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-13400 | Substitution in Big-O Estimate/Real Analysis | Let $f$ and $g$ be real-valued or complex-valued functions defined on a neighborhood of $+ \infty$ in $\R$.
Let $f = \map \OO g$, where $\OO$ denotes big-O notation.
Let $h$ be a real-valued defined on a neighborhood of $+ \infty$ in $\R$.
Let $\ds \lim_{x \mathop \to +\infty} \map h x = +\infty$.
Then:
:$f \circ h = \... | {{ProofWanted}}
Category:Big-O Notation
3p4oafq4g08ec0l09shqdiyoizg7y1n | Let $f$ and $g$ be [[Definition:Real-Valued Function|real-valued]] or [[Definition:Complex-Valued Function|complex-valued functions]] defined on a [[Definition:Neighborhood of Positive Infinity|neighborhood of $+ \infty$]] in $\R$.
Let $f = \map \OO g$, where $\OO$ denotes [[Definition:Big-O Notation|big-O notation]].... | {{ProofWanted}}
[[Category:Big-O Notation]]
3p4oafq4g08ec0l09shqdiyoizg7y1n | Substitution in Big-O Estimate/Real Analysis | https://proofwiki.org/wiki/Substitution_in_Big-O_Estimate/Real_Analysis | https://proofwiki.org/wiki/Substitution_in_Big-O_Estimate/Real_Analysis | [
"Big-O Notation"
] | [
"Definition:Real-Valued Function",
"Definition:Complex-Valued Function",
"Definition:Neighborhood of Infinity (Real Analysis)/Positive Infinity",
"Definition:Big-O Notation",
"Definition:Real-Valued Function",
"Definition:Neighborhood of Infinity (Real Analysis)/Positive Infinity"
] | [
"Category:Big-O Notation"
] |
proofwiki-13401 | Product of Big-O Estimates/Sequences | Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers.
Let:
:$a_n = \map \OO {b_n}$
:$c_n = \map \OO {d_n}$
where $\OO$ denotes big-$\OO$ notation.
Then:
:$a_n c_n = \map \OO {b_n d_n}$ | Since:
:$a_n = \map \OO {b_n}$
there exists a positive real number $C_1$ and natural number $N_1$ such that:
:$\size {a_n} \le C_1 \size {b_n}$
for all $n \ge N_1$.
Similarly, since:
:$c_n = \map \OO {d_n}$
there exists a positive real number $C_2$ and natural number $N_2$ such that:
:$\size {c_n} \le C_2 \size {d_... | Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
Let:
:$a_n = \map \OO {b_n}$
:$c_n = \map \OO {d_n}$
where $\OO$ denotes [[Definition:Big-O Notation|big-$\OO$ notation]].
... | Since:
:$a_n = \map \OO {b_n}$
there exists a [[Definition:Positive Real Number|positive real number]] $C_1$ and [[Definition:Natural Number|natural number]] $N_1$ such that:
:$\size {a_n} \le C_1 \size {b_n}$
for all $n \ge N_1$.
Similarly, since:
:$c_n = \map \OO {d_n}$
there exists a [[Definition:Positive ... | Product of Big-O Estimates/Sequences | https://proofwiki.org/wiki/Product_of_Big-O_Estimates/Sequences | https://proofwiki.org/wiki/Product_of_Big-O_Estimates/Sequences | [
"Big-O Notation"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Big-O Notation"
] | [
"Definition:Positive/Real Number",
"Definition:Natural Numbers",
"Definition:Positive/Real Number",
"Definition:Natural Numbers",
"Category:Big-O Notation"
] |
proofwiki-13402 | Equivalence of Definitions of Order of Entire Function | Let $f: \C \to \C$ be an entire function.
Let $\ln$ denote the natural logarithm.
{{TFAE|def = Order of Entire Function}} | Let:
:$\alpha_1 = \ds \limsup_{R \mathop \to \infty} \frac {\ds \ln \ln \max_{\cmod z \mathop \le R} \cmod f} {\ln R}$
:$\alpha_2 = \inf \set {\beta \ge 0: \ds \map \ln {\max_{\cmod z \mathop \le R} \cmod {\map f z} } = \map \OO {R^\beta} }$
:$\alpha_3 = \inf \set {\beta \ge 0: \map f z = \map \OO {\map \exp {\cmod z^\... | Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]].
Let $\ln$ denote the [[Definition:Natural Logarithm|natural logarithm]].
{{TFAE|def = Order of Entire Function}} | Let:
:$\alpha_1 = \ds \limsup_{R \mathop \to \infty} \frac {\ds \ln \ln \max_{\cmod z \mathop \le R} \cmod f} {\ln R}$
:$\alpha_2 = \inf \set {\beta \ge 0: \ds \map \ln {\max_{\cmod z \mathop \le R} \cmod {\map f z} } = \map \OO {R^\beta} }$
:$\alpha_3 = \inf \set {\beta \ge 0: \map f z = \map \OO {\map \exp {\cmod z^\... | Equivalence of Definitions of Order of Entire Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Order_of_Entire_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Order_of_Entire_Function | [
"Order of Entire Function"
] | [
"Definition:Entire Function",
"Definition:Natural Logarithm"
] | [] |
proofwiki-13403 | Sum of Big-O Estimates/Sequences | Let $\sequence {a_n},\sequence {b_n},\sequence {c_n},\sequence {d_n}$ be sequences of real or complex numbers.
Let:
:$a_n = \map \OO {b_n}$
:$c_n = \map \OO {d_n}$
where $\OO$ denotes big-$\OO$ notation.
Then:
:$a_n + c_n = \map \OO {\size {b_n} + \size {d_n} }$ | Since:
:$a_n = \map \OO {b_n}$
there exists a positive real number $C_1$ and natural number $N_1$ such that:
:$\size {a_n} \le C_1 \size {b_n}$
for all $n \ge N_1$.
Similarly, since:
:$c_n = \map \OO {d_n}$
there exists a positive real number $C_2$ and natural number $N_2$ such that:
:$\size {c_n} \le C_2 \size {d_... | Let $\sequence {a_n},\sequence {b_n},\sequence {c_n},\sequence {d_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
Let:
:$a_n = \map \OO {b_n}$
:$c_n = \map \OO {d_n}$
where $\OO$ denotes [[Definition:Big-O Notation|big-$\OO$ notation]].
Th... | Since:
:$a_n = \map \OO {b_n}$
there exists a [[Definition:Positive Real Number|positive real number]] $C_1$ and [[Definition:Natural Number|natural number]] $N_1$ such that:
:$\size {a_n} \le C_1 \size {b_n}$
for all $n \ge N_1$.
Similarly, since:
:$c_n = \map \OO {d_n}$
there exists a [[Definition:Positive ... | Sum of Big-O Estimates/Sequences | https://proofwiki.org/wiki/Sum_of_Big-O_Estimates/Sequences | https://proofwiki.org/wiki/Sum_of_Big-O_Estimates/Sequences | [
"Big-O Notation"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Big-O Notation"
] | [
"Definition:Positive/Real Number",
"Definition:Natural Numbers",
"Definition:Positive/Real Number",
"Definition:Natural Numbers",
"Triangle Inequality",
"Category:Big-O Notation"
] |
proofwiki-13404 | Little-O Times Big-O is Little-O/Sequences | Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers.
Let:
:$a_n = \map \OO {b_n}$
:$c_n = \map {\mathcal o} {d_n}$
where:
:$\OO$ denotes big-$\OO$ notation
:$\mathcal o$ denotes little-$\mathcal o$ notation.
Then:
:$a_n c_n = \map {\mathcal o} {b_n d_n}$ | Let $\epsilon \in \R_{> 0}$.
Since $a_n = \map \OO {b_n}$:
:$\exists c \in \R: c \ge 0: \exists n_0 \in \N: \paren {n \ge n_0 \implies \size {a_n} \le c \cdot \size {b_n} }$
Since $c_n = \map {\mathcal o} {d_n}$:
:$\exists n_1 \in \N: \paren {n \ge n_1 \implies \size {c_n} \le \dfrac \epsilon {c + 1} \cdot \size {d_n} ... | Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
Let:
:$a_n = \map \OO {b_n}$
:$c_n = \map {\mathcal o} {d_n}$
where:
:$\OO$ denotes [[Definition:Big-O Notation|big-$\OO$ no... | Let $\epsilon \in \R_{> 0}$.
Since $a_n = \map \OO {b_n}$:
:$\exists c \in \R: c \ge 0: \exists n_0 \in \N: \paren {n \ge n_0 \implies \size {a_n} \le c \cdot \size {b_n} }$
Since $c_n = \map {\mathcal o} {d_n}$:
:$\exists n_1 \in \N: \paren {n \ge n_1 \implies \size {c_n} \le \dfrac \epsilon {c + 1} \cdot \size {d_n... | Little-O Times Big-O is Little-O/Sequences | https://proofwiki.org/wiki/Little-O_Times_Big-O_is_Little-O/Sequences | https://proofwiki.org/wiki/Little-O_Times_Big-O_is_Little-O/Sequences | [
"Big-O Notation",
"Little-O Notation"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Big-O Notation",
"Definition:Little-O Notation"
] | [
"Category:Big-O Notation",
"Category:Little-O Notation"
] |
proofwiki-13405 | Field Norm of Complex Number Equals Field Norm | Let $z = a + i b$ be a complex number, where $a, b \in \R$.
Then the field norm of $z$ is the field norm with respect to the field extension $\C / \R$. | {{ProofWanted}}
Category:Field Extensions
Category:Field Norm of Complex Number
0i5v8vch6mvpd11ja0d3gdrgpmfcvbj | Let $z = a + i b$ be a [[Definition:Complex Number|complex number]], where $a, b \in \R$.
Then the [[Definition:Field Norm of Complex Number|field norm]] of $z$ is the [[Definition:Field Norm|field norm]] with respect to the [[Definition:Field Extension|field extension]] $\C / \R$. | {{ProofWanted}}
[[Category:Field Extensions]]
[[Category:Field Norm of Complex Number]]
0i5v8vch6mvpd11ja0d3gdrgpmfcvbj | Field Norm of Complex Number Equals Field Norm | https://proofwiki.org/wiki/Field_Norm_of_Complex_Number_Equals_Field_Norm | https://proofwiki.org/wiki/Field_Norm_of_Complex_Number_Equals_Field_Norm | [
"Field Extensions",
"Field Norm of Complex Number"
] | [
"Definition:Complex Number",
"Definition:Field Norm of Complex Number",
"Definition:Field Norm",
"Definition:Field Extension"
] | [
"Category:Field Extensions",
"Category:Field Norm of Complex Number"
] |
proofwiki-13406 | Big-O Notation for Sequences Coincides with General Definition | Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers.
Let $\N$ be given the discrete topology.
{{TFAE}}
:$(1): \quad a_n = \map \OO {b_n}$, where $\OO$ denotes big-$\OO$ notation for sequences
:$(2): \quad a_n = \map \OO {b_n}$, where $\OO$ stands for the general definition of big-$\OO$ n... | {{ProofWanted}}
Category:Big-O Notation
k2irag1srvyqutfx9i6bgs5dh0bu1mm | Let $\sequence {a_n}$ and $\sequence {b_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
Let $\N$ be given the [[Definition:Discrete Topology|discrete topology]].
{{TFAE}}
:$(1): \quad a_n = \map \OO {b_n}$, where $\OO$ denotes [[Definition... | {{ProofWanted}}
[[Category:Big-O Notation]]
k2irag1srvyqutfx9i6bgs5dh0bu1mm | Big-O Notation for Sequences Coincides with General Definition | https://proofwiki.org/wiki/Big-O_Notation_for_Sequences_Coincides_with_General_Definition | https://proofwiki.org/wiki/Big-O_Notation_for_Sequences_Coincides_with_General_Definition | [
"Big-O Notation"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Discrete Topology",
"Definition:Big-O Notation/Sequence",
"Definition:Big-O Notation/General Definition"
] | [
"Category:Big-O Notation"
] |
proofwiki-13407 | Penholodigital Square Equation | The following equations, which include each digit from $1$ to $9$ inclusive, are the only ones of their kind:
{{begin-eqn}}
{{eqn | l = 567^2
| r = 321 \, 489
}}
{{eqn | l = 854^2
| r = 729 \, 316
}}
{{end-eqn}} | The square of a $2$-digit integer cannot have more than $4$ digits:
:$99^2 = 9801$
The square of a $4$-digit integer has at least $7$ digits:
:$1000^2 = 1 \, 000 \, 000$
Hence we only need to inspect $3$-digit integers, with a corresponding $6$-digit square.
A lower bound is given by $\ceiling {\sqrt {123 \, 456}} = 35... | The following [[Definition:Equation|equations]], which include each [[Definition:Digit|digit]] from $1$ to $9$ inclusive, are the only ones of their kind:
{{begin-eqn}}
{{eqn | l = 567^2
| r = 321 \, 489
}}
{{eqn | l = 854^2
| r = 729 \, 316
}}
{{end-eqn}} | The [[Definition:Square (Algebra)|square]] of a $2$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] cannot have more than $4$ [[Definition:Digit|digits]]:
:$99^2 = 9801$
The [[Definition:Square (Algebra)|square]] of a $4$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] has at least $7$ [[Definition... | Penholodigital Square Equation | https://proofwiki.org/wiki/Penholodigital_Square_Equation | https://proofwiki.org/wiki/Penholodigital_Square_Equation | [
"Square Numbers",
"Recreational Mathematics",
"Penholodigital Sets",
"Penholodigital Integers"
] | [
"Definition:Equation",
"Definition:Digit"
] | [
"Definition:Square/Function",
"Definition:Digit",
"Definition:Integer",
"Definition:Digit",
"Definition:Square/Function",
"Definition:Digit",
"Definition:Integer",
"Definition:Digit",
"Definition:Digit",
"Definition:Integer",
"Definition:Digit",
"Definition:Square/Function",
"Definition:Digi... |
proofwiki-13408 | Bounded iff Big-O of 1/Sequences | Let $\sequence {a_n}$ be a sequence of real or complex numbers.
{{TFAE}}
:$(1): \quad a_n$ is bounded
:$(2): \quad a_n = \map \OO 1$, where $\OO$ denotes big-$\OO$ notation | {{begin-eqn}}
{{eqn | l = a_n
| o = \text {is}
| r = \text {bounded}
}}
{{eqn | ll= \leadstoandfrom
| q = \exists k \in \R
| l = \size {a_n}
| o = \le
| r = k
| c = {{Defof|Bounded Sequence}}
}}
{{eqn | ll= \leadstoandfrom
| q = \exists k \in \R
| l = \size {a_n}
... | Let $\sequence {a_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
{{TFAE}}
:$(1): \quad a_n$ is [[Definition:Bounded Sequence|bounded]]
:$(2): \quad a_n = \map \OO 1$, where $\OO$ denotes [[Definition:Big-O Notation for Sequences|big-$\O... | {{begin-eqn}}
{{eqn | l = a_n
| o = \text {is}
| r = \text {bounded}
}}
{{eqn | ll= \leadstoandfrom
| q = \exists k \in \R
| l = \size {a_n}
| o = \le
| r = k
| c = {{Defof|Bounded Sequence}}
}}
{{eqn | ll= \leadstoandfrom
| q = \exists k \in \R
| l = \size {a_n}
... | Bounded iff Big-O of 1/Sequences | https://proofwiki.org/wiki/Bounded_iff_Big-O_of_1/Sequences | https://proofwiki.org/wiki/Bounded_iff_Big-O_of_1/Sequences | [
"Big-O Notation"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Bounded Sequence",
"Definition:Big-O Notation/Sequence"
] | [
"Category:Big-O Notation"
] |
proofwiki-13409 | Sequence of 11 Primes by Trebling and Adding 16 | The process of multiplication by $3$ and then adding $16$ produces a sequence of $11$ primes when starting from $587$:
:$587, 1777, 5347, 16 \, 057, 48 \, 187, 144 \, 577, 433 \, 747, 1 \, 301 \, 257, 3 \, 903 \, 787, 11 \, 711 \, 377, 35 \, 134 \, 147$ | {{begin-eqn}}
{{eqn | o =
| r = 587
| c = is prime
}}
{{eqn | l = 3 \times 587 + 16
| r = 1777
| c = which is prime
}}
{{eqn | l = 3 \times 1777 + 16
| r = 5347
| c = which is prime
}}
{{eqn | l = 3 \times 5347 + 16
| r = 16 \, 057
| c = which is prime
}}
{{eqn | l = 3 \... | The process of [[Definition:Integer Multiplication|multiplication]] by $3$ and then adding $16$ produces a [[Definition:Integer Sequence|sequence]] of $11$ [[Definition:Prime Number|primes]] when starting from $587$:
:$587, 1777, 5347, 16 \, 057, 48 \, 187, 144 \, 577, 433 \, 747, 1 \, 301 \, 257, 3 \, 903 \, 787, 11 ... | {{begin-eqn}}
{{eqn | o =
| r = 587
| c = is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 3 \times 587 + 16
| r = 1777
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 3 \times 1777 + 16
| r = 5347
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 3 ... | Sequence of 11 Primes by Trebling and Adding 16 | https://proofwiki.org/wiki/Sequence_of_11_Primes_by_Trebling_and_Adding_16 | https://proofwiki.org/wiki/Sequence_of_11_Primes_by_Trebling_and_Adding_16 | [
"Prime Numbers",
"Polynomial Expressions for Primes"
] | [
"Definition:Multiplication/Integers",
"Definition:Integer Sequence",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",... |
proofwiki-13410 | Sum of Little-O Estimates/Sequences | Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers.
Let:
:$a_n = \map \oo {b_n}$
:$c_n = \map \oo {d_n}$
where $\oo$ denotes little-$\oo$ notation.
Then:
:$a_n + c_n = \map \oo {\size {b_n} + \size {d_n} }$ | Let $\epsilon > 0$.
Then by definition of little-$\oo$ notation:
:$\exists n_1 \in \N: \paren {n \ge n_1 \implies \size {a_n} \le \epsilon \cdot \size {b_n}}$
:$\exists n_2 \in \N: \paren {n \ge n_2 \implies \size {c_n} \le \epsilon \cdot \size {d_n}}$
For $n \ge \max \set {n_1, n_2}$:
{{begin-eqn}}
{{eqn | l = \size {... | Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
Let:
:$a_n = \map \oo {b_n}$
:$c_n = \map \oo {d_n}$
where $\oo$ denotes [[Definition:Little-O Notation for Sequences|little... | Let $\epsilon > 0$.
Then by definition of [[Definition:Little-O Notation for Sequences|little-$\oo$ notation]]:
:$\exists n_1 \in \N: \paren {n \ge n_1 \implies \size {a_n} \le \epsilon \cdot \size {b_n}}$
:$\exists n_2 \in \N: \paren {n \ge n_2 \implies \size {c_n} \le \epsilon \cdot \size {d_n}}$
For $n \ge \max \s... | Sum of Little-O Estimates/Sequences | https://proofwiki.org/wiki/Sum_of_Little-O_Estimates/Sequences | https://proofwiki.org/wiki/Sum_of_Little-O_Estimates/Sequences | [
"Little-O Notation"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Little-O Notation/Sequence"
] | [
"Definition:Little-O Notation/Sequence",
"Triangle Inequality",
"Definition:Absolute Value",
"Definition:Positive/Real Number",
"Definition:Little-O Notation/Sequence",
"Category:Little-O Notation"
] |
proofwiki-13411 | Equivalence of Definitions of Asymptotically Equal Sequences | Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences in $\R$.
{{TFAE|def = Asymptotically Equal Sequences}} | === $(1)$ iff $(2)$ ===
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \dfrac {a_n} {b_n}
| o = \to
| r = 1
}}
{{eqn | ll = \leadstoandfrom
| l = \lim_{n \mathop \to \infty} \paren {\dfrac {a_n} {b_n} - \dfrac {b_n} {b_n} }
| o = \to
| r = 0
}}
{{eqn | ll = \leadstoandfrom
| l... | Let $\sequence {a_n}$ and $\sequence {b_n}$ be [[Definition:Sequence|sequences in $\R$]].
{{TFAE|def = Asymptotically Equal Sequences}} | === $(1)$ iff $(2)$ ===
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \dfrac {a_n} {b_n}
| o = \to
| r = 1
}}
{{eqn | ll = \leadstoandfrom
| l = \lim_{n \mathop \to \infty} \paren {\dfrac {a_n} {b_n} - \dfrac {b_n} {b_n} }
| o = \to
| r = 0
}}
{{eqn | ll = \leadstoandfrom
| ... | Equivalence of Definitions of Asymptotically Equal Sequences | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Asymptotically_Equal_Sequences | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Asymptotically_Equal_Sequences | [
"Asymptotic Equality"
] | [
"Definition:Sequence"
] | [] |
proofwiki-13412 | Largest Number not Expressible as Sum of Multiples of 23 and 28 | The largest integer $n$ that cannot be expressed in the form:
:$n = 23 x + 28 y$
for $x, y \in \Z_{>0}$ is $593$. | By Largest Number not Expressible as Sum of Multiples of Coprime Integers, the largest such number is:
:$\paren {23 - 1} \times \paren {28 - 1} - 1 = 593$
{{ProofWanted|This is a specific "historical" example of a general result which I read in Polya and Szego some time back, which still needs to be added into {{ProofW... | The largest [[Definition:Positive Integer|integer]] $n$ that cannot be expressed in the form:
:$n = 23 x + 28 y$
for $x, y \in \Z_{>0}$ is $593$. | By [[Largest Number not Expressible as Sum of Multiples of Coprime Integers]], the largest such number is:
:$\paren {23 - 1} \times \paren {28 - 1} - 1 = 593$
{{ProofWanted|This is a specific "historical" example of a general result which I read in Polya and Szego some time back, which still needs to be added into {{P... | Largest Number not Expressible as Sum of Multiples of 23 and 28 | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Multiples_of_23_and_28 | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Multiples_of_23_and_28 | [
"Integer Combinations"
] | [
"Definition:Positive/Integer"
] | [
"Largest Number not Expressible as Sum of Multiples of Coprime Integers"
] |
proofwiki-13413 | Open implies There Exists Way Below Element | Let $L = \struct {S, \preceq, \tau}$ be a continuous topological lattice with Scott topology.
Let $p \in S, A \subseteq S$ such that:
:$A$ is open and $p \in A$.
Then:
:$\exists q \in A: q \ll p$
where $q \ll p$ denotes $q$ is way below $p$. | By definition of continuous ordered set:
:$p^\ll$ is directed
and
:$L$ satisfies the axiom of approximation.
By the axiom of approximation:
:$p = \map \sup {p^\ll}$
By definition of Scott topology:
:$A$ is inaccessible by directed suprema.
