that is, $8$:
:$40 \, 311, 40 \, 312, 40 \, 313, 40 \, 314, 40 \, 315$
This is the longest such [[Definition:Integer Sequence|sequence]] known. | In the below, $\sigma_0$ denotes the [[Definition:Divisor Count Function|divisor count function]].
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {40 \, 311}
| r = 8
| c = {{DCFLink|40,311|40 \, 311}}
}}
{{eqn | l = \map {\sigma_0} {40 \, 312}
| r = 8
| c = {{DCFLink|40,312|40 \, 312}}
}}
{{eqn | l... | Sequence of Consecutive Integers with Same Number of Divisors | https://proofwiki.org/wiki/Sequence_of_Consecutive_Integers_with_Same_Number_of_Divisors | https://proofwiki.org/wiki/Sequence_of_Consecutive_Integers_with_Same_Number_of_Divisors | [
"Divisor Count Function"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer Sequence"
] | [
"Definition:Divisor Count Function"
] |
proofwiki-13582 | Pentagonal and Hexagonal Numbers | The sequence of positive integers which are simultaneously pentagonal and hexagonal begins:
:$1, 40 \, 755, 1 \, 533 \, 776 \, 805, 57 \, 722 \, 156 \, 241 \, 751, \ldots$
{{OEIS|A046180}} | {{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \paren {3 \times 1 - 1} } 2
| c = Closed Form for Pentagonal Numbers
}}
{{eqn | r = 1 \paren {2 \times 1 - 1}
| c = Closed Form for Hexagonal Numbers
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 40 \, 755
| r = \dfrac {165 \paren {3 \times 165 - 1} } 2
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] which are simultaneously [[Definition:Pentagonal Number|pentagonal]] and [[Definition:Hexagonal Number|hexagonal]] begins:
:$1, 40 \, 755, 1 \, 533 \, 776 \, 805, 57 \, 722 \, 156 \, 241 \, 751, \ldots$
{{OEIS|A046180}} | {{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \paren {3 \times 1 - 1} } 2
| c = [[Closed Form for Pentagonal Numbers]]
}}
{{eqn | r = 1 \paren {2 \times 1 - 1}
| c = [[Closed Form for Hexagonal Numbers]]
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 40 \, 755
| r = \dfrac {165 \paren {3 \times 165 - ... | Pentagonal and Hexagonal Numbers | https://proofwiki.org/wiki/Pentagonal_and_Hexagonal_Numbers | https://proofwiki.org/wiki/Pentagonal_and_Hexagonal_Numbers | [
"Pentagonal Numbers",
"Hexagonal Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Pentagonal Number",
"Definition:Hexagonal Number"
] | [
"Closed Form for Pentagonal Numbers",
"Closed Form for Hexagonal Numbers",
"Closed Form for Pentagonal Numbers",
"Closed Form for Hexagonal Numbers"
] |
proofwiki-13583 | Carmichael Number with 4 Prime Factors | $41 \, 041$ is the smallest Carmichael number with $4$ prime factors:
:$41 \, 041 = 7 \times 11 \times 13 \times 41$ | From Sequence of Carmichael Numbers:
{{:Carmichael Number/Sequence}}
The sequence continues:
:$\ldots, 29 \, 341, 41 \, 041$
We now investigate their prime factors:
{{begin-eqn}}
{{eqn | l = 561
| r = 3 \times 11 \times 17
}}
{{eqn | l = 1105
| r = 5 \times 13 \times 17
}}
{{eqn | l = 1729
| r = 7 \ti... | $41 \, 041$ is the smallest [[Definition:Carmichael Number|Carmichael number]] with $4$ [[Definition:Prime Factor|prime factors]]:
:$41 \, 041 = 7 \times 11 \times 13 \times 41$ | From [[Carmichael Number/Sequence|Sequence of Carmichael Numbers]]:
{{:Carmichael Number/Sequence}}
The [[Definition:Integer Sequence|sequence]] continues:
:$\ldots, 29 \, 341, 41 \, 041$
We now investigate their [[Definition:Prime Factor|prime factors]]:
{{begin-eqn}}
{{eqn | l = 561
| r = 3 \times 11 \times... | Carmichael Number with 4 Prime Factors | https://proofwiki.org/wiki/Carmichael_Number_with_4_Prime_Factors | https://proofwiki.org/wiki/Carmichael_Number_with_4_Prime_Factors | [
"Carmichael Numbers",
"41,041"