By definition of inaccessible by directed suprema:
:$A \cap p^\ll \ne \O$
By def... | Let $L = \struct {S, \preceq, \tau}$ be a [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
Let $p \in S, A \subseteq S$ such that:
:$A$ is [[Definition:Open Set (Topology)|open]] and $p \in A$.
Then:
:$\exists q... | By definition of [[Definition:Continuous Ordered Set|continuous ordered set]]:
:$p^\ll$ is [[Definition:Directed Subset|directed]]
and
:$L$ satisfies the [[Axiom:Axiom of Approximation|axiom of approximation]].
By the [[Axiom:Axiom of Approximation|axiom of approximation]]:
:$p = \map \sup {p^\ll}$
By definition of [... | Open implies There Exists Way Below Element | https://proofwiki.org/wiki/Open_implies_There_Exists_Way_Below_Element | https://proofwiki.org/wiki/Open_implies_There_Exists_Way_Below_Element | [
"Topological Order Theory",
"Way Below Relation",
"Continuous Lattices",
"Scott Topology"
] | [
"Definition:Continuous Ordered Set",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Open Set/Topology",
"Definition:Element is Way Below"
] | [
"Definition:Continuous Ordered Set",
"Definition:Directed Subset",
"Axiom:Axiom of Approximation",
"Axiom:Axiom of Approximation",
"Definition:Scott Topology",
"Definition:Inaccessible by Directed Suprema",
"Definition:Inaccessible by Directed Suprema",
"Definition:Non-Empty Set",
"Definition:Set In... |
proofwiki-13414 | Smallest Fourth Power which is Sum of 5 Fourth Powers | $625$ is the smallest fourth power which is the sum of $5$ fourth powers:
:$625 = 5^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4$ | We check that for $n = 2, 3, 4$, $n^4$ is not a sum of $5$ smaller fourth powers.
We have:
:$5 \times 1^4 = 5 < 16 = 2^4$
:$5 \times 2^4 = 80 < 81 = 3^4$
so $2^4, 3^4$ are not sums of $5$ fourth powers.
For $n = 4$:
:$\dfrac {4^4} {3^4} < 4$
so such a sum can include at most $3$ $3^4$'s.
However:
:$3 \times 3^4 + 2^4 ... | $625$ is the smallest [[Definition:Fourth Power|fourth power]] which is the [[Definition:Integer Addition|sum]] of $5$ [[Definition:Fourth Power|fourth powers]]:
:$625 = 5^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4$ | We check that for $n = 2, 3, 4$, $n^4$ is not a [[Definition:Integer Addition|sum]] of $5$ smaller [[Definition:Fourth Power|fourth powers]].
We have:
:$5 \times 1^4 = 5 < 16 = 2^4$
:$5 \times 2^4 = 80 < 81 = 3^4$
so $2^4, 3^4$ are not [[Definition:Integer Addition|sums]] of $5$ [[Definition:Fourth Power|fourth power... | Smallest Fourth Power which is Sum of 5 Fourth Powers | https://proofwiki.org/wiki/Smallest_Fourth_Power_which_is_Sum_of_5_Fourth_Powers | https://proofwiki.org/wiki/Smallest_Fourth_Power_which_is_Sum_of_5_Fourth_Powers | [
"Fourth Powers",
"625"
] | [
"Definition:Fourth Power",
"Definition:Addition/Integers",
"Definition:Fourth Power"
] | [
"Definition:Addition/Integers",
"Definition:Fourth Power",
"Definition:Addition/Integers",
"Definition:Fourth Power",
"Definition:Addition/Integers",
"Definition:Addition/Integers",
"Definition:Fourth Power",
"Definition:Fourth Power",
"Definition:Addition/Integers",
"Definition:Fourth Power"
] |
proofwiki-13415 | Interior is Union of Way Above Closures | Let $\left({S, \preceq, \tau}\right)$ be a complete continuous topological lattice with Scott topology.
Let $X \subseteq S$.
Then $X^\circ = \bigcup \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$
where
:$X^\circ$ denotes the interior of $X$,
:$x^\gg$ denotes the way above closure of $x$. | We have:
:$\left\{ {G \in \left\{ {g^\gg: g \in S}\right\}: G \subseteq X}\right\} = \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$
By Way Above Closures Form Basis:
:$\left\{ {x^\gg: x \in S}\right\}$ is basis of $\left({S, \tau}\right)$.
By Interior is Union of Elements of Basis:
:$X^\circ = \bigcup \left\... | Let $\left({S, \preceq, \tau}\right)$ be a [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
Let $X \subseteq S$.
Then $X^\circ = \bigcup \left\{ {x^\gg: x \in S \land x^\... | We have:
:$\left\{ {G \in \left\{ {g^\gg: g \in S}\right\}: G \subseteq X}\right\} = \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$
By [[Way Above Closures Form Basis]]:
:$\left\{ {x^\gg: x \in S}\right\}$ is [[Definition:Analytic Basis|basis]] of $\left({S, \tau}\right)$.
By [[Interior is Union of Element... | Interior is Union of Way Above Closures | https://proofwiki.org/wiki/Interior_is_Union_of_Way_Above_Closures | https://proofwiki.org/wiki/Interior_is_Union_of_Way_Above_Closures | [
"Topological Order Theory",
"Continuous Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Continuous Ordered Set",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Interior (Topology)",
"Definition:Way Above Closure"
] | [
"Way Above Closures Form Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Interior is Union of Elements of Basis"
] |
proofwiki-13416 | Way Above Closures Form Basis | Let $L = \struct {S, \preceq, \tau}$ be a complete continuous topological lattice with Scott topology.
Then $\set {x^\gg: x \in S}$ is an (analytic) basis of $L$. | Define $B = \set {x^\gg: x \in S}$.
Thus by Way Above Closure is Open:
:$B \subseteq \tau$
We will prove that:
:for all $x \in S$: there exists a local basis $Q$ of $x$: $Q \subseteq B$
Let $x \in S$.
By Way Above Closures that Way Below Form Local Basis:
:$Q := \set {g^\gg: g \in S \land g \ll x}$ is a local basis at ... | Let $L = \struct {S, \preceq, \tau}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
Then $\set {x^\gg: x \in S}$ is an [[Definition:Analytic Basis|(analytic) basis]... | Define $B = \set {x^\gg: x \in S}$.
Thus by [[Way Above Closure is Open]]:
:$B \subseteq \tau$
We will prove that:
:for all $x \in S$: there exists a [[Definition:Local Basis|local basis]] $Q$ of $x$: $Q \subseteq B$
Let $x \in S$.
By [[Way Above Closures that Way Below Form Local Basis]]:
:$Q := \set {g^\gg: g \in... | Way Above Closures Form Basis | https://proofwiki.org/wiki/Way_Above_Closures_Form_Basis | https://proofwiki.org/wiki/Way_Above_Closures_Form_Basis | [
"Topological Order Theory",
"Continuous Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Continuous Ordered Set",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Basis (Topology)/Analytic Basis"
] | [
"Way Above Closure is Open",
"Definition:Local Basis",
"Way Above Closures that Way Below Form Local Basis",
"Definition:Local Basis",
"Definition:Subset",
"Characterization of Analytic Basis by Local Bases",
"Definition:Basis (Topology)/Analytic Basis"
] |
proofwiki-13417 | Numbers whose Product with Reverse are Equal | :$651 \times 156 = 372 \times 273$ | {{begin-eqn}}
{{eqn | l = 651 \times 156
| r = \paren {3 \times 7 \times 31} \times \paren {2^2 \times 3 \times 13}
| c =
}}
{{eqn | r = 2^2 \times 3^2 \times 7 \times 13 \times 31
| c =
}}
{{eqn | r = \paren {2^2 \times 3 \times 31} \times \paren {3 \times 7 \times 13}
| c =
}}
{{eqn | r = 3... | :$651 \times 156 = 372 \times 273$ | {{begin-eqn}}
{{eqn | l = 651 \times 156
| r = \paren {3 \times 7 \times 31} \times \paren {2^2 \times 3 \times 13}
| c =
}}
{{eqn | r = 2^2 \times 3^2 \times 7 \times 13 \times 31
| c =
}}
{{eqn | r = \paren {2^2 \times 3 \times 31} \times \paren {3 \times 7 \times 13}
| c =
}}
{{eqn | r = 3... | Numbers whose Product with Reverse are Equal | https://proofwiki.org/wiki/Numbers_whose_Product_with_Reverse_are_Equal | https://proofwiki.org/wiki/Numbers_whose_Product_with_Reverse_are_Equal | [
"Recreational Mathematics",
"651",
"156",
"372",
"273"
] | [] | [] |
proofwiki-13418 | Record Gaps between Twin Primes | The gaps between the following pairs of twin primes are larger than those for all smaller pairs:
{{begin-eqn}}
{{eqn | l = \tuple {3, 5}
| o = \to
| r = \tuple {5, 7}
| c = a gap of $0$
}}
{{eqn | l = \tuple {5, 7}
| o = \to
| r = \tuple {11, 13}
| c = a gap of $4$
}}
{{eqn | l = \tu... | By cases and inspection. | The gaps between the following [[Definition:Ordered Pair|pairs]] of [[Definition:Twin Primes|twin primes]] are larger than those for all smaller [[Definition:Ordered Pair|pairs]]:
{{begin-eqn}}
{{eqn | l = \tuple {3, 5}
| o = \to
| r = \tuple {5, 7}
| c = a gap of $0$
}}
{{eqn | l = \tuple {5, 7}
... | By cases and inspection. | Record Gaps between Twin Primes | https://proofwiki.org/wiki/Record_Gaps_between_Twin_Primes | https://proofwiki.org/wiki/Record_Gaps_between_Twin_Primes | [
"Twin Primes"
] | [
"Definition:Ordered Pair",
"Definition:Twin Primes",
"Definition:Ordered Pair"
] | [] |
proofwiki-13419 | Consecutive Sophie Germain Primes cannot be Pair of Twin Primes | Let $p$ and $p + 2$ be twin primes.
Then unless $p = 3$ it is not possible for both $p$ and $p + 2$ to be Sophie Germain primes. | First it is noted that $3$ and $5$ twin primes which are both Sophie Germain.
Prime numbers greater than $3$ are of the form $6 n - 1$ and $6 n + 1$.
Thus a pair of twin primes is of the form $\left({6 n - 1, 6 n + 1}\right)$.
The result follows from Sophie Germain Prime cannot be 6n+1.
{{qed}}
Category:Twin Primes
Cat... | Let $p$ and $p + 2$ be [[Definition:Twin Primes|twin primes]].
Then unless $p = 3$ it is not possible for both $p$ and $p + 2$ to be [[Definition:Sophie Germain Prime|Sophie Germain primes]]. | First it is noted that $3$ and $5$ [[Definition:Twin Primes|twin primes]] which are both [[Definition:Sophie Germain Prime|Sophie Germain]].
[[Definition:Prime Number|Prime numbers]] greater than $3$ are of the form $6 n - 1$ and $6 n + 1$.
Thus a pair of [[Definition:Twin Primes|twin primes]] is of the form $\left(... | Consecutive Sophie Germain Primes cannot be Pair of Twin Primes | https://proofwiki.org/wiki/Consecutive_Sophie_Germain_Primes_cannot_be_Pair_of_Twin_Primes | https://proofwiki.org/wiki/Consecutive_Sophie_Germain_Primes_cannot_be_Pair_of_Twin_Primes | [
"Twin Primes",
"Sophie Germain Primes"
] | [
"Definition:Twin Primes",
"Definition:Sophie Germain Prime"
] | [
"Definition:Twin Primes",
"Definition:Sophie Germain Prime",
"Definition:Prime Number",
"Definition:Twin Primes",
"Sophie Germain Prime cannot be 6n+1",
"Category:Twin Primes",
"Category:Sophie Germain Primes"
] |
proofwiki-13420 | Sophie Germain Prime cannot be 6n+1 | Let $p$ be a Sophie Germain prime.
Then $p$ cannot be of the form $6 n + 1$, where $n$ is a positive integer. | Let $p$ be a Sophie Germain prime.
Then, by definition, $2 p + 1$ is prime.
{{AimForCont}} $p = 6 n + 1$ for some $n \in \Z_{>0}$.
Then:
{{begin-eqn}}
{{eqn | l = 2 p + 1
| r = 2 \paren {6 n + 1} + 1
| c =
}}
{{eqn | r = 12 n + 3
| c =
}}
{{eqn | r = 3 \paren {4 n + 1}
| c =
}}
{{end-eqn}}
an... | Let $p$ be a [[Definition:Sophie Germain Prime|Sophie Germain prime]].
Then $p$ cannot be of the form $6 n + 1$, where $n$ is a [[Definition:Strictly Positive Integer|positive integer]]. | Let $p$ be a [[Definition:Sophie Germain Prime|Sophie Germain prime]].
Then, by definition, $2 p + 1$ is [[Definition:Prime Number|prime]].
{{AimForCont}} $p = 6 n + 1$ for some $n \in \Z_{>0}$.
Then:
{{begin-eqn}}
{{eqn | l = 2 p + 1
| r = 2 \paren {6 n + 1} + 1
| c =
}}
{{eqn | r = 12 n + 3
| c... | Sophie Germain Prime cannot be 6n+1 | https://proofwiki.org/wiki/Sophie_Germain_Prime_cannot_be_6n+1 | https://proofwiki.org/wiki/Sophie_Germain_Prime_cannot_be_6n+1 | [
"Sophie Germain Primes"
] | [
"Definition:Sophie Germain Prime",
"Definition:Strictly Positive/Integer"
] | [
"Definition:Sophie Germain Prime",
"Definition:Prime Number",
"Definition:Prime Number",
"Proof by Contradiction",
"Category:Sophie Germain Primes"
] |
proofwiki-13421 | Way Above Closure is Open | Let $L = \struct {S, \preceq, \tau}$ be a complete continuous topological lattice with Scott topology.
Let $x \in S$.
Then $x^\gg$ is open
where $x^\gg$ denotes the way above closure of $x$. | By Way Above Closure is Upper:
:$x^\gg$ is upper.
We will prove that
:$x^\gg$ is inaccessible by directed suprema.
Let $D$ be a directed subset of $S$ such that
:$\sup D \in x^\gg$
By definition of way above closure:
:$x \ll \sup D$
By Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element... | Let $L = \struct {S, \preceq, \tau}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
Let $x \in S$.
Then $x^\gg$ is [[Definition:Open Set (Topology)|open]]
where $... | By [[Way Above Closure is Upper]]:
:$x^\gg$ is [[Definition:Upper Section|upper]].
We will prove that
:$x^\gg$ is [[Definition:Inaccessible by Directed Suprema|inaccessible by directed suprema]].
Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
:$\sup D \in x^\gg$
By definition of [[Defin... | Way Above Closure is Open | https://proofwiki.org/wiki/Way_Above_Closure_is_Open | https://proofwiki.org/wiki/Way_Above_Closure_is_Open | [
"Topological Order Theory",
"Continuous Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Continuous Ordered Set",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Open Set/Topology",
"Definition:Way Above Closure"
] | [
"Way Above Closure is Upper",
"Definition:Upper Section",
"Definition:Inaccessible by Directed Suprema",
"Definition:Directed Subset",
"Definition:Way Above Closure",
"Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element of Directed Set First Operand Way Below Element",
"... |
proofwiki-13422 | Way Above Closure is Upper | Let $\struct {S, \preceq}$ be an ordered set.
Let $x \in S$.
Then $x^\gg$ is upper
where $x^\gg$ denotes the way above closure of $x$. | Let $y \in x^\gg$, $z \in S$ such that
:$y \preceq z$
By definition of way above closure:
:$x \ll y$
By Preceding and Way Below implies Way Below:
:$x \ll z$
Thus by definition of way above closure:
:$z \in x^\gg$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x \in S$.
Then $x^\gg$ is [[Definition:Upper Section|upper]]
where $x^\gg$ denotes the [[Definition:Way Above Closure|way above closure]] of $x$. | Let $y \in x^\gg$, $z \in S$ such that
:$y \preceq z$
By definition of [[Definition:Way Above Closure|way above closure]]:
:$x \ll y$
By [[Preceding and Way Below implies Way Below]]:
:$x \ll z$
Thus by definition of [[Definition:Way Above Closure|way above closure]]:
:$z \in x^\gg$
{{qed}} | Way Above Closure is Upper | https://proofwiki.org/wiki/Way_Above_Closure_is_Upper | https://proofwiki.org/wiki/Way_Above_Closure_is_Upper | [
"Way Below Relation"
] | [
"Definition:Ordered Set",
"Definition:Upper Section",
"Definition:Way Above Closure"
] | [
"Definition:Way Above Closure",
"Preceding and Way Below implies Way Below",
"Definition:Way Above Closure"
] |
proofwiki-13423 | Way Above Closures that Way Below Form Local Basis | {{rename|more descriptive of what the statement of the theorem says}}
Let $L = \struct {S, \preceq, \tau}$ be a complete continuous topological lattice with Scott topology.
Let $p \in S$.
Then $\set {q^\gg: q \in S \land q \ll p}$ is a local basis at $p$. | Define $B := \set {q^\gg: q \in S \land q \ll p}$
By Way Above Closure is Open:
:$B \subseteq \tau$
By definition of way above closure:
:$\forall X \in B: p \in X$
Thus by definition:
:$B$ is set of open neighborhoods.
{{explain|open neighborhoods of what?}}
Let $U$ be an open subset of $S$ such that
:$p \in U$
By Open... | {{rename|more descriptive of what the statement of the theorem says}}
Let $L = \struct {S, \preceq, \tau}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
Let $p \in... | Define $B := \set {q^\gg: q \in S \land q \ll p}$
By [[Way Above Closure is Open]]:
:$B \subseteq \tau$
By definition of [[Definition:Way Above Closure|way above closure]]:
:$\forall X \in B: p \in X$
Thus by definition:
:$B$ is [[Definition:Set of Sets|set]] of [[Definition:Open Neighborhood|open neighborhoods]].
{... | Way Above Closures that Way Below Form Local Basis | https://proofwiki.org/wiki/Way_Above_Closures_that_Way_Below_Form_Local_Basis | https://proofwiki.org/wiki/Way_Above_Closures_that_Way_Below_Form_Local_Basis | [
"Topological Order Theory",
"Way Below Relation",
"Continuous Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Continuous Ordered Set",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Local Basis"
] | [
"Way Above Closure is Open",
"Definition:Way Above Closure",
"Definition:Set of Sets",
"Definition:Open Neighborhood",
"Definition:Open Set/Topology",
"Definition:Subset",
"Open implies There Exists Way Below Element",
"Definition:Scott Topology",
"Definition:Upper Section",
"Definition:Way Above ... |
proofwiki-13424 | Characterization of Analytic Basis by Local Bases | Let $T = \struct {S, \tau}$ be a topological space.
Let $P$ be a set of subsets of $S$ such that
:$P \subseteq \tau$
and
:for all $p \in S$: there exists local basis $B$ at $p: B \subseteq P$
Then $P$ is basis of $T$. | By assumption:
:$P \subseteq \tau$
Let $U$ be an open subset of $S$.
Define:
:$X := \set {V \in P: V \subseteq U}$
By definition of subset:
:$X \subseteq P$
We will prove that:
:$\forall u \in S: u \in U \iff \exists Z \in X: u \in Z$
Let $u \in S$.
We will prove that:
:$u \in U \implies \exists Z \in X: u \in Z$
Assum... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $P$ be a [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $S$ such that
:$P \subseteq \tau$
and
:for all $p \in S$: there exists [[Definition:Local Basis|local basis]] $B$ at $p: B \subseteq P$
Then $P$ is [[De... | By assumption:
:$P \subseteq \tau$
Let $U$ be an [[Definition:Open Set (Topology)|open]] [[Definition:Subset|subset]] of $S$.
Define:
:$X := \set {V \in P: V \subseteq U}$
By definition of [[Definition:Subset|subset]]:
:$X \subseteq P$
We will prove that:
:$\forall u \in S: u \in U \iff \exists Z \in X: u \in Z$
L... | Characterization of Analytic Basis by Local Bases | https://proofwiki.org/wiki/Characterization_of_Analytic_Basis_by_Local_Bases | https://proofwiki.org/wiki/Characterization_of_Analytic_Basis_by_Local_Bases | [
"Topology"
] | [
"Definition:Topological Space",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Local Basis",
"Definition:Basis (Topology)/Analytic Basis"
] | [
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Subset",
"Definition:Local Basis",
"Definition:Local Basis",
"Definition:Subset",
"Definition:Local Basis",
"Definition:Subset",
"Definition:Set Union/Set of Sets",
"Definition:Basis (Topology)/Analytic Basis"
] |
proofwiki-13425 | Value of Golden Ratio using 666 | The Golden Ratio can be given by the following formula:
:$\phi = -2 \sin 666 \degrees = 1.61803 \, 39887 \, 5 \ldots$ | {{begin-eqn}}
{{eqn | l = -2 \sin 666 \degrees
| r = -2 \sin 306 \degrees
| c = Sine of Angle plus Full Angle
}}
{{eqn | r = 2 \sin 54 \degrees
| c = Sine of Conjugate Angle
}}
{{eqn | r = 2 \cos 36 \degrees
| c = Sine of Complement equals Cosine
}}
{{eqn | r = 2 \times \frac \phi 2
| c = ... | The [[Definition:Golden Ratio|Golden Ratio]] can be given by the following formula:
:$\phi = -2 \sin 666 \degrees = 1.61803 \, 39887 \, 5 \ldots$ | {{begin-eqn}}
{{eqn | l = -2 \sin 666 \degrees
| r = -2 \sin 306 \degrees
| c = [[Sine of Angle plus Full Angle]]
}}
{{eqn | r = 2 \sin 54 \degrees
| c = [[Sine of Conjugate Angle]]
}}
{{eqn | r = 2 \cos 36 \degrees
| c = [[Sine of Complement equals Cosine]]
}}
{{eqn | r = 2 \times \frac \phi 2
... | Value of Golden Ratio using 666 | https://proofwiki.org/wiki/Value_of_Golden_Ratio_using_666 | https://proofwiki.org/wiki/Value_of_Golden_Ratio_using_666 | [
"666",
"Golden Mean"
] | [
"Definition:Golden Mean"
] | [
"Sine of Angle plus Full Angle",
"Sine of Conjugate Angle",
"Sine of Complement equals Cosine",
"Cosine of 36 Degrees"
] |
proofwiki-13426 | Euler Phi Function of 666 equals Product of Digits | The number $666$ has the following interesting property:
:$\map \phi {666} = 6 \times 6 \times 6$
where $\phi$ denotes the Euler $\phi$ function. | From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$666 = 2 \times 3^2 \times 37$
Thus:
{{begin-eqn}}
{{eqn | l = \map \phi {666}
| r = 666 \paren {1 - \dfrac 1 2} \paren {1 - \dfra... | The number $666$ has the following interesting property:
:$\map \phi {666} = 6 \times 6 \times 6$
where $\phi$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]]. | From [[Euler Phi Function of Integer]]:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the [[Definition:Prime Number|primes]] which [[Definition:Divisor of Integer|divide]] $n$.
We have that:
:$666 = 2 \times 3^2 \times 37$
Thus:
{{begin-eqn}}
{{eqn | l = \ma... | Euler Phi Function of 666 equals Product of Digits | https://proofwiki.org/wiki/Euler_Phi_Function_of_666_equals_Product_of_Digits | https://proofwiki.org/wiki/Euler_Phi_Function_of_666_equals_Product_of_Digits | [
"Euler Phi Function",
"666"
] | [
"Definition:Euler Phi Function"
] | [
"Euler Phi Function of Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-13427 | Pair of Consecutive Powerful Numbers whose First is Odd | The only known pair of consecutive integers which are both powerful numbers such that the first of the pair is odd is:
:$\tuple {675, 676}$ | By investigation:
{{begin-eqn}}
{{eqn | l = 675
| r = 3^3 \times 5^2
}}
{{eqn | l = 676
| r = 2^2 \times 13^2
}}
{{end-eqn}}
That there are no smaller ones can be determined again by investigation.
{{qed}} | The only known [[Definition:Ordered Pair|pair]] of consecutive [[Definition:Integer|integers]] which are both [[Definition:Powerful Number|powerful numbers]] such that the first of the [[Definition:Ordered Pair|pair]] is [[Definition:Odd Integer|odd]] is:
:$\tuple {675, 676}$ | By investigation:
{{begin-eqn}}
{{eqn | l = 675
| r = 3^3 \times 5^2
}}
{{eqn | l = 676
| r = 2^2 \times 13^2
}}
{{end-eqn}}
That there are no smaller ones can be determined again by investigation.
{{qed}} | Pair of Consecutive Powerful Numbers whose First is Odd | https://proofwiki.org/wiki/Pair_of_Consecutive_Powerful_Numbers_whose_First_is_Odd | https://proofwiki.org/wiki/Pair_of_Consecutive_Powerful_Numbers_whose_First_is_Odd | [
"Powerful Numbers",
"675",
"676"
] | [
"Definition:Ordered Pair",
"Definition:Integer",
"Definition:Powerful Number",
"Definition:Ordered Pair",
"Definition:Odd Integer"
] | [] |
proofwiki-13428 | Interior is Union of Elements of Basis | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $B$ be a basis of $T$.
Let $V$ be a subset of $S$.
Then $V^\circ = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$
where $V^\circ$ denotes the interior of $V$. | By definition of interior:
:$\left\{ {G \in B: G \subseteq V}\right\} = \left\{ {G \in B: G \subseteq V^\circ}\right\}$
and
:$V^\circ$ is open.
Thus by Open Set is Union of Elements of Basis:
:$V^\circ = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$
{{qed}} | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $B$ be a [[Definition:Analytic Basis|basis]] of $T$.
Let $V$ be a [[Definition:Subset|subset]] of $S$.
Then $V^\circ = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$
where $V^\circ$ denotes the [[Definition:Interior (T... | By definition of [[Definition:Interior (Topology)|interior]]:
:$\left\{ {G \in B: G \subseteq V}\right\} = \left\{ {G \in B: G \subseteq V^\circ}\right\}$
and
:$V^\circ$ is [[Definition:Open Set (Topology)|open]].
Thus by [[Open Set is Union of Elements of Basis]]:
:$V^\circ = \bigcup \left\{ {G \in B: G \subseteq V}\... | Interior is Union of Elements of Basis | https://proofwiki.org/wiki/Interior_is_Union_of_Elements_of_Basis | https://proofwiki.org/wiki/Interior_is_Union_of_Elements_of_Basis | [
"Topological Bases"
] | [
"Definition:Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Subset",
"Definition:Interior (Topology)"
] | [
"Definition:Interior (Topology)",
"Definition:Open Set/Topology",
"Open Set is Union of Elements of Basis"
] |
proofwiki-13429 | Palindromic Squares with Non-Palindromic Roots | The sequence of palindromic squares with non-palindromic square roots begins:
:$676, 69 \, 696, 94 \, 249, 698 \, 896, 5 \, 221 \, 225, 6 \, 948 \, 496, 522 \, 808 \, 225, \ldots$
This sequence is not explicitly given in {{OEISLink}}.
The sequence of those corresponding non-palindromic square roots begins:
:$26, 264, 3... | By investigating all square numbers which are palindromic.
{{begin-eqn}}
{{eqn | l = 676
| r = 26^2
}}
{{eqn | l = 69 \, 696
| r = 264^2
}}
{{eqn | l = 94 \, 249
| r = 307^2
}}
{{eqn | l = 698 \, 896
| r = 836^2
}}
{{eqn | l = 5 \, 221 \, 225
| r = 2285^2
}}
{{eqn | l = 6 \, 948 \, 496
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Palindromic Number|palindromic]] [[Definition:Square Number|squares]] with non-[[Definition:Palindromic Number|palindromic]] [[Definition:Square Root|square roots]] begins:
:$676, 69 \, 696, 94 \, 249, 698 \, 896, 5 \, 221 \, 225, 6 \, 948 \, 496, 522 \, 808 ... | By investigating all [[Definition:Square Number|square numbers]] which are [[Definition:Palindromic Number|palindromic]].
{{begin-eqn}}
{{eqn | l = 676
| r = 26^2
}}
{{eqn | l = 69 \, 696
| r = 264^2
}}
{{eqn | l = 94 \, 249
| r = 307^2
}}
{{eqn | l = 698 \, 896
| r = 836^2
}}
{{eqn | l = 5 \, ... | Palindromic Squares with Non-Palindromic Roots | https://proofwiki.org/wiki/Palindromic_Squares_with_Non-Palindromic_Roots | https://proofwiki.org/wiki/Palindromic_Squares_with_Non-Palindromic_Roots | [
"Square Numbers",
"Palindromic Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Palindromic Number",
"Definition:Square Number",
"Definition:Palindromic Number",
"Definition:Square Root",
"Definition:Integer Sequence",
"Definition:Palindromic Number",
"Definition:Square Root"
] | [
"Definition:Square Number",
"Definition:Palindromic Number"
] |
proofwiki-13430 | Tetrahedral Numbers which are Sum of 2 Tetrahedral Numbers | The sequence of tetrahedral numbers which are the sum of two other tetrahedral numbers begins:
:$20, 680, 29260, 34220, 70300, \dots$
{{OEIS|A034404}} | {{begin-eqn}}
{{eqn | o =
| r = 20
| c = the $4$th tetrahedral number
}}
{{eqn | r = 10
| c = the $3$rd tetrahedral number
}}
{{eqn | o =
| ro= +
| r = 10
| c = the $3$rd tetrahedral number
}}
{{eqn | o =
| r = 680
| c = the $15$th tetrahedral number
}}
{{eqn | r = 120
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Tetrahedral Number|tetrahedral numbers]] which are the [[Definition:Integer Addition|sum]] of two other [[Definition:Tetrahedral Number|tetrahedral numbers]] begins:
:$20, 680, 29260, 34220, 70300, \dots$
{{OEIS|A034404}} | {{begin-eqn}}
{{eqn | o =
| r = 20
| c = the $4$th [[Definition:Tetrahedral Number|tetrahedral number]]
}}
{{eqn | r = 10
| c = the $3$rd [[Definition:Tetrahedral Number|tetrahedral number]]
}}
{{eqn | o =
| ro= +
| r = 10
| c = the $3$rd [[Definition:Tetrahedral Number|tetrahedral n... | Tetrahedral Numbers which are Sum of 2 Tetrahedral Numbers | https://proofwiki.org/wiki/Tetrahedral_Numbers_which_are_Sum_of_2_Tetrahedral_Numbers | https://proofwiki.org/wiki/Tetrahedral_Numbers_which_are_Sum_of_2_Tetrahedral_Numbers | [
"Tetrahedral Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Tetrahedral Number",
"Definition:Addition/Integers",
"Definition:Tetrahedral Number"
] | [
"Definition:Tetrahedral Number",
"Definition:Tetrahedral Number",
"Definition:Tetrahedral Number",
"Definition:Tetrahedral Number",
"Definition:Tetrahedral Number",
"Definition:Tetrahedral Number",
"Definition:Tetrahedral Number",
"Definition:Tetrahedral Number",
"Definition:Tetrahedral Number",
"... |
proofwiki-13431 | Open Set is Union of Elements of Basis | Let $T = \struct {S, \tau}$ be a topological space.
Let $B$ be a basis of $T$.
Let $V$ be an open subset of $S$.
Then $V = \bigcup \set {G \in B: G \subseteq V}$ | Let $x$ be arbitrary.
We will prove that:
:$x \in V \implies \exists Y \in \set {G \in B: G \subseteq V}: x \in Y$
Assume that:
:$x \in V$
By definition of basis:
:$\exists F \subseteq B: V = \bigcup F$
By definition of union:
:$\exists Y \in F: x \in Y$
By Set is Subset of Union/General Result:
:$Y \subseteq V$
Thus b... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $B$ be a [[Definition:Analytic Basis|basis]] of $T$.
Let $V$ be an [[Definition:Open Set (Topology)|open]] [[Definition:Subset|subset]] of $S$.
Then $V = \bigcup \set {G \in B: G \subseteq V}$ | Let $x$ be arbitrary.
We will prove that:
:$x \in V \implies \exists Y \in \set {G \in B: G \subseteq V}: x \in Y$
Assume that:
:$x \in V$
By definition of [[Definition:Analytic Basis|basis]]:
:$\exists F \subseteq B: V = \bigcup F$
By definition of [[Definition:Union of Set of Sets|union]]:
:$\exists Y \in F: x \i... | Open Set is Union of Elements of Basis | https://proofwiki.org/wiki/Open_Set_is_Union_of_Elements_of_Basis | https://proofwiki.org/wiki/Open_Set_is_Union_of_Elements_of_Basis | [
"Topological Bases"
] | [
"Definition:Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Open Set/Topology",
"Definition:Subset"
] | [
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Set Union/Set of Sets",
"Set is Subset of Union/General Result",
"Definition:Subset",
"Definition:Subset",
"Definition:Set Union/Set of Sets"
] |
proofwiki-13432 | Mapping at Element is Supremum of Compact Elements implies Mapping is Increasing | Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ be a lattice.
Let $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be a complete lattice.
Let $f: S \to T$ be a mapping such that
:$\forall x \in S: \map f x = \sup \leftset {\map f w : w \in S \land w \preceq_1 x \land w}$ is compact$\rightset{}$
Then $f$ is increasing. | Let $x, y \in S$ such that
:$x \preceq_1 y$
By Compact Closure is Increasing:
:$x^{\mathrm {compact} } \subseteq y^{\mathrm {compact} }$
By Image of Subset under Mapping is Subset of Image:
:$f \sqbrk {x^{\mathrm {compact} } } \subseteq f \sqbrk {y^{\mathrm {compact} } }$
By assumption:
:$\map f x = \sup \leftset {\map... | Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ be a [[Definition:Lattice (Order Theory)|lattice]].
Let $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
:$\forall x \in S: \map f x = \sup \leftset {\map f w ... | Let $x, y \in S$ such that
:$x \preceq_1 y$
By [[Compact Closure is Increasing]]:
:$x^{\mathrm {compact} } \subseteq y^{\mathrm {compact} }$
By [[Image of Subset under Mapping is Subset of Image]]:
:$f \sqbrk {x^{\mathrm {compact} } } \subseteq f \sqbrk {y^{\mathrm {compact} } }$
By assumption:
:$\map f x = \sup \le... | Mapping at Element is Supremum of Compact Elements implies Mapping is Increasing | https://proofwiki.org/wiki/Mapping_at_Element_is_Supremum_of_Compact_Elements_implies_Mapping_is_Increasing | https://proofwiki.org/wiki/Mapping_at_Element_is_Supremum_of_Compact_Elements_implies_Mapping_is_Increasing | [
"Way Below Relation"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Complete Lattice",
"Definition:Mapping",
"Definition:Compact Element",
"Definition:Increasing/Mapping"
] | [
"Compact Closure is Increasing",
"Image of Subset under Mapping is Subset of Image",
"Definition:Compact Element",
"Definition:Compact Element",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Compact Closure",
"Supremum of Subset",
"Definition:Complete Lattice"
] |
proofwiki-13433 | Equivalence of Definitions of Compatible Atlases | Let $M$ be a topological space.
Let $\mathscr F, \mathscr G$ be $d$-dimensional atlases of class $C^k$ on $M$.
{{TFAE|def = Compatible Atlases}} | === Definition $1$ implies Definition $2$ ===
Follows immediately from the definition of $C^k$-atlas.
{{qed|lemma}} | Let $M$ be a [[Definition:Topological Space|topological space]].
Let $\mathscr F, \mathscr G$ be $d$-[[Definition:Dimension of Atlas|dimensional]] [[Definition:Atlas|atlases]] of [[Definition:Class of Atlas|class]] $C^k$ on $M$.
{{TFAE|def = Compatible Atlases}} | === Definition $1$ implies Definition $2$ ===
Follows immediately from the definition of $C^k$-[[Definition:Atlas|atlas]].
{{qed|lemma}} | Equivalence of Definitions of Compatible Atlases | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Compatible_Atlases | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Compatible_Atlases | [
"Compatible Atlases"
] | [
"Definition:Topological Space",
"Definition:Atlas",
"Definition:Atlas",
"Definition:Atlas"
] | [
"Definition:Atlas",
"Definition:Atlas",
"Definition:Atlas",
"Definition:Atlas"
] |
proofwiki-13434 | Atlas is Contained in Unique Maximal Atlas | Let $M$ be a topological space.
Let $A$ be a $d$-dimensional atlas of class $C^k$.
Then $A$ is contained in a unique maximal atlas of class $C^k$. | === Existence === | Let $M$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a $d$-[[Definition:Dimension of Atlas|dimensional]] [[Definition:Atlas|atlas]] of [[Definition:Class of Atlas|class]] $C^k$.
Then $A$ is contained in a [[Definition:Unique|unique]] [[Definition:Maximal Atlas|maximal atlas]] of class $C^k$. | === Existence === | Atlas is Contained in Unique Maximal Atlas | https://proofwiki.org/wiki/Atlas_is_Contained_in_Unique_Maximal_Atlas | https://proofwiki.org/wiki/Atlas_is_Contained_in_Unique_Maximal_Atlas | [
"Manifolds",
"Atlases",
"Maximal Atlases"
] | [
"Definition:Topological Space",
"Definition:Atlas",
"Definition:Atlas",
"Definition:Atlas",
"Definition:Unique",
"Definition:Atlas/Maximal Atlas"
] | [] |
proofwiki-13435 | Locally Euclidean iff has C0-Atlas | Let $M$ be a topological space.
{{TFAE}}
:$(1): \quad$ $M$ is locally euclidean.
:$(2): \quad$ There exists a $C^0$-atlas on $M$. | {{ProofWanted|use Transition Mapping between Charts is Homeomorphism}}
Category:Atlases
Category:Topological Spaces
Category:Locally Euclidean Spaces
m4xz9zlgb5n2ga8wq14uqorb3lyv3n4 | Let $M$ be a [[Definition:Topological Space|topological space]].
{{TFAE}}
:$(1): \quad$ $M$ is [[Definition:Locally Euclidean Space|locally euclidean]].
:$(2): \quad$ There exists a [[Definition:Atlas|$C^0$-atlas]] on $M$. | {{ProofWanted|use [[Transition Mapping between Charts is Homeomorphism]]}}
[[Category:Atlases]]
[[Category:Topological Spaces]]
[[Category:Locally Euclidean Spaces]]
m4xz9zlgb5n2ga8wq14uqorb3lyv3n4 | Locally Euclidean iff has C0-Atlas | https://proofwiki.org/wiki/Locally_Euclidean_iff_has_C0-Atlas | https://proofwiki.org/wiki/Locally_Euclidean_iff_has_C0-Atlas | [
"Atlases",
"Topological Spaces",
"Locally Euclidean Spaces"
] | [
"Definition:Topological Space",
"Definition:Locally Euclidean Space",
"Definition:Atlas"
] | [
"Transition Mapping between Charts is Homeomorphism",
"Category:Atlases",
"Category:Topological Spaces",
"Category:Locally Euclidean Spaces"
] |
proofwiki-13436 | Differentiable Structure Contains Unique Maximal Atlas | Let $M$ be a topological space.
Let $k$ and $d$ be natural numbers.
Let $S$ be a $d$-dimensional differentiable structure of class $C^k$ on $M$.
Then $S$ contains a unique maximal $C^k$-atlas. | {{ProofWanted}}
Category:Topological Manifolds
Category:Maximal Atlases
alrzbmnkm9a5a5mei0vpkxszoneg71r | Let $M$ be a [[Definition:Topological Space|topological space]].
Let $k$ and $d$ be [[Definition:Natural Number|natural numbers]].
Let $S$ be a $d$-dimensional [[Definition:Differentiable Structure|differentiable structure]] of class $C^k$ on $M$.
Then $S$ contains a [[Definition:Unique|unique]] [[Definition:Maxima... | {{ProofWanted}}
[[Category:Topological Manifolds]]
[[Category:Maximal Atlases]]
alrzbmnkm9a5a5mei0vpkxszoneg71r | Differentiable Structure Contains Unique Maximal Atlas | https://proofwiki.org/wiki/Differentiable_Structure_Contains_Unique_Maximal_Atlas | https://proofwiki.org/wiki/Differentiable_Structure_Contains_Unique_Maximal_Atlas | [
"Topological Manifolds",
"Maximal Atlases"
] | [
"Definition:Topological Space",
"Definition:Natural Numbers",
"Definition:Differentiable Structure",
"Definition:Unique",
"Definition:Atlas/Maximal Atlas"
] | [
"Category:Topological Manifolds",
"Category:Maximal Atlases"
] |
proofwiki-13437 | Consecutive Integers whose Product is Primorial | The following primorials can be expressed as the product of consecutive integers:
:$2, 6, 30, 210, 510 \, 510$
{{OEIS|A161620}}
No others are known.
The corresponding indices of those primorials are:
:$2, 3, 5, 7, 17$
{{OEIS|A215658}}
The corresponding values of $n$ such that $p\# = \paren {n - 1} n$ are:
:$2, 3, 6, 15... | {{begin-eqn}}
{{eqn | l = 2 \#
| r = 1 \times 2
}}
{{eqn | r = 2
}}
{{eqn | l = 3 \#
| r = 2 \times 3
}}
{{eqn | r = 6
}}
{{eqn | l = 5 \#
| r = 2 \times 3 \times 5
}}
{{eqn | r = 5 \times \paren {2 \times 3}
}}
{{eqn | r = 5 \times 6
}}
{{eqn | r = 30
}}
{{eqn | l = 7 \#
| r = 2 \times 3 \times... | The following [[Definition:Primorial|primorials]] can be expressed as the [[Definition:Integer Multiplication|product]] of consecutive [[Definition:Integer|integers]]:
:$2, 6, 30, 210, 510 \, 510$
{{OEIS|A161620}}
No others are known.
The corresponding indices of those [[Definition:Primorial|primorials]] are:
:$2, 3... | {{begin-eqn}}
{{eqn | l = 2 \#
| r = 1 \times 2
}}
{{eqn | r = 2
}}
{{eqn | l = 3 \#
| r = 2 \times 3
}}
{{eqn | r = 6
}}
{{eqn | l = 5 \#
| r = 2 \times 3 \times 5
}}
{{eqn | r = 5 \times \paren {2 \times 3}
}}
{{eqn | r = 5 \times 6
}}
{{eqn | r = 30
}}
{{eqn | l = 7 \#
| r = 2 \times 3 \times... | Consecutive Integers whose Product is Primorial | https://proofwiki.org/wiki/Consecutive_Integers_whose_Product_is_Primorial | https://proofwiki.org/wiki/Consecutive_Integers_whose_Product_is_Primorial | [
"Primorials",
"Recreational Mathematics"
] | [
"Definition:Primorial",
"Definition:Multiplication/Integers",
"Definition:Integer",
"Definition:Primorial"
] | [] |
proofwiki-13438 | Implicit Function Theorem | {{:Implicit Function Theorem/Real Functions}} | ==== Reduction to $\tuple {a, b} = \tuple {0, 0}$ ====
We may assume $\tuple {a, b} = \tuple {0, 0}$.
{{explain|why}}
Define:
:$F: \Omega \to \R^k: \map F {x, y} = y - \map {D_2 \map f {a, b}^{-1} } {\map f {x, y} }$
By Linear Function is Continuous, $D_2 \map f {a, b}^{-1}$ is continuous on $\R^k$.
Thus $F$ is continu... | {{:Implicit Function Theorem/Real Functions}} | ==== Reduction to $\tuple {a, b} = \tuple {0, 0}$ ====
We may assume $\tuple {a, b} = \tuple {0, 0}$.
{{explain|why}}
Define:
:$F: \Omega \to \R^k: \map F {x, y} = y - \map {D_2 \map f {a, b}^{-1} } {\map f {x, y} }$
By [[Linear Function is Continuous]], $D_2 \map f {a, b}^{-1}$ is [[Definition:Continuous Real Func... | Implicit Function Theorem/Real Functions/Proof 1 | https://proofwiki.org/wiki/Implicit_Function_Theorem | https://proofwiki.org/wiki/Implicit_Function_Theorem/Real_Functions/Proof_1 | [
"Implicit Function Theorem",
"Implicit Functions"
] | [] | [
"Linear Function is Continuous",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function",
"Definition:Open Ball",
"Mean Value Inequality",
"Definition:Identity Mapping",
"Definition:Continuous Mapping",
"Definition:Uniform Contraction Mapping",
"Definition:Open Ball",
"Definit... |
proofwiki-13439 | Implicit Function Theorem/Real Functions | Let $n$ and $k$ be natural numbers.
Let $\Omega \subset \R^n \times \R^k$ be open.
Let $f: \Omega \to \R^k$ be continuous.
Let the partial derivatives of $f$ with respect to $\R^k$ be continuous.
Let $\tuple {a, b} \in \Omega$, with $a\in \R^n$ and $b\in \R^k$.
Let $\map f {a, b} = 0$.
For $\tuple {x_0, y_0} \in \Omega... | ==== Reduction to $\tuple {a, b} = \tuple {0, 0}$ ====
We may assume $\tuple {a, b} = \tuple {0, 0}$.
{{explain|why}}
Define:
:$F: \Omega \to \R^k: \map F {x, y} = y - \map {D_2 \map f {a, b}^{-1} } {\map f {x, y} }$
By Linear Function is Continuous, $D_2 \map f {a, b}^{-1}$ is continuous on $\R^k$.
Thus $F$ is continu... | Let $n$ and $k$ be [[Definition:Natural Number|natural numbers]].