] | [
"Definition:Carmichael Number",
"Definition:Prime Factor"
] | [
"Carmichael Number/Sequence",
"Definition:Integer Sequence",
"Definition:Prime Factor"
] |
proofwiki-13584 | Smallest Sequence of 5 Consecutive Numbers which are Happy | The smallest sequence of $5$ consecutive integers all of which are happy numbers is:
:$44 \, 488, 44 \, 489, 44 \, 490, 44 \, 491, 44 \, 492$ | {{ProofWanted|Exhaustive enumeration? I'm afraid recreational mathematics of this kind bores me.}} | The smallest [[Definition:Integer Sequence|sequence]] of $5$ consecutive [[Definition:Integer|integers]] all of which are [[Definition:Happy Number|happy numbers]] is:
:$44 \, 488, 44 \, 489, 44 \, 490, 44 \, 491, 44 \, 492$ | {{ProofWanted|Exhaustive enumeration? I'm afraid recreational mathematics of this kind bores me.}} | Smallest Sequence of 5 Consecutive Numbers which are Happy | https://proofwiki.org/wiki/Smallest_Sequence_of_5_Consecutive_Numbers_which_are_Happy | https://proofwiki.org/wiki/Smallest_Sequence_of_5_Consecutive_Numbers_which_are_Happy | [
"Happy Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Happy Number"
] | [] |
proofwiki-13585 | Reciprocal of 21 | :$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$ | Performing the calculation using long division:
<pre>
0.04761901...
--------------
21)1.00000000000
84
--
160
147
---
130
126
---
40
21
---
190
189
---
100
84
---
...
</pre>
{{qed}} | :$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.04761901...
--------------
21)1.00000000000
84
--
160
147
---
130
126
---
40
21
---
190
189
---
100
84
---
... | Reciprocal of 21 | https://proofwiki.org/wiki/Reciprocal_of_21 | https://proofwiki.org/wiki/Reciprocal_of_21 | [
"21",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division"
] |
proofwiki-13586 | Smallest Fourth Power as Sum of 5 Distinct Fourth Powers | The smallest $4$th power which can be expressed as the sum of $5$ distinct $4$th powers is:
:$15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$ | {{begin-eqn}}
{{eqn | l = 15^4
| r = 50 \, 625
| c =
}}
{{eqn | r = 256 + 1296 + 4096 + 6561 + 38 \, 416
| c =
}}
{{eqn | r = 4^4 + 6^4 + 8^4 + 9^4 + 14^4
| c =
}}
{{end-eqn}}
By Fermat's Little Theorem, for $5 \nmid a$:
:$a^4 \equiv 1 \pmod 5$
For $5 \divides a$:
:$a^4 \equiv 0 \pmod 5$
Ther... | The smallest [[Definition:Fourth Power|$4$th power]] which can be expressed as the [[Definition:Integer Addition|sum]] of $5$ [[Definition:Distinct|distinct]] [[Definition:Fourth Power|$4$th powers]] is:
:$15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$ | {{begin-eqn}}
{{eqn | l = 15^4
| r = 50 \, 625
| c =
}}
{{eqn | r = 256 + 1296 + 4096 + 6561 + 38 \, 416
| c =
}}
{{eqn | r = 4^4 + 6^4 + 8^4 + 9^4 + 14^4
| c =
}}
{{end-eqn}}
By [[Fermat's Little Theorem]], for $5 \nmid a$:
:$a^4 \equiv 1 \pmod 5$
For $5 \divides a$:
:$a^4 \equiv 0 \pmod 5... | Smallest Fourth Power as Sum of 5 Distinct Fourth Powers | https://proofwiki.org/wiki/Smallest_Fourth_Power_as_Sum_of_5_Distinct_Fourth_Powers | https://proofwiki.org/wiki/Smallest_Fourth_Power_as_Sum_of_5_Distinct_Fourth_Powers | [
"50,625",
"15",
"Fourth Powers"
] | [
"Definition:Fourth Power",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Fourth Power"
] | [
"Fermat's Little Theorem",
"Definition:Fourth Power",
"Definition:Divisor (Algebra)/Integer",
"Definition:Fourth Power",
"Definition:Divisor (Algebra)/Integer",
"Definition:Fourth Power",
"Definition:Divisor (Algebra)/Integer",
"Definition:Fourth Power",
"Definition:Divisor (Algebra)/Integer",
"De... |
proofwiki-13587 | Boolean Interpretation is Well-Defined | Let $\LL_0$ be the language of propositional logic.
Let $v: \LL_0 \to \set {\T, \F}$ be a boolean interpretation.
Then $v$ is well-defined. | By Language of Propositional Logic has Unique Parsability, $\LL_0$ is uniquely parsable.
Therefore, the Principle of Definition by Structural Induction can be applied to $\LL_0$.
By inspection, we see that the definition of the boolean interpretation $v$ follows the bottom-up specification of propositional logic.
Hence... | Let $\LL_0$ be the [[Definition:Language of Propositional Logic|language of propositional logic]].
Let $v: \LL_0 \to \set {\T, \F}$ be a [[Definition:Boolean Interpretation|boolean interpretation]].
Then $v$ is [[Definition:Well-Defined Mapping|well-defined]]. | By [[Language of Propositional Logic has Unique Parsability]], $\LL_0$ is [[Definition:Unique Parsability|uniquely parsable]].
Therefore, the [[Principle of Definition by Structural Induction]] can be applied to $\LL_0$.
By inspection, we see that the definition of the [[Definition:Boolean Interpretation|boolean int... | Boolean Interpretation is Well-Defined/Proof 1 | https://proofwiki.org/wiki/Boolean_Interpretation_is_Well-Defined | https://proofwiki.org/wiki/Boolean_Interpretation_is_Well-Defined/Proof_1 | [
"Boolean Interpretation is Well-Defined",
"Boolean Interpretations"
] | [
"Definition:Language of Propositional Logic",
"Definition:Boolean Interpretation",
"Definition:Well-Defined/Mapping"
] | [
"Language of Propositional Logic has Unique Parsability",
"Definition:Unique Parsability",
"Principle of Definition by Structural Induction",
"Definition:Boolean Interpretation",
"Definition:Language of Propositional Logic/Formal Grammar/Bottom-Up Specification",
"Principle of Definition by Structural Ind... |
proofwiki-13588 | Boolean Interpretation is Well-Defined | Let $\LL_0$ be the language of propositional logic.
Let $v: \LL_0 \to \set {\T, \F}$ be a boolean interpretation.
Then $v$ is well-defined. | This is to be done by strong induction on the length of WFFs.
By definition of $v$ being a boolean interpretation, $\map v p$ is well-defined for all $p \in \PP_0$, the vocabulary of $\LL_0$.
A WFF of length $1$ has (trivially) a unique parsing sequence.
Consequently, only a single defining rule for $v$ as a boolean in... | Let $\LL_0$ be the [[Definition:Language of Propositional Logic|language of propositional logic]].
Let $v: \LL_0 \to \set {\T, \F}$ be a [[Definition:Boolean Interpretation|boolean interpretation]].
Then $v$ is [[Definition:Well-Defined Mapping|well-defined]]. | This is to be done by [[Second Principle of Mathematical Induction|strong induction]] on the [[Definition:Length of String|length]] of [[Definition:WFF of Propositional Logic|WFFs]].