Let $\Omega \subset \R^n \times \R^k$ be [[Definition:Open Set|open]].
Let $f: \Omega \to \R^k$ be [[Definition:Continuous Function|continuous]].
Let the [[Definition:Partial Derivative|partial derivatives]] of $f$ with respect to $\R^k$ be [[Definiti... | ==== Reduction to $\tuple {a, b} = \tuple {0, 0}$ ====
We may assume $\tuple {a, b} = \tuple {0, 0}$.
{{explain|why}}
Define:
:$F: \Omega \to \R^k: \map F {x, y} = y - \map {D_2 \map f {a, b}^{-1} } {\map f {x, y} }$
By [[Linear Function is Continuous]], $D_2 \map f {a, b}^{-1}$ is [[Definition:Continuous Real Func... | Implicit Function Theorem/Real Functions/Proof 1 | https://proofwiki.org/wiki/Implicit_Function_Theorem/Real_Functions | https://proofwiki.org/wiki/Implicit_Function_Theorem/Real_Functions/Proof_1 | [
"Implicit Function Theorem"
] | [
"Definition:Natural Numbers",
"Definition:Open Set",
"Definition:Continuous Function",
"Definition:Partial Derivative",
"Definition:Continuous Function",
"Definition:Total Derivative",
"Definition:Linear Transformation",
"Definition:Inverse Linear Transformation",
"Definition:Neighborhood",
"Defin... | [
"Linear Function is Continuous",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function",
"Definition:Open Ball",
"Mean Value Inequality",
"Definition:Identity Mapping",
"Definition:Continuous Mapping",
"Definition:Uniform Contraction Mapping",
"Definition:Open Ball",
"Definit... |
proofwiki-13440 | Uniform Contraction Mapping Theorem | Let $M$ and $N$ be metric spaces.
Let $M$ be complete.
Let $f : M \times N \to M$ be a continuous uniform contraction.
Then for all $t \in N$ there exists a unique $\map g t \in M$ such that $\map f {\map g t, t} = \map g t$, and the mapping $g: N \to M$ is continuous. | For every $t\in N$, the mapping:
:$f_t: M \to M : x \mapsto \map f {x, t}$ is a contraction.
By the Banach Fixed-Point Theorem, there exists a unique $\map g t \in M$ such that $\map {f_t} {\map g t} = \map g t$.
We show that $g$ is continuous.
Let $K < 1$ be a uniform Lipschitz constant for $f$.
Let $s, t \in N$.
Then... | Let $M$ and $N$ be [[Definition:Metric Space|metric spaces]].
Let $M$ be [[Definition:Complete Metric Space|complete]].
Let $f : M \times N \to M$ be a [[Definition:Continuous Mapping|continuous]] [[Definition:Uniform Contraction Mapping|uniform contraction]].
Then for all $t \in N$ there exists a [[Definition:Uniq... | For every $t\in N$, the [[Definition:Mapping|mapping]]:
:$f_t: M \to M : x \mapsto \map f {x, t}$ is a [[Definition:Contraction Mapping (Metric Space)|contraction]].
By the [[Banach Fixed-Point Theorem]], there exists a [[Definition:unique|unique]] $\map g t \in M$ such that $\map {f_t} {\map g t} = \map g t$.
We sho... | Uniform Contraction Mapping Theorem | https://proofwiki.org/wiki/Uniform_Contraction_Mapping_Theorem | https://proofwiki.org/wiki/Uniform_Contraction_Mapping_Theorem | [
"Fixed Point Theorems",
"Implicit Functions",
"Metric Spaces",
"Named Theorems"
] | [
"Definition:Metric Space",
"Definition:Complete Metric Space",
"Definition:Continuous Mapping",
"Definition:Uniform Contraction Mapping",
"Definition:Unique",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Mapping",
"Definition:Contraction Mapping (Metric Space)",
"Banach Fixed-Point Theorem",
"Definition:unique",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Uniform Lipschitz Constant",
"Definition:Uniform Contraction Mapping",
"Definition:Euclidean Metric"
] |
proofwiki-13441 | Mapping at Element is Supremum of Compact Elements implies Mapping at Element is Supremum that Way Below | Let $\struct{S, \vee_1, \wedge_1, \preceq_1}$ and $\struct{T, \vee_2, \wedge_2, \preceq_2}$ be complete lattices.
Let $f: S \to T$ be a mapping such that
:$\forall x \in S: \map f x = \sup \leftset {\map f w: w \in S \land w \preceq_1 x \land w}$ is compact$\rightset{}$
Then
:$\forall x \in S: \map f x = \sup \set{ \ma... | Let $x \in S$.
Define $X = \leftset {\map f w: w \in S \land w \preceq_1 x \land w}$ is compact$\rightset{}$
Define $Y = \set { \map f w: w \in S \land w \ll x}$
We will prove that
:$X \subseteq Y$
Let $b \in X$.
By definition of $X$:
:$\exists w \in S: b = \map f w \land w \preceq_1 x \land w$ is compact.
By definitio... | Let $\struct{S, \vee_1, \wedge_1, \preceq_1}$ and $\struct{T, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
:$\forall x \in S: \map f x = \sup \leftset {\map f w: w \in S \land w \preceq_1 x \land w}$ is [[Definition:C... | Let $x \in S$.
Define $X = \leftset {\map f w: w \in S \land w \preceq_1 x \land w}$ is [[Definition:Compact Element|compact]]$\rightset{}$
Define $Y = \set { \map f w: w \in S \land w \ll x}$
We will prove that
:$X \subseteq Y$
Let $b \in X$.
By definition of $X$:
:$\exists w \in S: b = \map f w \land w \preceq_1... | Mapping at Element is Supremum of Compact Elements implies Mapping at Element is Supremum that Way Below | https://proofwiki.org/wiki/Mapping_at_Element_is_Supremum_of_Compact_Elements_implies_Mapping_at_Element_is_Supremum_that_Way_Below | https://proofwiki.org/wiki/Mapping_at_Element_is_Supremum_of_Compact_Elements_implies_Mapping_at_Element_is_Supremum_that_Way_Below | [
"Way Below Relation"
] | [
"Definition:Complete Lattice",
"Definition:Mapping",
"Definition:Compact Element"
] | [
"Definition:Compact Element",
"Definition:Compact Element",
"Definition:Compact Element",
"Preceding and Way Below implies Way Below",
"Definition:Upper Bound of Set",
"Way Below implies Preceding",
"Mapping at Element is Supremum implies Mapping is Increasing",
"Definition:Increasing/Mapping",
"Def... |
proofwiki-13442 | Inverse Function Theorem for Real Functions | Let $n \ge 1$ and $k \ge 1$ be natural numbers.
Let $\Omega \subset \R^n$ be open.
Let $f: \Omega \to \R^n$ be a vector-valued function of class $C^k$.
Let $a \in \Omega$.
Let the differential $D \map f a$ of $f$ at $a$ be invertible.
Then there exist open sets $U \subset \Omega$ and $V \subset \R^n$ such that:
:$a \in... | {{ProofWanted}}
Category:Implicit Functions
sg5dapv9eegzltg41sxmzl005tuvoft | Let $n \ge 1$ and $k \ge 1$ be [[Definition:Natural Number|natural numbers]].
Let $\Omega \subset \R^n$ be [[Definition:Open Set of Real Euclidean Space|open]].
Let $f: \Omega \to \R^n$ be a [[Definition:Vector-Valued Function|vector-valued function]] of [[Definition:Differentiability Class|class]] $C^k$.
Let $a \in... | {{ProofWanted}}
[[Category:Implicit Functions]]
sg5dapv9eegzltg41sxmzl005tuvoft | Inverse Function Theorem for Real Functions | https://proofwiki.org/wiki/Inverse_Function_Theorem_for_Real_Functions | https://proofwiki.org/wiki/Inverse_Function_Theorem_for_Real_Functions | [
"Implicit Functions"
] | [
"Definition:Natural Numbers",
"Definition:Open Set/Real Analysis/Real Euclidean Space",
"Definition:Vector-Valued Function",
"Definition:Differentiability Class",
"Definition:Differential of Mapping/Vector-Valued Function",
"Definition:Inverse Linear Operator",
"Definition:Open Set/Real Analysis/Real Eu... | [
"Category:Implicit Functions"
] |
proofwiki-13443 | Sum of 714 and 715 | The sum of $714$ and $715$ is a $4$-digit integer which has $6$ anagrams which are prime. | We have that:
:$714 + 715 = 1429$
Hence we investigate its anagrams.
We bother only to check those which do not end in either $2$ or $4$, as those are even.
{{begin-eqn}}
{{eqn | l = 1429
| o =
| c = is prime
}}
{{eqn | l = 1249
| o =
| c = is prime
}}
{{eqn | l = 4129
| o =
| c =... | The [[Definition:Integer Addition|sum]] of $714$ and $715$ is a [[Definition:Digit|$4$-digit]] [[Definition:Integer|integer]] which has $6$ [[Definition:Anagram|anagrams]] which are [[Definition:Prime Number|prime]]. | We have that:
:$714 + 715 = 1429$
Hence we investigate its [[Definition:Anagram|anagrams]].
We bother only to check those which do not end in either $2$ or $4$, as those are [[Definition:Even Integer|even]].
{{begin-eqn}}
{{eqn | l = 1429
| o =
| c = is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 1... | Sum of 714 and 715 | https://proofwiki.org/wiki/Sum_of_714_and_715 | https://proofwiki.org/wiki/Sum_of_714_and_715 | [
"Prime Numbers",
"714",
"715"
] | [
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Integer",
"Definition:Anagram",
"Definition:Prime Number"
] | [
"Definition:Anagram",
"Definition:Even Integer",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"D... |
proofwiki-13444 | 720 is Product of Consecutive Numbers in Two Ways | :$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$ | Follows from Factorial as Product of Two Factorials:
:$10! = 6! \times 7!$
and so:
:$\dfrac {10!} {7!} = 10 \times 9 \times 8 = 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Hence the result.
{{qed}} | :$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$ | Follows from [[Factorial as Product of Two Factorials]]:
:$10! = 6! \times 7!$
and so:
:$\dfrac {10!} {7!} = 10 \times 9 \times 8 = 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Hence the result.
{{qed}} | 720 is Product of Consecutive Numbers in Two Ways | https://proofwiki.org/wiki/720_is_Product_of_Consecutive_Numbers_in_Two_Ways | https://proofwiki.org/wiki/720_is_Product_of_Consecutive_Numbers_in_Two_Ways | [
"Factorials",
"720"
] | [] | [
"Factorial as Product of Two Factorials"
] |
proofwiki-13445 | Factorial which is Sum of Two Squares | The only factorial which can be expressed as the sum of two squares is:
{{begin-eqn}}
{{eqn | l = 6!
| r = 12^2 + 24^2
| c =
}}
{{end-eqn}} | We show that for $n \ge 7$, $n!$ cannot be expressed as the sum of two squares.
By refining the result in Interval containing Prime Number of forms 4n - 1, 4n + 1, 6n - 1, 6n + 1, one can show that:
:There exists a prime of the form $4 k + 3$ strictly between $m$ and $2 m$ whenever $m \ge 4$.
Let $n \ge 7$. Then $\ceil... | The only [[Definition:Factorial|factorial]] which can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]] is:
{{begin-eqn}}
{{eqn | l = 6!
| r = 12^2 + 24^2
| c =
}}
{{end-eqn}} | We show that for $n \ge 7$, $n!$ cannot be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]].
By refining the result in [[Interval containing Prime Number of forms 4n - 1, 4n + 1, 6n - 1, 6n + 1]], one can show that:
:There exists a [[Definition:Prime Number|prime]] of t... | Factorial which is Sum of Two Squares | https://proofwiki.org/wiki/Factorial_which_is_Sum_of_Two_Squares | https://proofwiki.org/wiki/Factorial_which_is_Sum_of_Two_Squares | [
"Factorials",
"Square Numbers",
"720"
] | [
"Definition:Factorial",
"Definition:Addition/Integers",
"Definition:Square Number"
] | [
"Definition:Addition/Integers",
"Definition:Square Number",
"Interval containing Prime Number of forms 4n - 1, 4n + 1, 6n - 1, 6n + 1",
"Definition:Prime Number",
"Definition:Prime Number",
"Integer as Sum of Two Squares",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Prime... |
proofwiki-13446 | Multiplicity of 720 in 720 Factorial | The multiplicity of $720$ in $720!$ is $178$.
That is:
:$720^{178} \divides 720!$
but:
:$720^{179} \nmid 720!$
where:
:$720!$ denotes $720$ factorial
:$\divides$ denotes divisibility
:$\nmid$ denotes non-divisibility. | We have that:
:$720 = 2^4 \times 3^2 \times 5$
It remains to inspect the divisibility of $2$, $3$ and $5$ in $720!$
Thus: | The [[Definition:Multiplicity of Prime Factor|multiplicity]] of $720$ in $720!$ is $178$.
That is:
:$720^{178} \divides 720!$
but:
:$720^{179} \nmid 720!$
where:
:$720!$ denotes [[Definition:Factorial|$720$ factorial]]
:$\divides$ denotes [[Definition:Divisor of Integer|divisibility]]
:$\nmid$ denotes non-[[Definition... | We have that:
:$720 = 2^4 \times 3^2 \times 5$
It remains to inspect the [[Definition:Divisor of Integer|divisibility]] of $2$, $3$ and $5$ in $720!$
Thus: | Multiplicity of 720 in 720 Factorial | https://proofwiki.org/wiki/Multiplicity_of_720_in_720_Factorial | https://proofwiki.org/wiki/Multiplicity_of_720_in_720_Factorial | [
"Factorials",
"720",
"De Polignac's Formula"
] | [
"Definition:Prime Decomposition/Multiplicity",
"Definition:Factorial",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-13447 | Continuous iff Mapping at Element is Supremum | Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be complete continuous topological lattices with Scott topologies.
Let $f: S \to T$ be a mapping.
Then $f$ is continuous {{iff}}
:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$ | === Sufficient Condition ===
Assume that
:$f$ is continuous.
By Continuous iff Directed Suprema Preserving:
:$f$ is preserves directed suprema.
Thus by Directed Suprema Preserving Mapping at Element is Supremum:
:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
{{qe... | Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Scott Topology|Scott topologies]].
Let $f: S \to T$ be a [[Definitio... | === Sufficient Condition ===
Assume that
:$f$ is [[Definition:Continuous (Topology)|continuous]].
By [[Continuous iff Directed Suprema Preserving]]:
:$f$ is [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
Thus by [[Directed Suprema Preserving Mapping at Element is Supremum]]:
:$\forall... | Continuous iff Mapping at Element is Supremum | https://proofwiki.org/wiki/Continuous_iff_Mapping_at_Element_is_Supremum | https://proofwiki.org/wiki/Continuous_iff_Mapping_at_Element_is_Supremum | [
"Topological Order Theory",
"Way Below Relation",
"Continuous Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Continuous Ordered Set",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Continuous Mapping (Topology)",
"Continuous iff Directed Suprema Preserving",
"Definition:Mapping Preserves Supremum/Directed",
"Directed Suprema Preserving Mapping at Element is Supremum",
"Definition:Continuous Mapping (Topology)"
] |
proofwiki-13448 | Equivalence of Definitions of Synthetic Basis | Let $S$ be a set.
{{TFAE|def = Synthetic Basis}} | We note that $\paren {\text B 1}$ and $\paren {\text B 1'}$ are the same for both definition $1$ and definition $2$.
It remains to demonstrate the equivalence of $\paren {\text B 2}$ and $\paren {\text B 2'}$. | Let $S$ be a [[Definition:Set|set]].
{{TFAE|def = Synthetic Basis}} | We note that $\paren {\text B 1}$ and $\paren {\text B 1'}$ are the same for both [[Definition:Synthetic Basis/Definition 1|definition $1$]] and [[Definition:Synthetic Basis/Definition 2|definition $2$]].
It remains to demonstrate the [[Definition:Logical Equivalence|equivalence]] of $\paren {\text B 2}$ and $\paren {... | Equivalence of Definitions of Synthetic Basis | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Synthetic_Basis | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Synthetic_Basis | [
"Synthetic Bases"
] | [
"Definition:Set"
] | [
"Definition:Synthetic Basis/Definition 1",
"Definition:Synthetic Basis/Definition 2",
"Definition:Logical Equivalence",
"Definition:Synthetic Basis/Definition 1",
"Definition:Synthetic Basis/Definition 1",
"Definition:Synthetic Basis/Definition 2",
"Definition:Synthetic Basis/Definition 1"
] |
proofwiki-13449 | Equivalence of Definitions of Generated Submodule over Ring with Unity | Let $R$ be a ring with unity..
Let $\struct {M, +, \circ}_R$ be a unitary $R$-module.
Let $S\subset M$ be a subset.
{{TFAE|def = Generated Submodule}} | Let:
:$\ds H_1 := \bigcap \set { M' \subseteq M : S \subseteq M', \textrm {$M'$ is a submodule of $M$} }$
be the intersection defined in first definition.
From Intersection of Submodules is Submodule:General Result, it follows that $H_1$ is a submodule of $M$.
Let:
:$\ds H_2 := \set { \sum_{i \mathop = 1}^n \lambda_i \... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]]..
Let $\struct {M, +, \circ}_R$ be a [[Definition:Unitary Module over Ring|unitary $R$-module]].
Let $S\subset M$ be a [[Definition:Subset|subset]].
{{TFAE|def = Generated Submodule}} | Let:
:$\ds H_1 := \bigcap \set { M' \subseteq M : S \subseteq M', \textrm {$M'$ is a submodule of $M$} }$
be the [[Definition:Set Intersection|intersection]] defined in [[Definition:Generated Submodule/Definition 1|first definition]].
From [[Intersection of Submodules is Submodule/General Result|Intersection of Subm... | Equivalence of Definitions of Generated Submodule over Ring with Unity | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generated_Submodule_over_Ring_with_Unity | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generated_Submodule_over_Ring_with_Unity | [
"Generators of Modules"
] | [
"Definition:Ring with Unity",
"Definition:Unitary Module over Ring",
"Definition:Subset"
] | [
"Definition:Set Intersection",
"Definition:Generated Submodule/Definition 1",
"Intersection of Submodules is Submodule/General Result",
"Definition:Submodule",
"Definition:Set",
"Definition:Generated Submodule/Unitary",
"Definition:Submodule",
"Definition:Set Intersection",
"Definition:Set",
"Defi... |
proofwiki-13450 | Equivalence of Definitions of Basis of Vector Space | Let $K$ be a division ring.
Let $\struct {G, +_G, \circ}_K$ be an vector space over $K$.
{{TFAE| def = Basis of Vector Space}} | === Definition 1 implies Definition 2 ===
Let $\BB$ be a linearly independent subset of $G$ which is a generator for $G$.
Suppose $\BB \subseteq \BB'$ is a linearly independent subset of $G$.
We aim to show that $\BB = \BB'$, proving maximality.
Suppose that $\BB \ne \BB'$.
Let $x \in \BB' \setminus \BB$.
Since $G$ ... | Let $K$ be a [[Definition:Division Ring|division ring]].
Let $\struct {G, +_G, \circ}_K$ be an [[Definition:Vector Space|vector space]] over $K$.
{{TFAE| def = Basis of Vector Space}} | === Definition 1 implies Definition 2 ===
Let $\BB$ be a [[Definition:Linearly Independent Set|linearly independent subset]] of $G$ which is a [[Definition:Generator of Module|generator]] for $G$.
Suppose $\BB \subseteq \BB'$ is a [[Definition:Linearly Independent Set|linearly independent subset]] of $G$.
We aim to ... | Equivalence of Definitions of Basis of Vector Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Basis_of_Vector_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Basis_of_Vector_Space | [
"Bases of Vector Spaces"
] | [
"Definition:Division Ring",
"Definition:Vector Space"
] | [
"Definition:Linearly Independent/Set",
"Definition:Generator of Module",
"Definition:Linearly Independent/Set",
"Definition:Maximal/Element",
"Definition:Generator of Module",
"Definition:Field (Abstract Algebra)",
"Definition:Linearly Dependent/Set",
"Definition:Maximal/Element",
"Definition:Linear... |
proofwiki-13451 | Equivalence of Definitions of Differentiable Real Function at Point | Let $\openint a b \subset \R$ be an open interval.
Let $\xi$ be a point in $\openint a b$.
{{TFAE|def = Differentiable Real Function at Point|view = differentiable real function at a point}} | {{ProofWanted|compare Equivalence of Definitions of Derivative}}
Category:Differentiable Real Functions
34ueqe29g1aqkgizzbncuvs99ypu7tq | Let $\openint a b \subset \R$ be an [[Definition:Open Real Interval|open interval]].
Let $\xi$ be a point in $\openint a b$.
{{TFAE|def = Differentiable Real Function at Point|view = differentiable real function at a point}} | {{ProofWanted|compare [[Equivalence of Definitions of Derivative]]}}
[[Category:Differentiable Real Functions]]
34ueqe29g1aqkgizzbncuvs99ypu7tq | Equivalence of Definitions of Differentiable Real Function at Point | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Differentiable_Real_Function_at_Point | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Differentiable_Real_Function_at_Point | [
"Differentiable Real Functions"
] | [
"Definition:Real Interval/Open"
] | [
"Equivalence of Definitions of Derivative",
"Category:Differentiable Real Functions"
] |
proofwiki-13452 | Continuous iff Mapping at Element is Supremum of Compact Elements | Let $L = \struct {S, \preceq_1, \tau_1}$ and $R = \struct {T, \preceq_2, \tau_2}$ be complete algebraic topological lattices with Scott topologies.
Let $f: S \to T$ be a mapping.
Then $f$ is continuous {{iff}}:
:$\forall x \in S: \map f x = \sup \leftset {\map f w: w \in S \land w \preceq_1 x \land w}$ is compact$\righ... | By Algebraic iff Continuous and For Every Way Below Exists Compact Between:
:$L$ and $R$ are continuous. | Let $L = \struct {S, \preceq_1, \tau_1}$ and $R = \struct {T, \preceq_2, \tau_2}$ be [[Definition:Complete Lattice|complete]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Scott Topology|Scott topologies]].