By definition of $v$ being a [[Definition:Boolean Interpretation|boolean interpretation]], $\map v p$ is well-defined for all $p \in \PP... | Boolean Interpretation is Well-Defined/Proof 2 | https://proofwiki.org/wiki/Boolean_Interpretation_is_Well-Defined | https://proofwiki.org/wiki/Boolean_Interpretation_is_Well-Defined/Proof_2 | [
"Boolean Interpretation is Well-Defined",
"Boolean Interpretations"
] | [
"Definition:Language of Propositional Logic",
"Definition:Boolean Interpretation",
"Definition:Well-Defined/Mapping"
] | [
"Second Principle of Mathematical Induction",
"Definition:Length of String",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Boolean Interpretation",
"Definition:Language of Propositional Logic/Alphabet/Letter",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
... |
proofwiki-13589 | Pandigital Pairs whose Squares are Pandigital | The elements of the following pandigital pairs of integers each have squares which are themselves pandigital:
:$\left({35 \, 172, 60 \, 984}\right), \left({57 \, 321, 60 \, 984}\right), \left({58 \, 413, 96 \, 702}\right), \left({59 \, 403, 76 \, 182}\right)$
{{OEIS|A085545|order = first}} | {{begin-eqn}}
{{eqn | l = 35 \, 172^2
| r = 1 \, 237 \, 069 \, 584
| c =
}}
{{eqn | l = 60 \, 984^2
| r = 3 \, 719 \, 048 \, 256
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 57 \, 321^2
| r = 3 \, 285 \, 697 \, 041
| c =
}}
{{eqn | l = 60 \, 984^2
| r = 3 \, 719 \, 048 \, ... | The [[Definition:Element|elements]] of the following [[Definition:Pandigital Set|pandigital]] [[Definition:Doubleton|pairs]] of [[Definition:Positive Integer|integers]] each have [[Definition:Square (Algebra)|squares]] which are themselves [[Definition:Pandigital Integer|pandigital]]:
:$\left({35 \, 172, 60 \, 984}\rig... | {{begin-eqn}}
{{eqn | l = 35 \, 172^2
| r = 1 \, 237 \, 069 \, 584
| c =
}}
{{eqn | l = 60 \, 984^2
| r = 3 \, 719 \, 048 \, 256
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 57 \, 321^2
| r = 3 \, 285 \, 697 \, 041
| c =
}}
{{eqn | l = 60 \, 984^2
| r = 3 \, 719 \, 048 \... | Pandigital Pairs whose Squares are Pandigital | https://proofwiki.org/wiki/Pandigital_Pairs_whose_Squares_are_Pandigital | https://proofwiki.org/wiki/Pandigital_Pairs_whose_Squares_are_Pandigital | [
"Square Numbers",
"Pandigital Sets"
] | [
"Definition:Element",
"Definition:Pandigital Set",
"Definition:Doubleton",
"Definition:Positive/Integer",
"Definition:Square/Function",
"Definition:Pandigital Set/Integer"
] | [] |
proofwiki-13590 | Language of Propositional Logic has Unique Parsability | The language of propositional logic $\LL_0$ has unique parsability. | It is to be demonstrated that each WFF arises by a unique rule of formation from the bottom-up specification of propositional logic.
The rules $\mathbf W : TF$ and $\mathbf W : \PP_0$ need no further treatment.
From inspection of the first character it is clear that the remaining $\mathbf W : \neg$ and $\mathbf W : Op$... | The [[Definition:Language of Propositional Logic|language of propositional logic]] $\LL_0$ has [[Definition:Unique Parsability|unique parsability]]. | It is to be demonstrated that each [[Definition:WFF of Propositional Logic|WFF]] arises by a unique [[Definition:Rule of Formation|rule of formation]] from the [[Definition:Bottom-Up Specification of Propositional Logic|bottom-up specification of propositional logic]].