Let $f: S \to T$ be a [[Definition:Ma... | By [[Algebraic iff Continuous and For Every Way Below Exists Compact Between]]:
:$L$ and $R$ are [[Definition:Continuous Ordered Set|continuous]]. | Continuous iff Mapping at Element is Supremum of Compact Elements | https://proofwiki.org/wiki/Continuous_iff_Mapping_at_Element_is_Supremum_of_Compact_Elements | https://proofwiki.org/wiki/Continuous_iff_Mapping_at_Element_is_Supremum_of_Compact_Elements | [
"Topological Order Theory",
"Way Below Relation",
"Continuous Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Algebraic Ordered Set",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Compact Element"
] | [
"Algebraic iff Continuous and For Every Way Below Exists Compact Between",
"Definition:Continuous Ordered Set"
] |
proofwiki-13453 | Solutions to Approximate Fermat Equation x^3 = y^3 + z^3 Plus or Minus 1 | The approximate Fermat equation:
:$x^3 = y^3 + z^3 \pm 1$
has the solutions:
{{begin-eqn}}
{{eqn | l = 9^3
| r = 6^3 + 8^3 + 1
}}
{{eqn | l = 103^3
| r = 64^3 + 94^3 - 1
| c =
}}
{{end-eqn}} | Performing the arithmetic:
{{begin-eqn}}
{{eqn | l = 6^3 + 8^3 + 1
| r = 216 + 512 + 1
}}
{{eqn | r = 729
| c =
}}
{{eqn | r = 9^3
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 64^3 + 94^3 - 1
| r = 262 \, 144 + 830 \, 584 - 1
}}
{{eqn | r = 1 \, 092 \, 727
| c =
}}
{{eqn | r = 103^3
... | The [[Definition:Approximate Fermat Equation|approximate Fermat equation]]:
:$x^3 = y^3 + z^3 \pm 1$
has the solutions:
{{begin-eqn}}
{{eqn | l = 9^3
| r = 6^3 + 8^3 + 1
}}
{{eqn | l = 103^3
| r = 64^3 + 94^3 - 1
| c =
}}
{{end-eqn}} | Performing the arithmetic:
{{begin-eqn}}
{{eqn | l = 6^3 + 8^3 + 1
| r = 216 + 512 + 1
}}
{{eqn | r = 729
| c =
}}
{{eqn | r = 9^3
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 64^3 + 94^3 - 1
| r = 262 \, 144 + 830 \, 584 - 1
}}
{{eqn | r = 1 \, 092 \, 727
| c =
}}
{{eqn | r = 103^... | Solutions to Approximate Fermat Equation x^3 = y^3 + z^3 Plus or Minus 1 | https://proofwiki.org/wiki/Solutions_to_Approximate_Fermat_Equation_x^3_=_y^3_+_z^3_Plus_or_Minus_1 | https://proofwiki.org/wiki/Solutions_to_Approximate_Fermat_Equation_x^3_=_y^3_+_z^3_Plus_or_Minus_1 | [
"Approximate Fermat Equations"
] | [
"Definition:Approximate Fermat Equation"
] | [] |
proofwiki-13454 | Cubes which are Sum of Five Cubes | The following cube numbers can be expressed as the sum of $5$ positive cube numbers:
:$9^3, \ldots$
{{expand|More terms needed. It seems that: </br> $4$ and all numbers $> 8$ can be so expressed </br> only $4, 8, 10, 11, 13$ require repeated cubes}} | {{begin-eqn}}
{{eqn | l = 9^3
| r = 729
| c =
}}
{{eqn | r = 1 + 27 + 64 + 125 + 512
| c =
}}
{{eqn | r = 1^3 + 3^3 + 4^3 + 5^3 + 8^3
| c =
}}
{{end-eqn}}
{{expand|Add the proof based on $9^3 {{=}} 1^3 + 6^3 + 8^3$ and $6^3 {{=}} 3^3 + 4^3 + 5^3$ from Cubes which are Sum of Three Cubes}} | The following [[Definition:Cube Number|cube numbers]] can be expressed as the [[Definition:Integer Addition|sum]] of $5$ [[Definition:Positive Integer|positive]] [[Definition:Cube Number|cube numbers]]:
:$9^3, \ldots$
{{expand|More terms needed. It seems that: </br> $4$ and all numbers $> 8$ can be so expressed </br> ... | {{begin-eqn}}
{{eqn | l = 9^3
| r = 729
| c =
}}
{{eqn | r = 1 + 27 + 64 + 125 + 512
| c =
}}
{{eqn | r = 1^3 + 3^3 + 4^3 + 5^3 + 8^3
| c =
}}
{{end-eqn}}
{{expand|Add the proof based on $9^3 {{=}} 1^3 + 6^3 + 8^3$ and $6^3 {{=}} 3^3 + 4^3 + 5^3$ from [[Cubes which are Sum of Three Cubes]]}} | Cubes which are Sum of Five Cubes | https://proofwiki.org/wiki/Cubes_which_are_Sum_of_Five_Cubes | https://proofwiki.org/wiki/Cubes_which_are_Sum_of_Five_Cubes | [
"Sums of Cubes"
] | [
"Definition:Cube Number",
"Definition:Addition/Integers",
"Definition:Positive/Integer",
"Definition:Cube Number"
] | [
"Cubes which are Sum of Three Cubes"
] |
proofwiki-13455 | Period of Reciprocal of 729 is 81 | The decimal expansion of the reciprocal of $729$ has $\dfrac 1 9$ the maximum period, that is, $81$:
:$\dfrac 1 {729} = 0 \cdotp \dot 00137 \, 17421 \, 12482 \, 85322 \, 35939 \, 64334 \, 70507 \, 54458 \, 16186 \, 55692 \, 72976 \, 68038 \, 40877 \, 91495 \, 19890 \, 26063 \, \dot 1$
The recurring part can be arranged... | Performing the calculation using long division:
<pre>
0.00137174211248285322359396433470507544581618655692729766803840877914951989026063100
---------------------------------------------------------------------------------------
729)1.0000000000000000000000000000000000000000000000000000000000000000000000000000000... | The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $729$ has $\dfrac 1 9$ the maximum [[Definition:Period of Recurrence|period]], that is, $81$:
:$\dfrac 1 {729} = 0 \cdotp \dot 00137 \, 17421 \, 12482 \, 85322 \, 35939 \, 64334 \, 70507 \, 54458 \, 16186 \, 55692 \, ... | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.00137174211248285322359396433470507544581618655692729766803840877914951989026063100
---------------------------------------------------------------------------------------
729)1.0000000000000000000000000000000000000000000000000... | Period of Reciprocal of 729 is 81 | https://proofwiki.org/wiki/Period_of_Reciprocal_of_729_is_81 | https://proofwiki.org/wiki/Period_of_Reciprocal_of_729_is_81 | [
"729",
"Examples of Reciprocals"
] | [
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Basis Expansion/Recurrence"
] | [
"Definition:Classical Algorithm/Division"
] |
proofwiki-13456 | Implicit Function Theorem for Differentiable Real Functions | Let $\Omega \subset \R^{n+k}$ be open.
Let $f : \Omega \to \R^k$ be differentiable.
Let the $i$th partial derivatives of $f$ be continuous in $\Omega$ for $n+1 \leq i \leq n+k$.
Let $(a,b) \in \Omega$, with $a\in \R^n$ and $b\in \R^k$.
Let $f(a,b) = 0$.
For $(x_0,y_0)\in\Omega$, let $D_2 f(x_0,y_0)$ denote the differen... | {{ProofWanted}}
Category:Implicit Functions
4t3eiy6pzrf811n7rr0487hxp0tgrea | Let $\Omega \subset \R^{n+k}$ be [[Definition:Open Set of Real Euclidean Space|open]].
Let $f : \Omega \to \R^k$ be [[Definition:Differentiable Vector-Valued Function|differentiable]].
Let the $i$th [[Definition:Partial Derivative of Real-Valued Function|partial derivatives]] of $f$ be [[Definition:Continuous Functio... | {{ProofWanted}}
[[Category:Implicit Functions]]
4t3eiy6pzrf811n7rr0487hxp0tgrea | Implicit Function Theorem for Differentiable Real Functions | https://proofwiki.org/wiki/Implicit_Function_Theorem_for_Differentiable_Real_Functions | https://proofwiki.org/wiki/Implicit_Function_Theorem_for_Differentiable_Real_Functions | [
"Implicit Functions"
] | [
"Definition:Open Set/Real Analysis/Real Euclidean Space",
"Definition:Differentiable Mapping/Vector-Valued Function",
"Definition:Partial Derivative/Real Analysis",
"Definition:Continuous Function",
"Definition:Differential of Mapping/Vector-Valued Function",
"Definition:Linear Transformation",
"Definit... | [
"Category:Implicit Functions"
] |
proofwiki-13457 | Implicit Function Theorem for Smooth Real Functions | Let $\Omega \subset \R^{n+k}$ be open.
Let $f : \Omega \to \R^k$ be smooth.
Let $(a,b) \in \Omega$, with $a\in \R^n$ and $b\in \R^k$.
Let $f(a,b) = 0$.
For $(x_0,y_0)\in\Omega$, let $D_2 f(x_0,y_0)$ denote the differential of the function $y\mapsto f(x_0, y)$ at $y_0$.
Let the linear map $D_2 f(a,b)$ be invertible.
The... | {{ProofWanted}}
Category:Implicit Functions
op8r7n6voyto8kaqajoonhlny7b7fra | Let $\Omega \subset \R^{n+k}$ be [[Definition:Open Set of Real Euclidean Space|open]].
Let $f : \Omega \to \R^k$ be [[Definition:Smooth Vector-Valued Function|smooth]].
Let $(a,b) \in \Omega$, with $a\in \R^n$ and $b\in \R^k$.
Let $f(a,b) = 0$.
For $(x_0,y_0)\in\Omega$, let $D_2 f(x_0,y_0)$ denote the [[Definition:... | {{ProofWanted}}
[[Category:Implicit Functions]]
op8r7n6voyto8kaqajoonhlny7b7fra | Implicit Function Theorem for Smooth Real Functions | https://proofwiki.org/wiki/Implicit_Function_Theorem_for_Smooth_Real_Functions | https://proofwiki.org/wiki/Implicit_Function_Theorem_for_Smooth_Real_Functions | [
"Implicit Functions"
] | [
"Definition:Open Set/Real Analysis/Real Euclidean Space",
"Definition:Smooth Vector-Valued Function",
"Definition:Differential of Mapping/Vector-Valued Function",
"Definition:Linear Transformation",
"Definition:Invertible Linear Mapping",
"Definition:Neighborhood",
"Definition:Function",
"Definition:S... | [
"Category:Implicit Functions"
] |
proofwiki-13458 | Sum of 4 Consecutive Binomial Coefficients forming Square | Consider the Diophantine equation:
:$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$
where:
:$\dbinom a b$ denotes a binomial coefficient
:$n$ is an integer
:$m$ is a non-negative integer.
Then $n$ has one of the following values:
:$-1, 0, 2, 7, 15, 74, 767$
{{OEIS|A047694}}
The corresponding values of $m$... | {{begin-eqn}}
{{eqn | r = \dbinom {-1} 0 + \dbinom {-1} 1 + \dbinom {-1} 2 + \dbinom {-1} 3
| o =
| c =
}}
{{eqn | r = \left({-1}\right)^0 \dbinom 0 0 + \left({-1}\right)^1 \dbinom 1 1 + \left({-1}\right)^2 \dbinom 2 2 + \left({-1}\right)^3 \dbinom 3 3
| c = Negated Upper Index of Binomial Coefficie... | Consider the [[Definition:Diophantine Equation|Diophantine equation]]:
:$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$
where:
:$\dbinom a b$ denotes a [[Definition:Binomial Coefficient|binomial coefficient]]
:$n$ is an [[Definition:Integer|integer]]
:$m$ is a [[Definition:Positive Integer|non-negative in... | {{begin-eqn}}
{{eqn | r = \dbinom {-1} 0 + \dbinom {-1} 1 + \dbinom {-1} 2 + \dbinom {-1} 3
| o =
| c =
}}
{{eqn | r = \left({-1}\right)^0 \dbinom 0 0 + \left({-1}\right)^1 \dbinom 1 1 + \left({-1}\right)^2 \dbinom 2 2 + \left({-1}\right)^3 \dbinom 3 3
| c = [[Negated Upper Index of Binomial Coeffic... | Sum of 4 Consecutive Binomial Coefficients forming Square | https://proofwiki.org/wiki/Sum_of_4_Consecutive_Binomial_Coefficients_forming_Square | https://proofwiki.org/wiki/Sum_of_4_Consecutive_Binomial_Coefficients_forming_Square | [
"Binomial Coefficients",
"Square Numbers"
] | [
"Definition:Diophantine Equation",
"Definition:Binomial Coefficient",
"Definition:Integer",
"Definition:Positive/Integer"
] | [
"Negated Upper Index of Binomial Coefficient/Corollary 1",
"Binomial Coefficient with Self",
"Binomial Coefficient with Zero",
"Binomial Coefficient with Zero",
"Binomial Coefficient with One",
"Binomial Coefficient with Self"
] |
proofwiki-13459 | Implicit Function Theorem for Lipschitz Contractions | Let $M$ and $N$ be metric spaces.
Let $M$ be complete.
Let $f : M \times N \to M$ be a Lipschitz continuous uniform contraction.
Then for all $t\in N$ there exists a unique $\map g t \in M$ such that $\map f {\map g t, t} = \map g t$, and the mapping $g : N \to M$ is Lipschitz continuous. | For every $t\in N$, the mapping:
:$f_t : M \to M : x \mapsto \map f {x, t}$ is a contraction mapping.
By the Banach Fixed-Point Theorem, there exists a unique $\map g t \in M$ such that $\map {f_t} {\map g t} = \map g t$.
We show that $g$ is Lipschitz continuous.
Let $K<1$ be a uniform Lipschitz constant for $f$.
Let $... | Let $M$ and $N$ be [[Definition:Metric Space|metric spaces]].
Let $M$ be [[Definition:Complete Metric Space|complete]].
Let $f : M \times N \to M$ be a [[Definition:Lipschitz Continuous|Lipschitz continuous]] [[Definition:Uniform Contraction Mapping|uniform contraction]].
Then for all $t\in N$ there exists a [[Defi... | For every $t\in N$, the [[Definition:Mapping|mapping]]:
:$f_t : M \to M : x \mapsto \map f {x, t}$ is a [[Definition:Contraction Mapping (Metric Space)|contraction mapping]].
By the [[Banach Fixed-Point Theorem]], there exists a [[Definition:unique|unique]] $\map g t \in M$ such that $\map {f_t} {\map g t} = \map g t$... | Implicit Function Theorem for Lipschitz Contractions | https://proofwiki.org/wiki/Implicit_Function_Theorem_for_Lipschitz_Contractions | https://proofwiki.org/wiki/Implicit_Function_Theorem_for_Lipschitz_Contractions | [
"Implicit Functions"
] | [
"Definition:Metric Space",
"Definition:Complete Metric Space",
"Definition:Lipschitz Continuity",
"Definition:Uniform Contraction Mapping",
"Definition:Unique",
"Definition:Mapping",
"Definition:Lipschitz Continuity"
] | [
"Definition:Mapping",
"Definition:Contraction Mapping (Metric Space)",
"Banach Fixed-Point Theorem",
"Definition:unique",
"Definition:Lipschitz Continuity",
"Definition:Uniform Lipschitz Constant",
"Definition:Lipschitz Continuity/Lipschitz Constant",
"Definition:Uniform Contraction Mapping",
"Defin... |
proofwiki-13460 | Implicit Function Theorem for Lipschitz Contraction at Point | Let $M$ and $N$ be metric spaces.
Let $M$ be complete.
Let $f: M \times N \to M$ be a uniform contraction.
Then for all $t \in N$ there exists a unique $\map g t \in M$ such that $\map f {\map g t, t} = \map g t$, and if $f$ is Lipschitz continuous at a point $\tuple {\map g t, t}$, then $g$ is Lipschitz continuous at ... | For every $t \in N$, the mapping:
:$f_t : M \to M : x \mapsto \map f {x, t}$ is a contraction mapping.
{{explain|Source for above statement? What exactly does $f_t$ mean here?}}
By the Banach Fixed-Point Theorem, there exists a unique $\map g t \in M$ such that $\map {f_t} {\map g t} = \map g t$.
Let $f$ be Lipschitz c... | Let $M$ and $N$ be [[Definition:Metric Space|metric spaces]].
Let $M$ be [[Definition:Complete Metric Space|complete]].
Let $f: M \times N \to M$ be a [[Definition:Uniform Contraction Mapping|uniform contraction]].
Then for all $t \in N$ there exists a [[Definition:Unique|unique]] $\map g t \in M$ such that $\map f... | For every $t \in N$, the [[Definition:Mapping|mapping]]:
:$f_t : M \to M : x \mapsto \map f {x, t}$ is a [[Definition:Contraction Mapping (Metric Space)|contraction mapping]].
{{explain|Source for above statement? What exactly does $f_t$ mean here?}}
By the [[Banach Fixed-Point Theorem]], there exists a [[Definition:... | Implicit Function Theorem for Lipschitz Contraction at Point | https://proofwiki.org/wiki/Implicit_Function_Theorem_for_Lipschitz_Contraction_at_Point | https://proofwiki.org/wiki/Implicit_Function_Theorem_for_Lipschitz_Contraction_at_Point | [
"Implicit Functions"
] | [
"Definition:Metric Space",
"Definition:Complete Metric Space",
"Definition:Uniform Contraction Mapping",
"Definition:Unique",
"Definition:Lipschitz Continuity/Point",
"Definition:Lipschitz Continuity/Point"
] | [
"Definition:Mapping",
"Definition:Contraction Mapping (Metric Space)",
"Banach Fixed-Point Theorem",
"Definition:unique",
"Definition:Lipschitz Continuity/Point",
"Definition:Lipschitz Continuity/Point",
"Definition:Uniform Lipschitz Constant",
"Definition:Lipschitz Continuity/Lipschitz Constant",
"... |
proofwiki-13461 | Smallest Square Inscribed in Two Pythagorean Triangles | The smallest square with integer sides that can be inscribed within two different Pythagorean triangles so that one side of the square lies on the hypotenuse has side length $780$.
The two Pythagorean triangles in question have side lengths $\tuple {1443, 1924, 2405}$ and $\tuple {1145, 2748, 2977}$. | By Inscribed Squares in Right-Angled Triangle/Side Lengths/Side Lies on Hypotenuse:
:For a Pythagorean triangle with side lengths $a, b, c$, the required inscribed square has side length given by:
::$\dfrac {abc}{ab + c^2}$
For primitive Pythagorean triples, $a, b, c$ are pairwise coprime, so the above fraction is in c... | The smallest [[Definition:Square (Geometry)|square]] with [[Definition:Integer|integer]] [[Definition:Side of Polygon|sides]] that can be [[Definition:Polygon Inscribed within Polygon|inscribed]] within two different [[Definition:Pythagorean Triangle|Pythagorean triangles]] so that one [[Definition:Side of Polygon|side... | By [[Inscribed Squares in Right-Angled Triangle/Side Lengths/Side Lies on Hypotenuse]]:
:For a [[Definition:Pythagorean Triangle|Pythagorean triangle]] with [[Definition:Side of Polygon|side]] [[Definition:Length of Line|lengths]] $a, b, c$, the required [[Definition:Polygon Inscribed within Polygon|inscribed]] [[Defi... | Smallest Square Inscribed in Two Pythagorean Triangles | https://proofwiki.org/wiki/Smallest_Square_Inscribed_in_Two_Pythagorean_Triangles | https://proofwiki.org/wiki/Smallest_Square_Inscribed_in_Two_Pythagorean_Triangles | [
"Pythagorean Triangles"
] | [
"Definition:Quadrilateral/Square",
"Definition:Integer",
"Definition:Polygon/Side",
"Definition:Inscribe/Polygon in Polygon",
"Definition:Pythagorean Triangle",
"Definition:Polygon/Side",
"Definition:Quadrilateral/Square",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Polygon... | [
"Inscribed Squares in Right-Angled Triangle/Side Lengths/Side Lies on Hypotenuse",
"Definition:Pythagorean Triangle",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Inscribe/Polygon in Polygon",
"Definition:Quadrilateral/Square",
"Definition:Polygon/Side",
"Definition:Linea... |
proofwiki-13462 | Sequence of Numbers Divisible by Sequence of Primes | The integers in this sequence:
:$788, 789, 790, 791, 792, 793$
are divisible by:
:$2, 3, 5, 7, 11, 13$
respectively. | {{begin-eqn}}
{{eqn | l = 788
| r = 2 \times 394
}}
{{eqn | l = 789
| r = 3 \times 263
}}
{{eqn | l = 790
| r = 5 \times 158
}}
{{eqn | l = 791
| r = 7 \times 113
}}
{{eqn | l = 792
| r = 11 \times 72
}}
{{eqn | l = 793
| r = 13 \times 61
}}
{{end-eqn}}
{{qed}} | The [[Definition:Positive Integer|integers]] in this [[Definition:Integer Sequence|sequence]]:
:$788, 789, 790, 791, 792, 793$
are [[Definition:Divisor of Integer|divisible]] by:
:$2, 3, 5, 7, 11, 13$
respectively. | {{begin-eqn}}
{{eqn | l = 788
| r = 2 \times 394
}}
{{eqn | l = 789
| r = 3 \times 263
}}
{{eqn | l = 790
| r = 5 \times 158
}}
{{eqn | l = 791
| r = 7 \times 113
}}
{{eqn | l = 792
| r = 11 \times 72
}}
{{eqn | l = 793
| r = 13 \times 61
}}
{{end-eqn}}
{{qed}} | Sequence of Numbers Divisible by Sequence of Primes | https://proofwiki.org/wiki/Sequence_of_Numbers_Divisible_by_Sequence_of_Primes | https://proofwiki.org/wiki/Sequence_of_Numbers_Divisible_by_Sequence_of_Primes | [
"Prime Numbers",
"Divisors",
"Recreational Mathematics",
"788",
"789",
"790",
"791",
"792",
"793"
] | [
"Definition:Positive/Integer",
"Definition:Integer Sequence",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-13463 | Numbers whose Square is Palindromic with Even Number of Digits | The sequence of positive integers whose square is a palindromic number with an even number of digits begins:
:$836, 798 \, 644, 64 \, 030 \, 648, 83 \, 163 \, 115 \, 486, 6 \, 360 \, 832 \, 925 \, 898, \ldots$
{{OEIS|A016113}} | {{begin-eqn}}
{{eqn | l = 836^2
| r = 698 \, 896
| c = $6$ digits
}}
{{eqn | l = 798 \, 644^2
| r = 637 \, 832 \, 238 \, 736
| c = $12$ digits
}}
{{eqn | l = 64 \, 030 \, 648^2
| r = 4 \, 099 \, 923 \, 883 \, 299 \, 904
| c = $16$ digits
}}
{{eqn | l = 83 \, 163 \, 115 \, 486^2
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] whose [[Definition:Square (Algebra)|square]] is a [[Definition:Palindromic Number|palindromic number]] with an [[Definition:Even Integer|even number]] of [[Definition:Digit|digits]] begins:
:$836, 798 \, 644, 64 \, 030 \, ... | {{begin-eqn}}
{{eqn | l = 836^2
| r = 698 \, 896
| c = $6$ [[Definition:Digit|digits]]
}}
{{eqn | l = 798 \, 644^2
| r = 637 \, 832 \, 238 \, 736
| c = $12$ [[Definition:Digit|digits]]
}}
{{eqn | l = 64 \, 030 \, 648^2
| r = 4 \, 099 \, 923 \, 883 \, 299 \, 904
| c = $16$ [[Definitio... | Numbers whose Square is Palindromic with Even Number of Digits | https://proofwiki.org/wiki/Numbers_whose_Square_is_Palindromic_with_Even_Number_of_Digits | https://proofwiki.org/wiki/Numbers_whose_Square_is_Palindromic_with_Even_Number_of_Digits | [
"Square Numbers",
"Palindromic Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Square/Function",
"Definition:Palindromic Number",
"Definition:Even Integer",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit"
] |
proofwiki-13464 | Sum of Sequence of Factorials | The sequence $S = \sequence {s_n}$ defined as:
:$\ds s_n = \sum_{k \mathop = 1}^n k!$
begins:
:$1, 3, 9, 33, 153, 873, 5913, 46 \, 233, 409 \, 113, 4 \, 037 \, 913, \ldots$
{{OEIS|A007489}} | {{begin-eqn}}
{{eqn | l = s_1
| r = 1!
| c =
}}
{{eqn | r = 1
| c = {{Defof|Factorial}}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = s_2
| r = s_1 + 2!
| c =
}}
{{eqn | r = 1 + 2
| c = {{Defof|Factorial}}
}}
{{eqn | r = 3
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = s_3
... | The [[Definition:Integer Sequence|sequence]] $S = \sequence {s_n}$ defined as:
:$\ds s_n = \sum_{k \mathop = 1}^n k!$
begins:
:$1, 3, 9, 33, 153, 873, 5913, 46 \, 233, 409 \, 113, 4 \, 037 \, 913, \ldots$
{{OEIS|A007489}} | {{begin-eqn}}
{{eqn | l = s_1
| r = 1!
| c =
}}
{{eqn | r = 1
| c = {{Defof|Factorial}}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = s_2
| r = s_1 + 2!
| c =
}}
{{eqn | r = 1 + 2
| c = {{Defof|Factorial}}
}}
{{eqn | r = 3
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = s_3
... | Sum of Sequence of Factorials | https://proofwiki.org/wiki/Sum_of_Sequence_of_Factorials | https://proofwiki.org/wiki/Sum_of_Sequence_of_Factorials | [
"Factorials",
"Sums of Sequences"
] | [
"Definition:Integer Sequence"
] | [
"Category:Factorials",
"Category:Sums of Sequences"
] |
proofwiki-13465 | Primitive Semiperfect Numbers which are not Primitive Abundant | The sequence of primitive semiperfect numbers which are not also primitive abundant starts:
:$6, 28, 350, 490, 496, 770, 910, 1190, \ldots$
These are semiperfect numbers which are either:
: perfect
or:
: whose only abundant aliquot parts are weird. | A primitive semiperfect number is a semiperfect number which has no aliquot parts which are themselves semiperfect.
Thus by definition a primitive semiperfect number is either perfect or abundant.
A primitive abundant number is an abundant number whose aliquot parts are all deficient.
Thus the perfect numbers:
:$6, 28,... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Primitive Semiperfect Number|primitive semiperfect numbers]] which are not also [[Definition:Primitive Abundant Number|primitive abundant]] starts:
:$6, 28, 350, 490, 496, 770, 910, 1190, \ldots$
These are [[Definition:Semiperfect Number|semiperfect numbers]... | A [[Definition:Primitive Semiperfect Number|primitive semiperfect number]] is a [[Definition:Semiperfect Number|semiperfect number]] which has no [[Definition:Aliquot Part|aliquot parts]] which are themselves [[Definition:Semiperfect Number|semiperfect]].
Thus by definition a [[Definition:Primitive Semiperfect Number|... | Primitive Semiperfect Numbers which are not Primitive Abundant | https://proofwiki.org/wiki/Primitive_Semiperfect_Numbers_which_are_not_Primitive_Abundant | https://proofwiki.org/wiki/Primitive_Semiperfect_Numbers_which_are_not_Primitive_Abundant | [
"Primitive Semiperfect Numbers",
"Primitive Abundant Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Primitive Semiperfect Number",
"Definition:Primitive Abundant Number",
"Definition:Semiperfect Number",
"Definition:Perfect Number",
"Definition:Abundant Number",
"Definition:Divisor (Algebra)/Integer/Aliquot Part",
"Definition:Weird Number"
] | [
"Definition:Primitive Semiperfect Number",
"Definition:Semiperfect Number",
"Definition:Divisor (Algebra)/Integer/Aliquot Part",
"Definition:Semiperfect Number",
"Definition:Primitive Semiperfect Number",
"Definition:Perfect Number",
"Definition:Abundant Number",
"Definition:Primitive Abundant Number"... |
proofwiki-13466 | Set of 5 Triplets whose Sums and Products are Equal | The following set of $5$ triplets of integers have the property that:
:the sum of the integers in each triplet are equal
and:
:the product of the integers in each triplet are equal:
:$\tuple {6, 480, 495}$, $\tuple {11, 160, 810}$, $\tuple {12, 144, 825}$, $\tuple {20, 81, 880}$, $\tuple {33, 48, 900}$
The sum is $981$... | {{begin-eqn}}
{{eqn | l = 6 + 480 + 495
| r = 981
}}
{{eqn | l = 11 + 160 + 810
| r = 981
}}
{{eqn | l = 12 + 144 + 825
| r = 981
}}
{{eqn | l = 20 + 81 + 880
| r = 981
}}
{{eqn | l = 33 + 48 + 900
| r = 981
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 480 \times 495
| r = \pare... | The following [[Definition:Set|set]] of $5$ [[Definition:Ordered Triple|triplets]] of [[Definition:Positive Integer|integers]] have the property that:
:the [[Definition:Integer Addition|sum]] of the [[Definition:Positive Integer|integers]] in each [[Definition:Ordered Triple|triplet]] are equal
and:
:the [[Definition:I... | {{begin-eqn}}
{{eqn | l = 6 + 480 + 495
| r = 981
}}
{{eqn | l = 11 + 160 + 810
| r = 981
}}
{{eqn | l = 12 + 144 + 825
| r = 981
}}
{{eqn | l = 20 + 81 + 880
| r = 981
}}
{{eqn | l = 33 + 48 + 900
| r = 981
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 480 \times 495
| r = \pa... | Set of 5 Triplets whose Sums and Products are Equal | https://proofwiki.org/wiki/Set_of_5_Triplets_whose_Sums_and_Products_are_Equal | https://proofwiki.org/wiki/Set_of_5_Triplets_whose_Sums_and_Products_are_Equal | [
"Recreational Mathematics",
"981",
"1,425,600"
] | [
"Definition:Set",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Positive/Integer",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Multiplication/Integers",
"Definition:Positive/Integer",
"De... | [] |
proofwiki-13467 | Sum of Pandigital Triplet of 3-Digit Primes | The smallest integer which is the sum of a set of $3$ three-digit primes using all $9$ digits from $1$ to $9$ once each is $999$:
:$149 + 263 + 587 = 999$ | All three-digit primes end in $1, 3, 7, 9$.
Suppose $1$ is used as the units digit of a prime.
Since the digit $1$ cannot be used again, the sum of the primes is at least:
:$221 + 333 + 447 = 1001$
so $1$ cannot be used as a units digit .
The units digits of the primes are $3, 7, 9$.
To minimise the sum, the hundreds d... | The smallest [[Definition:Integer|integer]] which is the [[Definition:Integer Addition|sum]] of a [[Definition:Set|set]] of $3$ [[Definition:Digit|three-digit]] [[Definition:Prime Number|primes]] using all $9$ [[Definition:Digit|digits]] from $1$ to $9$ once each is $999$:
:$149 + 263 + 587 = 999$ | All [[Definition:Digit|three-digit]] [[Definition:Prime Number|primes]] end in $1, 3, 7, 9$.
Suppose $1$ is used as the units digit of a [[Definition:Prime Number|prime]].
Since the digit $1$ cannot be used again, the sum of the [[Definition:Prime Number|primes]] is at least:
:$221 + 333 + 447 = 1001$
so $1$ cannot... | Sum of Pandigital Triplet of 3-Digit Primes | https://proofwiki.org/wiki/Sum_of_Pandigital_Triplet_of_3-Digit_Primes | https://proofwiki.org/wiki/Sum_of_Pandigital_Triplet_of_3-Digit_Primes | [
"Prime Numbers",
"999"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Set",
"Definition:Digit",
"Definition:Prime Number",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-13468 | Divisibility Test for 7, 11 and 13 | Mark off the integer $N$ being tested into groups of $3$ digits.
Because of the standard way of presenting integers, this may already be done, for example:
:$N = 22 \, 846 \, 293 \, 462 \, 733 \, 356$
Number the groups of $3$ from the right:
:$N = \underbrace{22}_6 \, \underbrace{846}_5 \, \underbrace{293}_4 \, \under... | Let $N$ be expressed as:
:$N = \ds \sum_{k \mathop = 0}^n a_k 1000^k = a_0 + a_1 1000 + a_2 1000^2 + \cdots + a_n 1000^n$
where $n$ is the number of groups of $3$ digits.
We have that:
:$1000 \equiv -1 \pmod {1001}$
Hence from Congruence of Powers:
:$1000^r \equiv \paren {-1}^r \pmod {1001}$
Thus:
:$N \equiv a_0 + \par... | Mark off the [[Definition:Integer|integer]] $N$ being tested into groups of $3$ [[Definition:Digit|digits]].
Because of the standard way of presenting [[Definition:Integer|integers]], this may already be done, for example:
:$N = 22 \, 846 \, 293 \, 462 \, 733 \, 356$
Number the groups of $3$ from the right:
:$N = \u... | Let $N$ be expressed as:
:$N = \ds \sum_{k \mathop = 0}^n a_k 1000^k = a_0 + a_1 1000 + a_2 1000^2 + \cdots + a_n 1000^n$
where $n$ is the number of groups of $3$ [[Definition:Digit|digits]].
We have that:
:$1000 \equiv -1 \pmod {1001}$
Hence from [[Congruence of Powers]]:
:$1000^r \equiv \paren {-1}^r \pmod {1001}$
... | Divisibility Test for 7, 11 and 13 | https://proofwiki.org/wiki/Divisibility_Test_for_7,_11_and_13 | https://proofwiki.org/wiki/Divisibility_Test_for_7,_11_and_13 | [
"Divisibility Tests",
"7",
"11",
"13",
"1001"
] | [
"Definition:Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Even Integer",
"Definition:Subtraction/Integers",
"Definition:Odd Integer",
"Definition:Sign of Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Digit",
"Congruence of Powers",
"Definition:Modulo Addition"
] |
proofwiki-13469 | Solutions to Diophantine Equation x (x + 1) = y (y + 5) (y + 10) (y + 15) | The Diophantine equation
: $n = x \paren {x + 1} = y \paren {y + 5} \paren {y + 10} \paren {y + 15}$
has exactly $2$ solutions in $\N \setminus \set 0$:
{{begin-eqn}}
{{eqn | l = 1056
| r = 32 \times 33 = 1 \times 6 \times 11 \times 16
}}
{{eqn | l = 43 \, 056
| r = 207 \times 208 = 8 \times 13 \times 18 \t... | First, we observe that:
{{begin-eqn}}
{{eqn | l = x \paren {x + 1}
| r = \paren {x + \dfrac 1 2 }^2 - \paren {\dfrac 1 2}^2
}}
{{eqn | ll = \leadsto
| l = x \paren {x + 1}
| o = \lt
| r = \paren {x + \dfrac 1 2 }^2
}}
{{eqn | ll = \leadsto
| l = \sqrt {x \paren {x + 1} }
| o = \lt
... | The [[Definition:Diophantine Equation|Diophantine equation]]
: $n = x \paren {x + 1} = y \paren {y + 5} \paren {y + 10} \paren {y + 15}$
has exactly $2$ solutions in $\N \setminus \set 0$:
{{begin-eqn}}
{{eqn | l = 1056
| r = 32 \times 33 = 1 \times 6 \times 11 \times 16
}}
{{eqn | l = 43 \, 056
| r = 207 ... | First, we observe that:
{{begin-eqn}}
{{eqn | l = x \paren {x + 1}
| r = \paren {x + \dfrac 1 2 }^2 - \paren {\dfrac 1 2}^2
}}
{{eqn | ll = \leadsto
| l = x \paren {x + 1}
| o = \lt
| r = \paren {x + \dfrac 1 2 }^2
}}
{{eqn | ll = \leadsto
| l = \sqrt {x \paren {x + 1} }
| o = \lt
... | Solutions to Diophantine Equation x (x + 1) = y (y + 5) (y + 10) (y + 15) | https://proofwiki.org/wiki/Solutions_to_Diophantine_Equation_x_(x_+_1)_=_y_(y_+_5)_(y_+_10)_(y_+_15) | https://proofwiki.org/wiki/Solutions_to_Diophantine_Equation_x_(x_+_1)_=_y_(y_+_5)_(y_+_10)_(y_+_15) | [
"Diophantine Equations"
] | [
"Definition:Diophantine Equation"
] | [
"Definition:Square Root",
"Definition:Square Root",
"Definition:Fractional Part",
"Definition:Fractional Part",
"Definition:Square Root",
"Definition:Integer",
"Definition:Fractional Part",
"Definition:Fractional Part",
"Definition:Square Root",
"Solution to Quadratic Equation",
"File:Diophantin... |
proofwiki-13470 | Numbers Reversed when Multiplying by 9 | Numbers of the form $\sqbrk {10 (9) 89}_{10}$ are reversed when they are multiplied by $9$:
{{begin-eqn}}
{{eqn | l = 1089 \times 9
| r = 9801
}}
{{eqn | l = 10 \, 989 \times 9
| r = 98 \, 901
}}
{{eqn | l = 109 \, 989 \times 9
| r = 989 \, 901
}}
{{end-eqn}}
and so on. | Let k represent the number of $9$s in the middle of the number.
For $k > 0$ We can rewrite the number as follows:
{{begin-eqn}}
{{eqn | l = \sqbrk {10 (9) 89}_{10}
| r = 10 \times 10^{k + 2 } + 900 \sum_{i \mathop = 0}^{k - 1} 10^i + 89
| c = {{Defof|Geometric Series}}
}}
{{end-eqn}}
Taking numbers of this ... | Numbers of the form $\sqbrk {10 (9) 89}_{10}$ are [[Definition:Reversal|reversed]] when they are [[Definition:Integer Multiplication|multiplied]] by $9$:
{{begin-eqn}}
{{eqn | l = 1089 \times 9
| r = 9801
}}
{{eqn | l = 10 \, 989 \times 9
| r = 98 \, 901
}}
{{eqn | l = 109 \, 989 \times 9
| r = 989 \... | Let k represent the number of $9$s in the middle of the number.
For $k > 0$ We can rewrite the number as follows:
{{begin-eqn}}
{{eqn | l = \sqbrk {10 (9) 89}_{10}
| r = 10 \times 10^{k + 2 } + 900 \sum_{i \mathop = 0}^{k - 1} 10^i + 89
| c = {{Defof|Geometric Series}}
}}
{{end-eqn}}
Taking numbers of th... | Numbers Reversed when Multiplying by 9 | https://proofwiki.org/wiki/Numbers_Reversed_when_Multiplying_by_9 | https://proofwiki.org/wiki/Numbers_Reversed_when_Multiplying_by_9 | [
"Reversals",
"1089"
] | [
"Definition:Reversal",
"Definition:Multiplication/Integers"
] | [
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit"
] |
proofwiki-13471 | Reciprocal of 1089 | :$\dfrac 1 {1089} = 0 \cdotp \dot 00091 \, 82736 \, 45546 \, 37281 \, 9 \dot 1$ | Performing the calculation using long division:
<pre>
0.00091827364554637281910009...
--------------------------------------------
1089)1.00000000000000000000000000...
9801 5445 1089
---- ---- ----
1990 5950 9910
1089 5445 9801
---- ---- ----
... | :$\dfrac 1 {1089} = 0 \cdotp \dot 00091 \, 82736 \, 45546 \, 37281 \, 9 \dot 1$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.00091827364554637281910009...
--------------------------------------------
1089)1.00000000000000000000000000...
9801 5445 1089
---- ---- ----
1990 5950 9910
1089 5445 9801... | Reciprocal of 1089 | https://proofwiki.org/wiki/Reciprocal_of_1089 | https://proofwiki.org/wiki/Reciprocal_of_1089 | [
"1089",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division"
] |
proofwiki-13472 | Square which is Difference between Square and Square of Reversal | $33^2 = 65^2 - 56^2$
This is the only square of a $2$-digit number which has this property. | {{begin-eqn}}
{{eqn | l = 33^2
| r = 1089
| c =
}}
{{eqn | l = 65^2 - 56^2
| r = 4225 - 3136
| c =
}}
{{eqn | r = 1089
| c =
}}
{{end-eqn}}
Let $\sqbrk {xy}$ be a $2$-digit number such that $x \ge y$ and $\sqbrk {xy}^2 - \sqbrk {yx}^2$ is a square of a $2$-digit number.
The case $x = y$... | $33^2 = 65^2 - 56^2$
This is the only [[Definition:Square Number|square]] of a $2$-[[Definition:Digit|digit]] number which has this property. | {{begin-eqn}}
{{eqn | l = 33^2
| r = 1089
| c =
}}
{{eqn | l = 65^2 - 56^2
| r = 4225 - 3136
| c =
}}
{{eqn | r = 1089
| c =
}}
{{end-eqn}}
Let $\sqbrk {xy}$ be a $2$-[[Definition:Digit|digit]] number such that $x \ge y$ and $\sqbrk {xy}^2 - \sqbrk {yx}^2$ is a [[Definition:Square Num... | Square which is Difference between Square and Square of Reversal | https://proofwiki.org/wiki/Square_which_is_Difference_between_Square_and_Square_of_Reversal | https://proofwiki.org/wiki/Square_which_is_Difference_between_Square_and_Square_of_Reversal | [
"Square Numbers",
"Reversals"
] | [
"Definition:Square Number",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Square Number",
"Definition:Digit",
"Definition:Square Number",
"Definition:Digit",
"Euclid's Lemma for Prime Divisors",
"Absolute Value of Integer is not less than Divisors",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number",
"Def... |
proofwiki-13473 | Numbers for which Sixth Power plus 1091 is Composite | The number $1091$ has the property that:
:$x^6 + 1091$
is composite for all integer values of $x$ from $1$ to $3905$. | We check the result and show that it cannot be improved further by showing:
:$3906$ is the smallest $x$ such that $x^6 + 1091$ is prime.
Suppose $x^6 + 1091$ is prime.
Then:
:$x$ is a multiple of $42$
:$x$ ends in $0$, $4$ or $6$ in decimal notation
:$x \not \equiv \pm 1, \pm 3, \pm 4 \pmod {13}$
:$x \not \equiv \pm 4,... | The number $1091$ has the property that:
:$x^6 + 1091$
is [[Definition:Composite Number|composite]] for all [[Definition:Integer|integer]] values of $x$ from $1$ to $3905$. | We check the result and show that it cannot be improved further by showing:
:$3906$ is the smallest $x$ such that $x^6 + 1091$ is [[Definition:Prime Number|prime]].
Suppose $x^6 + 1091$ is [[Definition:Prime Number|prime]].
Then:
:$x$ is a multiple of $42$
:$x$ ends in $0$, $4$ or $6$ in decimal notation
:$x \not \e... | Numbers for which Sixth Power plus 1091 is Composite | https://proofwiki.org/wiki/Numbers_for_which_Sixth_Power_plus_1091_is_Composite | https://proofwiki.org/wiki/Numbers_for_which_Sixth_Power_plus_1091_is_Composite | [
"Sixth Powers"
] | [
"Definition:Composite Number",
"Definition:Integer"
] | [
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-13474 | 1105 as Sum of Two Squares | $1105$ can be expressed as the sum of two squares in more ways than any smaller integer:
{{begin-eqn}}
{{eqn | l = 1105
| m = 1089 + 16
| mo= =
| r = 33^2 + 4^2
| c =
}}
{{eqn | m = 1024 + 81
| mo= =
| r = 32^2 + 9^2
| c =
}}
{{eqn | m = 961 + 144
| mo= =
| r = 31... | Here is the source code of a program in Python that finds all positive integers up to $1105$ that can be written as a sum of two squares in more ways than any smaller positive integer:
<syntaxhighlight lang="python">
import numpy as np
def two_sq_decomp_rich(n):
bound = int(np.floor(np.sqrt(n)))
count_of_two_sq... | $1105$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]] in more ways than any smaller [[Definition:Integer|integer]]:
{{begin-eqn}}
{{eqn | l = 1105
| m = 1089 + 16
| mo= =
| r = 33^2 + 4^2
| c =
}}
{{eqn | m = 1024 + 81
| mo= =
... | Here is the source code of a program in Python that finds all [[Definition:Positive Integer|positive integers]] up to $1105$ that can be written as a [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]] in more ways than any smaller [[Definition:Positive Integer|positive integer]]:
<syntaxhi... | 1105 as Sum of Two Squares | https://proofwiki.org/wiki/1105_as_Sum_of_Two_Squares | https://proofwiki.org/wiki/1105_as_Sum_of_Two_Squares | [
"Sums of Squares",
"1105",
"1105 as Sum of Two Squares"
] | [
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Integer"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Positive/Integer"
] |
proofwiki-13475 | Difference between Two Squares equal to Repunit/Corollary 1 | {{begin-eqn}}
{{eqn | l = 6^2 - 5^2
| r = 11
| c =
}}
{{eqn | l = 56^2 - 45^2
| r = 1111
| c =
}}
{{eqn | l = 556^2 - 445^2
| r = 111 \, 111
| c =
}}
{{eqn | o = :
| c =
}}
{{end-eqn}}
and in general for integer $n$:
:$R_{2 n} = {\underbrace {55 \ldots 56}_{\text {$n - 1$ $... | From Difference between Two Squares equal to Repunit, $R_{2 n} = x^2 - y^2$ exactly when $R_{2 n} = a b$ where $x = \dfrac {a + b} 2$ and $y = \dfrac {a - b} 2$.
By the Basis Representation Theorem
{{begin-eqn}}
{{eqn | l = R_{2n}
| r = \sum_{k \mathop = 0}^{2 n - 1} 10^k
| c =
}}
{{eqn | r = \sum_{k \math... | {{begin-eqn}}
{{eqn | l = 6^2 - 5^2
| r = 11
| c =
}}
{{eqn | l = 56^2 - 45^2
| r = 1111
| c =
}}
{{eqn | l = 556^2 - 445^2
| r = 111 \, 111
| c =
}}
{{eqn | o = :
| c =
}}
{{end-eqn}}
and in general for [[Definition:Integer|integer]] $n$:
:$R_{2 n} = {\underbrace {55 \ldo... | From [[Difference between Two Squares equal to Repunit]], $R_{2 n} = x^2 - y^2$ exactly when $R_{2 n} = a b$ where $x = \dfrac {a + b} 2$ and $y = \dfrac {a - b} 2$.