The rules $\mathbf W : TF$ and $\mathbf W : \PP_0... | Language of Propositional Logic has Unique Parsability | https://proofwiki.org/wiki/Language_of_Propositional_Logic_has_Unique_Parsability | https://proofwiki.org/wiki/Language_of_Propositional_Logic_has_Unique_Parsability | [
"Language of Propositional Logic"
] | [
"Definition:Language of Propositional Logic",
"Definition:Unique Parsability"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Rule of Formation",
"Definition:Language of Propositional Logic/Formal Grammar/Bottom-Up Specification",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Rule of Formation"
] |
proofwiki-13591 | Powers of 2 not containing Digit Power of 2 | $2^{16} = 65 \, 536$ is the only known power of $2$, up to $2^{31 \, 000}$, whose digits do not contain $1$, $2$, $4$ or $8$. | This has been demonstrated by an exhaustive search.
{{qed}} | $2^{16} = 65 \, 536$ is the only known [[Definition:Integer Power|power]] of $2$, up to $2^{31 \, 000}$, whose [[Definition:Digit|digits]] do not contain $1$, $2$, $4$ or $8$. | This has been demonstrated by an exhaustive search.
{{qed}} | Powers of 2 not containing Digit Power of 2 | https://proofwiki.org/wiki/Powers_of_2_not_containing_Digit_Power_of_2 | https://proofwiki.org/wiki/Powers_of_2_not_containing_Digit_Power_of_2 | [
"Powers of 2",
"65,536"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Digit"
] | [] |
proofwiki-13592 | Construction of Regular 65,537-Gon | It is possible to construct a regular polygon with $65 \, 537$ sides) using a compass and straightedge construction. | From Construction of Regular Prime $p$-Gon Exists iff $p$ is Fermat Prime it is known that this construction is possible.
{{ProofWanted}} | It is possible to construct a [[Definition:Regular Polygon|regular polygon]] with $65 \, 537$ [[Definition:Side of Polygon|sides]]) using a [[Definition:Compass and Straightedge Construction|compass and straightedge construction]]. | From [[Construction of Regular Prime p-Gon Exists iff p is Fermat Prime|Construction of Regular Prime $p$-Gon Exists iff $p$ is Fermat Prime]] it is known that this construction is possible.
{{ProofWanted}} | Construction of Regular 65,537-Gon | https://proofwiki.org/wiki/Construction_of_Regular_65,537-Gon | https://proofwiki.org/wiki/Construction_of_Regular_65,537-Gon | [
"Compass and Straightedge Constructions",
"Regular Polygons",
"65,537"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"Definition:Compass and Straightedge Construction"
] | [
"Construction of Regular Prime p-Gon Exists iff p is Fermat Prime"
] |
proofwiki-13593 | There are no Odd Unitary Perfect Numbers | No unitary perfect numbers exist which are odd. | Let $n$ be an odd number with prime decomposition $n = p_1^{a_1} \cdots p_k^{a_k}$.
By Sum of Unitary Divisors of Integer, the sum of its unitary divisors is $\ds \prod_{1 \mathop \le i \mathop \le k} \paren {1 + p_i^{a_i} }$.
To be a unitary perfect number, this must be equal to $2 n$.
Since each $p$ is odd, each $1 +... | No [[Definition:Unitary Perfect Number|unitary perfect numbers]] exist which are [[Definition:Odd Integer|odd]]. | Let $n$ be an [[Definition:Odd Integer|odd number]] with [[Definition:Prime Decomposition|prime decomposition]] $n = p_1^{a_1} \cdots p_k^{a_k}$.
By [[Sum of Unitary Divisors of Integer]], the sum of its [[Definition:Unitary Divisor|unitary divisors]] is $\ds \prod_{1 \mathop \le i \mathop \le k} \paren {1 + p_i^{a_i}... | There are no Odd Unitary Perfect Numbers | https://proofwiki.org/wiki/There_are_no_Odd_Unitary_Perfect_Numbers | https://proofwiki.org/wiki/There_are_no_Odd_Unitary_Perfect_Numbers | [
"Unitary Perfect Numbers"
] | [
"Definition:Unitary Perfect Number",
"Definition:Odd Integer"
] | [
"Definition:Odd Integer",
"Definition:Prime Decomposition",
"Sum of Unitary Divisors of Integer",
"Definition:Unitary Divisor",
"Definition:Unitary Perfect Number",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Odd Integer",
"Definition:Di... |
proofwiki-13594 | Automorphic Numbers with 5 Digits | The only $5$-digit automorphic number which does not begin with a zero is $90 \, 625$. | We have:
:$90 \, 625^2 = 8 \, 212 \, 8 \mathbf {90 \, 625}$
thus demonstrating it is automorphic.