By the [[Basis Representation Theorem]]
{{begin-eqn}}
{{eqn | l = R_{2n}
| r = \sum_{k \mathop = 0}^{2 n - 1} 10^k
| c =
}}
{{eqn | r = \s... | Difference between Two Squares equal to Repunit/Corollary 1 | https://proofwiki.org/wiki/Difference_between_Two_Squares_equal_to_Repunit/Corollary_1 | https://proofwiki.org/wiki/Difference_between_Two_Squares_equal_to_Repunit/Corollary_1 | [
"Difference between Two Squares equal to Repunit"
] | [
"Definition:Integer"
] | [
"Difference between Two Squares equal to Repunit",
"Basis Representation Theorem",
"Definition:Summation"
] |
proofwiki-13476 | Difference between Two Squares equal to Repunit/Corollary 2 | {{begin-eqn}}
{{eqn | l = 6^2 - 5^2
| r = 11
| c =
}}
{{eqn | l = 56^2 - 45^2
| r = 1111
| c =
}}
{{eqn | l = 5056^2 - 5045^2
| r = 111 \, 111
| c =
}}
{{eqn | o = :
| c =
}}
{{end-eqn}}
and in general for integer $n$:
:$R_{2 n} = {\underbrace{5050 \ldots 56}_{n - 1 \ 5 \te... | From Difference between Two Squares equal to Repunit, $R_{2n} = x^2 - y^2$ exactly when $R_{2n} = a b$ where $x = \dfrac {a + b} 2$ and $y = \dfrac {a - b} 2$.
By the Basis Representation Theorem
{{begin-eqn}}
{{eqn | l = R_{2n}
| r = \sum_{0 \mathop \le k \mathop < 2 n} 10^k
| c =
}}
{{eqn | r = \sum_{\su... | {{begin-eqn}}
{{eqn | l = 6^2 - 5^2
| r = 11
| c =
}}
{{eqn | l = 56^2 - 45^2
| r = 1111
| c =
}}
{{eqn | l = 5056^2 - 5045^2
| r = 111 \, 111
| c =
}}
{{eqn | o = :
| c =
}}
{{end-eqn}}
and in general for [[Definition:Integer|integer]] $n$:
:$R_{2 n} = {\underbrace{5050 \... | From [[Difference between Two Squares equal to Repunit]], $R_{2n} = x^2 - y^2$ exactly when $R_{2n} = a b$ where $x = \dfrac {a + b} 2$ and $y = \dfrac {a - b} 2$.
By the [[Basis Representation Theorem]]
{{begin-eqn}}
{{eqn | l = R_{2n}
| r = \sum_{0 \mathop \le k \mathop < 2 n} 10^k
| c =
}}
{{eqn | r ... | Difference between Two Squares equal to Repunit/Corollary 2 | https://proofwiki.org/wiki/Difference_between_Two_Squares_equal_to_Repunit/Corollary_2 | https://proofwiki.org/wiki/Difference_between_Two_Squares_equal_to_Repunit/Corollary_2 | [
"Difference between Two Squares equal to Repunit"
] | [
"Definition:Integer"
] | [
"Difference between Two Squares equal to Repunit",
"Basis Representation Theorem",
"Category:Difference between Two Squares equal to Repunit"
] |
proofwiki-13477 | Sixth Power as Sum of 7 Sixth Powers | The smallest known integer whose $6$th power can be expressed as the sum of $7$ smaller $6$th powers is $1141$:
:$1141^6 = 74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6$ | {{begin-eqn}}
{{eqn | r = 74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6
| o =
| c =
}}
{{eqn | r = 164 \, 206 \, 490 \, 176
| c =
}}
{{eqn | o =
| ro= +
| r = 164 \, 170 \, 508 \, 913 \, 216
| c =
}}
{{eqn | o =
| ro= +
| r = 4 \, 220 \, 426 \, 278 \, 476 \,... | The smallest known [[Definition:Integer|integer]] whose [[Definition:Sixth Power|$6$th power]] can be expressed as the [[Definition:Integer Addition|sum]] of $7$ smaller [[Definition:Sixth Power|$6$th powers]] is $1141$:
:$1141^6 = 74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6$ | {{begin-eqn}}
{{eqn | r = 74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6
| o =
| c =
}}
{{eqn | r = 164 \, 206 \, 490 \, 176
| c =
}}
{{eqn | o =
| ro= +
| r = 164 \, 170 \, 508 \, 913 \, 216
| c =
}}
{{eqn | o =
| ro= +
| r = 4 \, 220 \, 426 \, 278 \, 476 \,... | Sixth Power as Sum of 7 Sixth Powers | https://proofwiki.org/wiki/Sixth_Power_as_Sum_of_7_Sixth_Powers | https://proofwiki.org/wiki/Sixth_Power_as_Sum_of_7_Sixth_Powers | [
"Sixth Powers",
"1141"
] | [
"Definition:Integer",
"Definition:Sixth Power",
"Definition:Addition/Integers",
"Definition:Sixth Power"
] | [] |
proofwiki-13478 | Square Numbers which are Sum of Sequence of Odd Cubes | The sequence of square numbers which can be expressed as the sum of a sequence of odd cubes from $1$ begins:
:$1, 1225, 1 \, 413 \, 721, 1 \, 631 \, 432 \, 881, \dotsc$
{{OEIS|A046177}}
The sequence of square roots of this sequence is:
:$1, 35, 1189, 40 \, 391, \dotsc$
{{OEIS|A046176}} | We have that:
{{begin-eqn}}
{{eqn | l = 1225
| r = 35^2
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^5 \paren {2 k - 1}^3 = 1^3 + 3^3 + 5^3 + 7^3 + 9^3
| c =
}}
{{eqn | l = 1 \, 413 \, 721
| r = 1189^2
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^{29} \paren {2 k - 1}^3 = 1^3 + 3^3 + 5^3 + \d... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Square Number|square numbers]] which can be expressed as the [[Definition:Integer Addition|sum]] of a [[Definition:Integer Sequence|sequence]] of [[Definition:Odd Number|odd]] [[Definition:Cube Number|cubes]] from $1$ begins:
:$1, 1225, 1 \, 413 \, 721, 1 \,... | We have that:
{{begin-eqn}}
{{eqn | l = 1225
| r = 35^2
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^5 \paren {2 k - 1}^3 = 1^3 + 3^3 + 5^3 + 7^3 + 9^3
| c =
}}
{{eqn | l = 1 \, 413 \, 721
| r = 1189^2
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^{29} \paren {2 k - 1}^3 = 1^3 + 3^3 + 5^3 + \... | Square Numbers which are Sum of Sequence of Odd Cubes | https://proofwiki.org/wiki/Square_Numbers_which_are_Sum_of_Sequence_of_Odd_Cubes | https://proofwiki.org/wiki/Square_Numbers_which_are_Sum_of_Sequence_of_Odd_Cubes | [
"Square Numbers",
"Sums of Sequences",
"Cube Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Square Number",
"Definition:Addition/Integers",
"Definition:Integer Sequence",
"Definition:Odd Integer",
"Definition:Cube Number",
"Definition:Integer Sequence",
"Definition:Square Root",
"Definition:Integer Sequence"
] | [
"Sum of Sequence of Odd Cubes",
"Definition:Square Number",
"Definition:Pell's Equation",
"Pell's Equation/Examples/2/-1",
"Definition:Positive/Integer",
"Definition:Multiplication/Integers",
"Definition:Integer Sequence",
"Definition:Square Root"
] |
proofwiki-13479 | Retract of Injective Space is Injective | Let $T = \struct {S, \tau}$ be an injective topological space.
Let $R = \struct {Z, \tau'}$ be a retract of $T$.
Then $R$ is injective. | By definition of retract:
:there exists a continuous retraction $r: S \to Z$ of $T$.
Let $\YY = \struct {Y, \sigma}$ be a topological space.
Let $f: Y \to Z$ be a continuous mapping.
Let $\XX = \struct {X, \sigma'}$ such that
:$\YY$ is topological subspace of $\XX$.
By Inclusion Mapping is Continuous:
:$i_Z$ is continu... | Let $T = \struct {S, \tau}$ be an [[Definition:Injective Space|injective topological space]].
Let $R = \struct {Z, \tau'}$ be a [[Definition:Retract (Topology)|retract]] of $T$.
Then $R$ is [[Definition:Injective Space|injective]]. | By definition of [[Definition:Retract (Topology)|retract]]:
:there exists a [[Definition:Continuous (Topology)|continuous]] [[Definition:Retraction (Topology)|retraction]] $r: S \to Z$ of $T$.
Let $\YY = \struct {Y, \sigma}$ be a [[Definition:Topological Space|topological space]].
Let $f: Y \to Z$ be a [[Definition:C... | Retract of Injective Space is Injective | https://proofwiki.org/wiki/Retract_of_Injective_Space_is_Injective | https://proofwiki.org/wiki/Retract_of_Injective_Space_is_Injective | [
"Topology"
] | [
"Definition:Injective Space",
"Definition:Retract (Topology)",
"Definition:Injective Space"
] | [
"Definition:Retract (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Retraction (Topology)",
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Mapping",
"Definition:Topological Subspace",
"Inclusion Mapping is Continuous",
"Definition:Continu... |
proofwiki-13480 | Numbers equal to Sum of Squares of two Parts | Integers that can be split into two parts whose squares add up to it include:
:$1233 = 12^2 + 33^2$
:$8833 = 88^2 + 33^2$
{{expand|Need to establish the parameters of this}} | {{ProofWanted|Need to establish exactly what is to be proved}} | [[Definition:Integer|Integers]] that can be split into two parts whose [[Definition:Square (Algebra)|squares]] add up to it include:
:$1233 = 12^2 + 33^2$
:$8833 = 88^2 + 33^2$
{{expand|Need to establish the parameters of this}} | {{ProofWanted|Need to establish exactly what is to be proved}} | Numbers equal to Sum of Squares of two Parts | https://proofwiki.org/wiki/Numbers_equal_to_Sum_of_Squares_of_two_Parts | https://proofwiki.org/wiki/Numbers_equal_to_Sum_of_Squares_of_two_Parts | [
"Sums of Squares",
"Recreational Mathematics"
] | [
"Definition:Integer",
"Definition:Square/Function"
] | [] |
proofwiki-13481 | Triples of Consecutive Sphenic Numbers | The sequence of triplets of consecutive sphenic numbers starts:
:$\tuple {1309, 1310, 1311}, \tuple {1885, 1886, 1887}, \tuple {2013, 2014, 2015}, \ldots$
{{OEIS|A066509|order = first}}
{{OEIS|A248202|order = middle}} | Note that there cannot be quadruplets of such numbers, since one of the quadruplets must be divisible by $4$, making it non-sphenic.
We have:
{{begin-eqn}}
{{eqn | l = 1309
| r = 7 \times 11 \times 17
}}
{{eqn | l = 1310
| r = 2 \times 5 \times 131
}}
{{eqn | l = 1311
| r = 3 \times 19 \times 23
}}
{{... | The [[Definition:Sequence|sequence]] of [[Definition:Ordered Triple|triplets]] of consecutive [[Definition:Sphenic Number|sphenic numbers]] starts:
:$\tuple {1309, 1310, 1311}, \tuple {1885, 1886, 1887}, \tuple {2013, 2014, 2015}, \ldots$
{{OEIS|A066509|order = first}}
{{OEIS|A248202|order = middle}} | Note that there cannot be [[Definition:Ordered Quadruple|quadruplets]] of such numbers, since one of the [[Definition:Ordered Quadruple|quadruplets]] must be [[Definition:Divisor of Integer|divisible]] by $4$, making it non-[[Definition:Sphenic Number|sphenic]].
We have:
{{begin-eqn}}
{{eqn | l = 1309
| r = 7 \t... | Triples of Consecutive Sphenic Numbers | https://proofwiki.org/wiki/Triples_of_Consecutive_Sphenic_Numbers | https://proofwiki.org/wiki/Triples_of_Consecutive_Sphenic_Numbers | [
"Sphenic Numbers"
] | [
"Definition:Sequence",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Sphenic Number"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Quadruple",
"Definition:Ordered Tuple as Ordered Set/Ordered Quadruple",
"Definition:Divisor (Algebra)/Integer",
"Definition:Sphenic Number",
"Definition:Sphenic Number"
] |
proofwiki-13482 | Cube Number as Sum of Three Consecutive Odd Squares | :$1331 = 11^3 = 19^2 + 21^2 + 23^2$
No other such sequence of $3$ consecutive odd squares has the same property. | {{begin-eqn}}
{{eqn | l = 19^2 + 21^2 + 23^2
| r = 361 + 441 + 529
| c =
}}
{{eqn | r = 1331
| c =
}}
{{end-eqn}}
Any sequence of $3$ consecutive odd integers that have squares that sum to a cube would satisfy:
:$m^3 = \paren {n - 2}^2 + n^2 + \paren {n + 2}^2$
where $n$ is the middle number of the ... | :$1331 = 11^3 = 19^2 + 21^2 + 23^2$
No other such [[Definition:Integer Sequence|sequence]] of $3$ consecutive [[Definition:Odd Integer|odd]] [[Definition:Square Number|squares]] has the same property. | {{begin-eqn}}
{{eqn | l = 19^2 + 21^2 + 23^2
| r = 361 + 441 + 529
| c =
}}
{{eqn | r = 1331
| c =
}}
{{end-eqn}}
Any [[Definition:Integer Sequence|sequence]] of $3$ consecutive [[Definition:Odd Integer|odd integers]] that have [[Definition:Square Number|squares]] that [[Definition:Integer Addition... | Cube Number as Sum of Three Consecutive Odd Squares | https://proofwiki.org/wiki/Cube_Number_as_Sum_of_Three_Consecutive_Odd_Squares | https://proofwiki.org/wiki/Cube_Number_as_Sum_of_Three_Consecutive_Odd_Squares | [
"Cube Numbers",
"Sums of Squares",
"1331"
] | [
"Definition:Integer Sequence",
"Definition:Odd Integer",
"Definition:Square Number"
] | [
"Definition:Integer Sequence",
"Definition:Odd Integer",
"Definition:Square Number",
"Definition:Addition/Integers",
"Definition:Cube Number",
"Definition:Integer Sequence",
"Square of Sum",
"Square of Difference",
"Definition:Elliptic Curve",
"Definition:Elliptic Curve",
"Definition:Lattice Poi... |
proofwiki-13483 | Smallest Triplet of Consecutive Integers Divisible by Cube | The smallest sequence of triplets of consecutive integers each of which is divisible by a cube greater than $1$ is:
:$\tuple {1375, 1376, 1377}$ | We will show that:
{{begin-eqn}}
{{eqn | l = 1375
| r = 11 \times 5^3
| c =
}}
{{eqn | l = 1376
| r = 172 \times 2^3
| c =
}}
{{eqn | l = 1377
| r = 51 \times 3^3
| c =
}}
{{end-eqn}}
is the smallest such triplet.
Each number in such triplets of consecutive integers is divisible b... | The smallest [[Definition:Sequence|sequence]] of [[Definition:Ordered Triple|triplets]] of consecutive [[Definition:Integer|integers]] each of which is [[Definition:Divisor of Integer|divisible]] by a [[Definition:Cube Number|cube]] greater than $1$ is:
:$\tuple {1375, 1376, 1377}$ | We will show that:
{{begin-eqn}}
{{eqn | l = 1375
| r = 11 \times 5^3
| c =
}}
{{eqn | l = 1376
| r = 172 \times 2^3
| c =
}}
{{eqn | l = 1377
| r = 51 \times 3^3
| c =
}}
{{end-eqn}}
is the smallest such [[Definition:Ordered Triple|triplet]].
Each number in such [[Definition:... | Smallest Triplet of Consecutive Integers Divisible by Cube | https://proofwiki.org/wiki/Smallest_Triplet_of_Consecutive_Integers_Divisible_by_Cube | https://proofwiki.org/wiki/Smallest_Triplet_of_Consecutive_Integers_Divisible_by_Cube | [
"Cube Numbers"
] | [
"Definition:Sequence",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Cube Number"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Cube Number",
"Definition:Prime Number",
"Chinese Remainder Theorem",
"Definition:Divisor (Algebra)/Integer",
... |
proofwiki-13484 | Smallest Quadruplet of Consecutive Integers Divisible by Cube | The smallest sequence of quadruplets of consecutive integers each of which is divisible by a cube greater than $1$ is:
:$\tuple {22 \, 624, 22 \, 625, 22 \, 626, 22 \, 627}$ | {{begin-eqn}}
{{eqn | l = 22 \, 624
| r = 2828 \times 2^3
| c =
}}
{{eqn | l = 22 \, 625
| r = 181 \times 5^3
| c =
}}
{{eqn | l = 22 \, 626
| r = 838 \times 3^3
| c =
}}
{{eqn | l = 22 \, 627
| r = 17 \times 11^3
| c =
}}
{{end-eqn}}
{{ProofWanted|It remains to be sh... | The smallest [[Definition:Sequence|sequence]] of [[Definition:Ordered Quadruple|quadruplets]] of consecutive [[Definition:Integer|integers]] each of which is [[Definition:Divisor of Integer|divisible]] by a [[Definition:Cube Number|cube]] greater than $1$ is:
:$\tuple {22 \, 624, 22 \, 625, 22 \, 626, 22 \, 627}$ | {{begin-eqn}}
{{eqn | l = 22 \, 624
| r = 2828 \times 2^3
| c =
}}
{{eqn | l = 22 \, 625
| r = 181 \times 5^3
| c =
}}
{{eqn | l = 22 \, 626
| r = 838 \times 3^3
| c =
}}
{{eqn | l = 22 \, 627
| r = 17 \times 11^3
| c =
}}
{{end-eqn}}
{{ProofWanted|It remains to be s... | Smallest Quadruplet of Consecutive Integers Divisible by Cube | https://proofwiki.org/wiki/Smallest_Quadruplet_of_Consecutive_Integers_Divisible_by_Cube | https://proofwiki.org/wiki/Smallest_Quadruplet_of_Consecutive_Integers_Divisible_by_Cube | [
"Cube Numbers"
] | [
"Definition:Sequence",
"Definition:Ordered Tuple as Ordered Set/Ordered Quadruple",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Cube Number"
] | [] |
proofwiki-13485 | Riemann Zeta Function as a Multiple Integral | For $n \in \Z_{> 0}$, the Riemann zeta function is given by:
:$\ds \map \zeta n = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i$
where $\closedint 0 1^n$ denotes the Cartesian $n$th power of the closed real interval $\closedint 0 1$. | {{begin-eqn}}
{{eqn | l = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i
| r = \int_{\closedint 0 1^n} \sum_{j \mathop = 1}^\infty \paren {\prod_{i \mathop = 1}^n x_i}^{j - 1} \prod_{i \mathop = 1}^n \rd x_i
| c = Sum of Infinite Geometric Sequence
}}
{{e... | For $n \in \Z_{> 0}$, the [[Definition:Riemann Zeta Function|Riemann zeta function]] is given by:
:$\ds \map \zeta n = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i$
where $\closedint 0 1^n$ denotes the [[Definition:Cartesian Space|Cartesian $n$th power]] of the [[D... | {{begin-eqn}}
{{eqn | l = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i
| r = \int_{\closedint 0 1^n} \sum_{j \mathop = 1}^\infty \paren {\prod_{i \mathop = 1}^n x_i}^{j - 1} \prod_{i \mathop = 1}^n \rd x_i
| c = [[Sum of Infinite Geometric Sequence]]
}}... | Riemann Zeta Function as a Multiple Integral | https://proofwiki.org/wiki/Riemann_Zeta_Function_as_a_Multiple_Integral | https://proofwiki.org/wiki/Riemann_Zeta_Function_as_a_Multiple_Integral | [
"Riemann Zeta Function",
"Analytic Number Theory"
] | [
"Definition:Riemann Zeta Function",
"Definition:Cartesian Product/Cartesian Space",
"Definition:Real Interval/Closed"
] | [
"Sum of Infinite Geometric Sequence",
"Integral of Series of Positive Measurable Functions",
"Fubini's Theorem",
"Fubini's Theorem",
"Integral of Power",
"Category:Riemann Zeta Function",
"Category:Analytic Number Theory"
] |
proofwiki-13486 | Closed Form for Hexagonal Pyramidal Numbers | The closed-form expression for the $n$th hexagonal pyramidal number is:
:$S_n = \dfrac {n \paren {n + 1} \paren {4 n - 1} } 6$ | {{begin-eqn}}
{{eqn | l = S_n
| r = \sum_{k \mathop = 1}^n H_k
| c = {{Defof|Hexagonal Pyramidal Number}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n k \paren {2 k - 1}
| c = Closed Form for Hexagonal Numbers
}}
{{eqn | r = 3 \sum_{k \mathop = 1}^n 2 k^2 - \sum_{k \mathop = 1}^n k
| c =
}}
{{eqn | r ... | The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Hexagonal Pyramidal Number|hexagonal pyramidal number]] is:
:$S_n = \dfrac {n \paren {n + 1} \paren {4 n - 1} } 6$ | {{begin-eqn}}
{{eqn | l = S_n
| r = \sum_{k \mathop = 1}^n H_k
| c = {{Defof|Hexagonal Pyramidal Number}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n k \paren {2 k - 1}
| c = [[Closed Form for Hexagonal Numbers]]
}}
{{eqn | r = 3 \sum_{k \mathop = 1}^n 2 k^2 - \sum_{k \mathop = 1}^n k
| c =
}}
{{eqn ... | Closed Form for Hexagonal Pyramidal Numbers | https://proofwiki.org/wiki/Closed_Form_for_Hexagonal_Pyramidal_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Hexagonal_Pyramidal_Numbers | [
"Closed Forms",
"Pyramidal Numbers"
] | [
"Definition:Closed Form Expression",
"Definition:Hexagonal Pyramidal Number"
] | [
"Closed Form for Hexagonal Numbers",
"Sum of Sequence of Squares",
"Closed Form for Triangular Numbers",
"Category:Closed Forms",
"Category:Pyramidal Numbers"
] |
proofwiki-13487 | Tetrahedral and Triangular Numbers | The only positive integers which are simultaneously tetrahedral and triangular are:
:$1, 10, 120, 1540, 7140$ | {{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6
| c = Closed Form for Tetrahedral Numbers
}}
{{eqn | r = \dfrac {1 \times \paren {1 + 1} } 2
| c = Closed Form for Triangular Numbers
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 10
| r = \dfrac {3 \paren {3 + 1} \paren {3... | The only [[Definition:Positive Integer|positive integers]] which are simultaneously [[Definition:Tetrahedral Number|tetrahedral]] and [[Definition:Triangular Number|triangular]] are:
:$1, 10, 120, 1540, 7140$ | {{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6
| c = [[Closed Form for Tetrahedral Numbers]]
}}
{{eqn | r = \dfrac {1 \times \paren {1 + 1} } 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 10
| r = \dfrac {3 \paren {3 + 1}... | Tetrahedral and Triangular Numbers | https://proofwiki.org/wiki/Tetrahedral_and_Triangular_Numbers | https://proofwiki.org/wiki/Tetrahedral_and_Triangular_Numbers | [
"Triangular Numbers",
"Pyramidal Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Tetrahedral Number",
"Definition:Triangular Number"
] | [
"Closed Form for Tetrahedral Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Tetrahedral Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Tetrahedral Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Tetrahedral Numbers",
"Closed Form for Triangular Numb... |
proofwiki-13488 | Restriction of Composition is Composition of Restriction | Let $X, Y, Z$ be sets.