By Automorphic Numbers in Base 10, the only other possible candidate is $6^{5^4}$.
However:
:$6^{5^4} \equiv 09 \, 376 \pmod {10^5}$
begins with a zero.
Hence there are no others.
{{qed}} | The only $5$-[[Definition:Digit|digit]] [[Definition:Automorphic Number|automorphic number]] which does not begin with a [[Definition:Zero Digit|zero]] is $90 \, 625$. | We have:
:$90 \, 625^2 = 8 \, 212 \, 8 \mathbf {90 \, 625}$
thus demonstrating it is [[Definition:Automorphic Number|automorphic]].
By [[Automorphic Numbers in Base 10]], the only other possible candidate is $6^{5^4}$.
However:
:$6^{5^4} \equiv 09 \, 376 \pmod {10^5}$
begins with a [[Definition:Zero Digit|zero]].
H... | Automorphic Numbers with 5 Digits | https://proofwiki.org/wiki/Automorphic_Numbers_with_5_Digits | https://proofwiki.org/wiki/Automorphic_Numbers_with_5_Digits | [
"Automorphic Numbers"
] | [
"Definition:Digit",
"Definition:Automorphic Number",
"Definition:Zero Digit"
] | [
"Definition:Automorphic Number",
"Automorphic Numbers in Base 10",
"Definition:Zero Digit"
] |
proofwiki-13595 | Kaprekar's Process on 5 Digit Number | Let $n$ be a $5$-digit integer whose digits are not all the same.
Kaprekar's process, when applied to $n$, results in one of the following $3$ cycles:
:$53 \, 955 \to 59 \, 994 \to 53 \, 955$
:$61 \, 974 \to 82 \, 962 \to 75 \, 933 \to 63 \, 954 \to 61 \, 974$
:$62 \, 964 \to 71 \, 973 \to 83 \, 952 \to 74 \, 943 \to 6... | We have:
{{begin-eqn}}
{{eqn | l = 95 \, 553 - 35 \, 559
| r = 59 \, 994
}}
{{eqn | l = 99 \, 954 - 45 \, 999
| r = 53 \, 995
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 97 \, 641 - 14 \, 679
| r = 82 \, 962
}}
{{eqn | l = 98 \, 622 - 22 \, 689
| r = 75 \, 933
}}
{{eqn | l = 97 \, 533 - 33 \, 579
... | Let $n$ be a $5$-[[Definition:Digit|digit]] [[Definition:Positive Integer|integer]] whose [[Definition:Digit|digits]] are not all the same.
[[Definition:Kaprekar's Process|Kaprekar's process]], when applied to $n$, results in one of the following $3$ cycles:
:$53 \, 955 \to 59 \, 994 \to 53 \, 955$
:$61 \, 974 \to 8... | We have:
{{begin-eqn}}
{{eqn | l = 95 \, 553 - 35 \, 559
| r = 59 \, 994
}}
{{eqn | l = 99 \, 954 - 45 \, 999
| r = 53 \, 995
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 97 \, 641 - 14 \, 679
| r = 82 \, 962
}}
{{eqn | l = 98 \, 622 - 22 \, 689
| r = 75 \, 933
}}
{{eqn | l = 97 \, 533 - 33 \, 57... | Kaprekar's Process on 5 Digit Number | https://proofwiki.org/wiki/Kaprekar's_Process_on_5_Digit_Number | https://proofwiki.org/wiki/Kaprekar's_Process_on_5_Digit_Number | [
"Kaprekar's Process"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Digit",
"Definition:Kaprekar's Process"
] | [] |
proofwiki-13596 | Tableau Extension Lemma/General Statement | Let $\mathbf H'$ be another finite set of WFFs.
Then there exists a finished finite propositional tableau $T'$ such that:
$(1):\quad$ the root of $T'$ is $\mathbf H \cup \mathbf H'$;
$(2):\quad$ $T$ is a rooted subtree of $T'$. | Let $T_{\mathbf H'}$ be the finite propositional tableau obtained by replacing the hypothesis set $\mathbf H$ of $T$ with $\mathbf H \cup \mathbf H'$.
By the Tableau Extension Lemma, $T_{\mathbf H'}$ has a finished extension $T'$.
By definition of extension, $T_{\mathbf H'}$ is a rooted subtree of $T'$.
But $T_{\mathbf... | Let $\mathbf H'$ be another [[Definition:Finite Set|finite set]] of [[Definition:WFF of Propositional Logic|WFFs]].
Then there exists a [[Definition:Finished Propositional Tableau|finished]] [[Definition:Finite Propositional Tableau|finite propositional tableau]] $T'$ such that:
$(1):\quad$ the [[Definition:Root of ... | Let $T_{\mathbf H'}$ be the [[Definition:Finite Propositional Tableau|finite propositional tableau]] obtained by replacing the [[Definition:Hypothesis Set|hypothesis set]] $\mathbf H$ of $T$ with $\mathbf H \cup \mathbf H'$.
By the [[Tableau Extension Lemma]], $T_{\mathbf H'}$ has a [[Definition:Finished Propositional... | Tableau Extension Lemma/General Statement/Proof 1 | https://proofwiki.org/wiki/Tableau_Extension_Lemma/General_Statement | https://proofwiki.org/wiki/Tableau_Extension_Lemma/General_Statement/Proof_1 | [
"Tableau Extension Lemma",
"Propositional Tableaux"
] | [
"Definition:Finite Set",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Finished Propositional Tableau",
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Labeled Tree for Propositional Logic/Hypothesis Set",
"Definition:Rooted Subtree"
] | [
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Labeled Tree for Propositional Logic/Hypothesis Set",
"Tableau Extension Lemma",
"Definition:Finished Propositional Tableau",
"Definition:Extension of Propositional Tableau",
"Definition:Extension of Propositional Tableau",
"Definition:... |
proofwiki-13597 | Tableau Extension Lemma/General Statement | Let $\mathbf H'$ be another finite set of WFFs.
Then there exists a finished finite propositional tableau $T'$ such that:
$(1):\quad$ the root of $T'$ is $\mathbf H \cup \mathbf H'$;
$(2):\quad$ $T$ is a rooted subtree of $T'$. | The proof uses induction on the number $n$ of elements of $\mathbf H$.
Suppose we are given the result for the case $n = 1$, that is, when $\mathbf H$ is a singleton.
Suppose also that we are given the result for all sets $\mathbf H'$ with $n$ elements.
Now, given a set $\mathbf H' = \set {\mathbf A_1, \ldots, \mathbf ... | Let $\mathbf H'$ be another [[Definition:Finite Set|finite set]] of [[Definition:WFF of Propositional Logic|WFFs]].
Then there exists a [[Definition:Finished Propositional Tableau|finished]] [[Definition:Finite Propositional Tableau|finite propositional tableau]] $T'$ such that:
$(1):\quad$ the [[Definition:Root of ... | The proof uses [[Principle of Mathematical Induction|induction]] on the number $n$ of elements of $\mathbf H$.
Suppose we are given the result for the case $n = 1$, that is, when $\mathbf H$ is a [[Definition:Singleton|singleton]].
Suppose also that we are given the result for all sets $\mathbf H'$ with $n$ [[Definit... | Tableau Extension Lemma/General Statement/Proof 2 | https://proofwiki.org/wiki/Tableau_Extension_Lemma/General_Statement | https://proofwiki.org/wiki/Tableau_Extension_Lemma/General_Statement/Proof_2 | [
"Tableau Extension Lemma",
"Propositional Tableaux"
] | [
"Definition:Finite Set",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Finished Propositional Tableau",
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Labeled Tree for Propositional Logic/Hypothesis Set",
"Definition:Rooted Subtree"
] | [
"Principle of Mathematical Induction",
"Definition:Singleton",
"Definition:Element",
"Definition:Set",
"Definition:Element",
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Finished Propositional Tableau",
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Subgra... |
proofwiki-13598 | Tableau Extension Lemma/General Statement/Proof 1 | Let $T$ be a finite propositional tableau.