Let $f: X \to Y$ and $g: Y \to Z$ be mappings.
Let $S \subseteq X$.
Then:
:$\paren {g \circ f} \restriction S = g \circ \paren {f \restriction S}$ | By definitions of composition of mappings and restriction of mapping:
:$\paren {g \circ f} \restriction S: S \to Z$ and $g \circ \paren {f \restriction S}: S \to Z$
Let $s \in S$.
By definition of restriction of mapping:
:$\map {\paren {\paren {g \circ f} \restriction S} } s = \map {\paren {g \circ f} } s$
Thus
{{begin... | Let $X, Y, Z$ be [[Definition:Set|sets]].
Let $f: X \to Y$ and $g: Y \to Z$ be [[Definition:Mapping|mappings]].
Let $S \subseteq X$.
Then:
:$\paren {g \circ f} \restriction S = g \circ \paren {f \restriction S}$ | By definitions of [[Definition:Composition of Mappings|composition of mappings]] and [[Definition:Restriction of Mapping|restriction of mapping]]:
:$\paren {g \circ f} \restriction S: S \to Z$ and $g \circ \paren {f \restriction S}: S \to Z$
Let $s \in S$.
By definition of [[Definition:Restriction of Mapping|restrict... | Restriction of Composition is Composition of Restriction | https://proofwiki.org/wiki/Restriction_of_Composition_is_Composition_of_Restriction | https://proofwiki.org/wiki/Restriction_of_Composition_is_Composition_of_Restriction | [
"Mapping Theory"
] | [
"Definition:Set",
"Definition:Mapping"
] | [
"Definition:Composition of Mappings",
"Definition:Restriction/Mapping",
"Definition:Restriction/Mapping"
] |
proofwiki-13489 | Odd Numbers not Sum of Prime and Power | The sequence of odd numbers which cannot be expressed as the sum of a perfect power and a prime number begins:
:$1, 5, 1549, 1 \, 771 \, 561, \ldots$
{{OEIS|A119747}}
It is not known if there are any more terms. | The cases $1$ and $5$ are trivial.
Now we show that $1549 - a^b$ is never prime for $a \ge 1$ and $b \ge 2$.
It suffices to show the result for prime values of $b$.
We first prove:
{{begin-eqn}}
{{eqn | n = 1
| l = 2
| o = \divides
| r = a
}}
{{eqn | n = 2
| l = 3
| o = \divides
| r ... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Odd Integer|odd numbers]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of a [[Definition:Perfect Power|perfect power]] and a [[Definition:Prime Number|prime number]] begins:
:$1, 5, 1549, 1 \, 771 \, 561, \ldots$
{{OEIS|A119747}}
It i... | The cases $1$ and $5$ are trivial.
Now we show that $1549 - a^b$ is never [[Definition:Prime Number|prime]] for $a \ge 1$ and $b \ge 2$.
It suffices to show the result for [[Definition:Prime Number|prime]] values of $b$.
We first prove:
{{begin-eqn}}
{{eqn | n = 1
| l = 2
| o = \divides
| r = a
}}
... | Odd Numbers not Sum of Prime and Power | https://proofwiki.org/wiki/Odd_Numbers_not_Sum_of_Prime_and_Power | https://proofwiki.org/wiki/Odd_Numbers_not_Sum_of_Prime_and_Power | [
"Powers",
"Prime Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Odd Integer",
"Definition:Addition/Integers",
"Definition:Perfect Power",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Prime Number",
"Definition:Perfect Power",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Square Number",
"Definition:Prime Number",
"Definition:... |
proofwiki-13490 | One-Digit Number is Harshad | Let $n$ be a $1$-digit positive integer.
Then $n$ is a harshad number. | By definition, a harshad number is divisible by the sum of its digits base $10$.
Let $n$ be a $1$-digit positive integer.
The sum of the digits of $n$ is trivially $n$.
The result follows from Integer Divides Itself.
{{qed}}
Category:Harshad Numbers
kxykvvanwxcz0ibl71hfbpqgniu0v7p | Let $n$ be a [[Definition:Digit|$1$-digit]] [[Definition:Positive Integer|positive integer]].
Then $n$ is a [[Definition:Harshad Number|harshad number]]. | By definition, a [[Definition:Harshad Number|harshad number]] is [[Definition:Divisor of Integer|divisible]] by the [[Definition:Integer Addition|sum]] of its [[Definition:Digit|digits]] [[Definition:Decimal Notation|base $10$]].
Let $n$ be a [[Definition:Digit|$1$-digit]] [[Definition:Positive Integer|positive intege... | One-Digit Number is Harshad | https://proofwiki.org/wiki/One-Digit_Number_is_Harshad | https://proofwiki.org/wiki/One-Digit_Number_is_Harshad | [
"Harshad Numbers"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Harshad Number"
] | [
"Definition:Harshad Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Decimal Notation",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Digit",
"Integer Divisor Results/Integer Divides It... |
proofwiki-13491 | Smallest Fermat Pseudoprime to Bases 2, 3 and 5 | The smallest Fermat pseudoprime to bases $2$, $3$ and $5$ is $1729$. | {{ProofWanted|We have the list of Poulet numbers and Fermat pseudoprimes base $3$, but not of base $5$. Once we get that list, we can find the numbers on the list for both.}} | The smallest [[Definition:Fermat Pseudoprime|Fermat pseudoprime]] to bases $2$, $3$ and $5$ is $1729$. | {{ProofWanted|We have the list of [[Definition:Poulet Number|Poulet numbers]] and [[Definition:Fermat Pseudoprime/Base 3|Fermat pseudoprimes base $3$]], but not of base $5$. Once we get that list, we can find the numbers on the list for both.}} | Smallest Fermat Pseudoprime to Bases 2, 3 and 5 | https://proofwiki.org/wiki/Smallest_Fermat_Pseudoprime_to_Bases_2,_3_and_5 | https://proofwiki.org/wiki/Smallest_Fermat_Pseudoprime_to_Bases_2,_3_and_5 | [
"Fermat Pseudoprimes",
"1729"
] | [
"Definition:Fermat Pseudoprime"
] | [
"Definition:Poulet Number",
"Definition:Fermat Pseudoprime/Base 3"
] |
proofwiki-13492 | Numbers that Factorise into Sum of Digits and Reversal | The following positive integers can each be expressed as the product of the sum of its digits and the reversal of the sum of its digits:
:$1, 81, 1458, 1729$
{{OEIS|A110921}} | {{begin-eqn}}
{{eqn | l = 1
| r = 1 \times 1
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 81
| r = 9 \times 9
}}
{{eqn | r = 9 \times \paren {8 + 1}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1458
| r = 81 \times 18
}}
{{eqn | r = 81 \times \paren {1 + 4 + 5 + 8}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 172... | The following [[Definition:Positive Integer|positive integers]] can each be expressed as the [[Definition:Integer Multiplication|product]] of the [[Definition:Integer Addition|sum]] of its [[Definition:Digit|digits]] and the [[Definition:Reversal|reversal]] of the [[Definition:Integer Addition|sum]] of its [[Definition... | {{begin-eqn}}
{{eqn | l = 1
| r = 1 \times 1
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 81
| r = 9 \times 9
}}
{{eqn | r = 9 \times \paren {8 + 1}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1458
| r = 81 \times 18
}}
{{eqn | r = 81 \times \paren {1 + 4 + 5 + 8}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l... | Numbers that Factorise into Sum of Digits and Reversal | https://proofwiki.org/wiki/Numbers_that_Factorise_into_Sum_of_Digits_and_Reversal | https://proofwiki.org/wiki/Numbers_that_Factorise_into_Sum_of_Digits_and_Reversal | [
"Fermat Pseudoprimes",
"1729"
] | [
"Definition:Positive/Integer",
"Definition:Multiplication/Integers",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Reversal",
"Definition:Addition/Integers",
"Definition:Digit"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Reversal",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Integer",
"Equal Numbers are Congruent",
"Congruence of Sum of Digits to Base Less 1",
"Definition:Digit",
"Definition:Integer... |
proofwiki-13493 | 1782 is 3 Times Sum of all 2-Digit Numbers from its Digits | $1782$ equals $3$ multiplied by the sum of all the $2$-digit integers that can be formed from its digits. | The number of $2$-digit integers that can be formed from the digits of $1782$ equals the number of $2$-permutations of $\set {1, 7, 8, 2}$.
That is:
:$\set {17, 18, 12, 71, 78, 72, 81, 87, 82, 21, 27, 28}$
Hence:
:$17 + 18 + 12 + 71 + 78 + 72 + 81 + 87 + 82 + 21 + 27 + 28 = 594 = \dfrac {1782} 3$
{{qed}} | $1782$ equals $3$ [[Definition:Integer Multiplication|multiplied by]] the [[Definition:Integer Addition|sum]] of all the [[Definition:Digit|$2$-digit]] [[Definition:Integer|integers]] that can be formed from its [[Definition:Digit|digits]]. | The number of [[Definition:Digit|$2$-digit]] [[Definition:Integer|integers]] that can be formed from the [[Definition:Digit|digits]] of $1782$ equals the number of [[Definition:Permutation (Ordered Selection)|$2$-permutations]] of $\set {1, 7, 8, 2}$.
That is:
:$\set {17, 18, 12, 71, 78, 72, 81, 87, 82, 21, 27, 28}$
... | 1782 is 3 Times Sum of all 2-Digit Numbers from its Digits | https://proofwiki.org/wiki/1782_is_3_Times_Sum_of_all_2-Digit_Numbers_from_its_Digits | https://proofwiki.org/wiki/1782_is_3_Times_Sum_of_all_2-Digit_Numbers_from_its_Digits | [
"Recreational Mathematics",
"1782"
] | [
"Definition:Multiplication/Integers",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Integer",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Integer",
"Definition:Digit",
"Definition:Permutation/Ordered Selection"
] |
proofwiki-13494 | Triple of Consecutive Happy Numbers | The smallest triple of consecutive integers all of which are happy is:
:$\left({1880, 1881, 1882}\right)$ | {{begin-eqn}}
{{eqn | o =
| r = 1880
| c =
}}
{{eqn | ll= \leadsto
| l = 1^2 + 8^2 + 8^2 + 0^2
| r = 1 + 64 + 64 + 0
| c =
}}
{{eqn | r = 129
| c =
}}
{{eqn | ll= \leadsto
| l = 1^2 + 2^2 + 9^2
| r = 1 + 4 + 81
| c =
}}
{{eqn | r = 86
| c =
}}
{{eqn | ll... | The smallest [[Definition:Ordered Triple|triple]] of consecutive [[Definition:Integer|integers]] all of which are [[Definition:Happy Number|happy]] is:
:$\left({1880, 1881, 1882}\right)$ | {{begin-eqn}}
{{eqn | o =
| r = 1880
| c =
}}
{{eqn | ll= \leadsto
| l = 1^2 + 8^2 + 8^2 + 0^2
| r = 1 + 64 + 64 + 0
| c =
}}
{{eqn | r = 129
| c =
}}
{{eqn | ll= \leadsto
| l = 1^2 + 2^2 + 9^2
| r = 1 + 4 + 81
| c =
}}
{{eqn | r = 86
| c =
}}
{{eqn | ll... | Triple of Consecutive Happy Numbers | https://proofwiki.org/wiki/Triple_of_Consecutive_Happy_Numbers | https://proofwiki.org/wiki/Triple_of_Consecutive_Happy_Numbers | [
"Happy Numbers"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Integer",
"Definition:Happy Number"
] | [
"Definition:Happy Number",
"Definition:Happy Number",
"Definition:Happy Number"
] |
proofwiki-13495 | Numbers whose Digits are Unchanged when Subtracting Reversal | The following sequence consists of the integers which have the property that subtraction of their reversals results in anagrams of them:
:$954, 1980, 2961, 3870, 5823, 7641, 9108, 19980, 29880, 29961, 32760, \ldots$
{{OEIS|A121969}} | {{begin-eqn}}
{{eqn | l = 954 - 459
| r = 495
}}
{{eqn | l = 1980 - 0891
| r = 1089
}}
{{eqn | l = 2961 - 1692
| r = 1269
}}
{{eqn | l = 3870 - 0783
| r = 3087
}}
{{eqn | l = 5823 - 3285
| r = 2538
}}
{{eqn | l = 7641 - 1467
| r = 6174
}}
{{eqn | l = 9108 - 8019
| r = 1089
}}
{... | The following [[Definition:Integer Sequence|sequence]] consists of the [[Definition:Integer|integers]] which have the property that [[Definition:Integer Subtraction|subtraction]] of their [[Definition:Reversal|reversals]] results in [[Definition:Anagram|anagrams]] of them:
:$954, 1980, 2961, 3870, 5823, 7641, 9108, 199... | {{begin-eqn}}
{{eqn | l = 954 - 459
| r = 495
}}
{{eqn | l = 1980 - 0891
| r = 1089
}}
{{eqn | l = 2961 - 1692
| r = 1269
}}
{{eqn | l = 3870 - 0783
| r = 3087
}}
{{eqn | l = 5823 - 3285
| r = 2538
}}
{{eqn | l = 7641 - 1467
| r = 6174
}}
{{eqn | l = 9108 - 8019
| r = 1089
}}
{... | Numbers whose Digits are Unchanged when Subtracting Reversal | https://proofwiki.org/wiki/Numbers_whose_Digits_are_Unchanged_when_Subtracting_Reversal | https://proofwiki.org/wiki/Numbers_whose_Digits_are_Unchanged_when_Subtracting_Reversal | [
"Reversals",
"Anagrams"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Subtraction/Integers",
"Definition:Reversal",
"Definition:Anagram"
] | [] |
proofwiki-13496 | Sequence of Composite Mersenne Numbers | The sequence of Mersenne numbers which are composite begins:
:$2047, 8 \, 388 \, 607, 536 \, 870 \, 911, 137 \, 438 \, 953 \, 471, 2 \, 199 \, 023 \, 255 \, 551,\ldots$
{{OEIS|A065341}}
The sequence of corresponding indices $p$ such that $2^p - 1$ is composite begins:
:$11, 23, 29, 37, 41, 43, 47, 53, 59, 67, 71, 73, 7... | Established by inspecting the sequence of Mersenne numbers:
:$3, 7, 31, 127, 2047, 8191, 131 \, 071, 524 \, 287, 8 \, 388 \, 607, 536 \, 870 \, 911, 2 \, 147 \, 483 \, 647, \ldots$
and removing from it the sequence of Mersenne primes:
:$3, 7, 31, 127, 8191, 131 \, 071, 524 \, 287, 2 \, 147 \, 483 \, 647, \ldots$
{{qed}... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Mersenne Number|Mersenne numbers]] which are [[Definition:Composite Number|composite]] begins:
:$2047, 8 \, 388 \, 607, 536 \, 870 \, 911, 137 \, 438 \, 953 \, 471, 2 \, 199 \, 023 \, 255 \, 551,\ldots$
{{OEIS|A065341}}
The [[Definition:Integer Sequence|sequ... | Established by inspecting the [[Definition:Mersenne Number/Sequence|sequence of Mersenne numbers]]:
:$3, 7, 31, 127, 2047, 8191, 131 \, 071, 524 \, 287, 8 \, 388 \, 607, 536 \, 870 \, 911, 2 \, 147 \, 483 \, 647, \ldots$
and removing from it the [[Mersenne Prime/Current Status|sequence of Mersenne primes]]:
:$3, 7, 31... | Sequence of Composite Mersenne Numbers | https://proofwiki.org/wiki/Sequence_of_Composite_Mersenne_Numbers | https://proofwiki.org/wiki/Sequence_of_Composite_Mersenne_Numbers | [
"Mersenne Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Mersenne Number",
"Definition:Composite Number",
"Definition:Integer Sequence",
"Definition:Composite Number",
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Prime Number",
"Definition:Composite Number"
] | [
"Definition:Mersenne Number/Sequence",
"Mersenne Prime/Current Status"
] |
proofwiki-13497 | Numbers Reversed when Multiplying by 4 | Numbers of the form $\sqbrk {21 \paren 9 78}_{10}$ are reversed when they are multiplied by $4$:
{{begin-eqn}}
{{eqn | l = 2178 \times 4
| r = 8712
}}
{{eqn | l = 21 \, 978 \times 4
| r = 87 \, 912
}}
{{eqn | l = 219 \, 978 \times 4
| r = 879 \, 912
}}
{{end-eqn}}
and so on. | Let k represent the number of $9$s in the middle of the number.
For $k > 0$ We can rewrite the number as follows:
{{begin-eqn}}
{{eqn | l = \sqbrk {21 (9) 78}_{10}
| r = 21 \times 10^{k + 2 } + 900 \sum_{i \mathop = 0}^{k - 1} 10^i + 78
| c = {{Defof|Geometric Series}}
}}
{{end-eqn}}
Taking numbers of this ... | Numbers of the form $\sqbrk {21 \paren 9 78}_{10}$ are [[Definition:Reversal|reversed]] when they are [[Definition:Integer Multiplication|multiplied]] by $4$:
{{begin-eqn}}
{{eqn | l = 2178 \times 4
| r = 8712
}}
{{eqn | l = 21 \, 978 \times 4
| r = 87 \, 912
}}
{{eqn | l = 219 \, 978 \times 4
| r = ... | Let k represent the number of $9$s in the middle of the number.
For $k > 0$ We can rewrite the number as follows:
{{begin-eqn}}
{{eqn | l = \sqbrk {21 (9) 78}_{10}
| r = 21 \times 10^{k + 2 } + 900 \sum_{i \mathop = 0}^{k - 1} 10^i + 78
| c = {{Defof|Geometric Series}}
}}
{{end-eqn}}
Taking numbers of th... | Numbers Reversed when Multiplying by 4 | https://proofwiki.org/wiki/Numbers_Reversed_when_Multiplying_by_4 | https://proofwiki.org/wiki/Numbers_Reversed_when_Multiplying_by_4 | [
"Reversals",
"2178"
] | [
"Definition:Reversal",
"Definition:Multiplication/Integers"
] | [
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit"
] |
proofwiki-13498 | 17 Consecutive Integers each with Common Factor with Product of other 16 | The $17$ consecutive integers from $2184$ to $2200$ have the property that each one is not coprime with the product of the other $16$. | We obtain the prime decomposition of all $17$ of these integers:
{{begin-eqn}}
{{eqn | l = 2184
| r = 2^3 \times 3 \times 7 \times 13
}}
{{eqn | l = 2185
| r = 5 \times 19 \times 23
}}
{{eqn | l = 2186
| r = 2 \times 1093
}}
{{eqn | l = 2187
| r = 3^7
}}
{{eqn | l = 2188
| r = 2^2 \times 5... | The $17$ consecutive [[Definition:Integer|integers]] from $2184$ to $2200$ have the property that each one is not [[Definition:Coprime Integers|coprime]] with the [[Definition:Integer Multiplication|product]] of the other $16$. | We obtain the [[Definition:Prime Decomposition|prime decomposition]] of all $17$ of these [[Definition:Integer|integers]]:
{{begin-eqn}}
{{eqn | l = 2184
| r = 2^3 \times 3 \times 7 \times 13
}}
{{eqn | l = 2185
| r = 5 \times 19 \times 23
}}
{{eqn | l = 2186
| r = 2 \times 1093
}}
{{eqn | l = 2187
... | 17 Consecutive Integers each with Common Factor with Product of other 16 | https://proofwiki.org/wiki/17_Consecutive_Integers_each_with_Common_Factor_with_Product_of_other_16 | https://proofwiki.org/wiki/17_Consecutive_Integers_each_with_Common_Factor_with_Product_of_other_16 | [
"Recreational Mathematics"
] | [
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Multiplication/Integers"
] | [
"Definition:Prime Decomposition",
"Definition:Integer",
"Definition:Integer",
"Definition:Integer Sequence",
"Definition:Prime Factor",
"Definition:Integer Sequence"
] |
proofwiki-13499 | Relational Structure admits Lower Topology | Let $R = \left({S, \preceq}\right)$ be a relational structure.
Then there exists a relational structure with lower topology $T = \left({S, \preceq, \tau}\right)$ such that $T$ is a topological space. | Define $B := \left\{ {\complement_S\left({x^\succeq}\right): x \in S}\right\}$
where $x^\succeq$ denotes the upper closure of $x$.
By definition of generated topology:
:$\tau\left({B}\right)$ is a topology on $S$
where $B$ is a sub-basis of $\tau \left({B}\right)$.
Thus by definition of lower topology:
:$T := \left({S,... | Let $R = \left({S, \preceq}\right)$ be a [[Definition:Relational Structure|relational structure]].
Then there exists a [[Definition:Relational Structure with Topology|relational structure]] with [[Definition:Lower Topology|lower topology]] $T = \left({S, \preceq, \tau}\right)$ such that $T$ is a [[Definition:Topologi... | 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 $x$.
By definition of [[Definition:Topology Generated by Synthetic Sub-Basis|generated topology]]:
:$\tau\left({B}\right)$ is a [[Definition:Topology|to... | Relational Structure admits Lower Topology | https://proofwiki.org/wiki/Relational_Structure_admits_Lower_Topology | https://proofwiki.org/wiki/Relational_Structure_admits_Lower_Topology | [
"Topological Order Theory"
] | [
"Definition:Relational Structure",
"Definition:Relational Structure with Topology",
"Definition:Lower Topology",
"Definition:Topological Space"
] | [
"Definition:Upper Closure/Element",
"Definition:Topology Generated by Synthetic Sub-Basis",
"Definition:Topology",
"Definition:Sub-Basis/Synthetic Sub-Basis",
"Definition:Lower Topology",
"Definition:Lower Topology",
"Definition:Topological Space"
] |
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