Let its hypothesis set $\mathbf H$ be finite.
{{:Tableau Extension Lemma/General Statement}} | Let $T_{\mathbf H'}$ be the finite propositional tableau obtained by replacing the hypothesis set $\mathbf H$ of $T$ with $\mathbf H \cup \mathbf H'$.
By the Tableau Extension Lemma, $T_{\mathbf H'}$ has a finished extension $T'$.
By definition of extension, $T_{\mathbf H'}$ is a rooted subtree of $T'$.
But $T_{\mathbf... | Let $T$ be a [[Definition:Finite Propositional Tableau|finite propositional tableau]].
Let its [[Definition:Hypothesis Set|hypothesis set]] $\mathbf H$ be [[Definition:Finite Set|finite]].
{{:Tableau Extension Lemma/General Statement}} | Let $T_{\mathbf H'}$ be the [[Definition:Finite Propositional Tableau|finite propositional tableau]] obtained by replacing the [[Definition:Hypothesis Set|hypothesis set]] $\mathbf H$ of $T$ with $\mathbf H \cup \mathbf H'$.
By the [[Tableau Extension Lemma]], $T_{\mathbf H'}$ has a [[Definition:Finished Propositional... | Tableau Extension Lemma/General Statement/Proof 1 | https://proofwiki.org/wiki/Tableau_Extension_Lemma/General_Statement/Proof_1 | https://proofwiki.org/wiki/Tableau_Extension_Lemma/General_Statement/Proof_1 | [
"Tableau Extension Lemma"
] | [
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Labeled Tree for Propositional Logic/Hypothesis Set",
"Definition:Finite Set"
] | [
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Labeled Tree for Propositional Logic/Hypothesis Set",
"Tableau Extension Lemma",
"Definition:Finished Propositional Tableau",
"Definition:Extension of Propositional Tableau",
"Definition:Extension of Propositional Tableau",
"Definition:... |
proofwiki-13599 | Tableau Extension Lemma/General Statement/Proof 2 | Let $T$ be a finite propositional tableau.
Let its hypothesis set $\mathbf H$ be finite.
{{:Tableau Extension Lemma/General Statement}} | The proof uses induction on the number $n$ of elements of $\mathbf H$.
Suppose we are given the result for the case $n = 1$, that is, when $\mathbf H$ is a singleton.
Suppose also that we are given the result for all sets $\mathbf H'$ with $n$ elements.
Now, given a set $\mathbf H' = \set {\mathbf A_1, \ldots, \mathbf ... | Let $T$ be a [[Definition:Finite Propositional Tableau|finite propositional tableau]].
Let its [[Definition:Hypothesis Set|hypothesis set]] $\mathbf H$ be [[Definition:Finite Set|finite]].
{{:Tableau Extension Lemma/General Statement}} | The proof uses [[Principle of Mathematical Induction|induction]] on the number $n$ of elements of $\mathbf H$.
Suppose we are given the result for the case $n = 1$, that is, when $\mathbf H$ is a [[Definition:Singleton|singleton]].
Suppose also that we are given the result for all sets $\mathbf H'$ with $n$ [[Definit... | Tableau Extension Lemma/General Statement/Proof 2 | https://proofwiki.org/wiki/Tableau_Extension_Lemma/General_Statement/Proof_2 | https://proofwiki.org/wiki/Tableau_Extension_Lemma/General_Statement/Proof_2 | [
"Tableau Extension Lemma"
] | [
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Labeled Tree for Propositional Logic/Hypothesis Set",
"Definition:Finite Set"
] | [
"Principle of Mathematical Induction",
"Definition:Singleton",
"Definition:Element",
"Definition:Set",
"Definition:Element",
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Finished Propositional Tableau",
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Subgra... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.