nyuuzyou/wikis huggingface dataset:
3069

== Number ==
$3069$ (three thousand and sixty-nine) is:
:$3^2 \times 11 \times 31$
:The $17$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$, $105$, $160$, $661$, $2653$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$ is prime
== Also see ==
* 
Category:Specific Numbers
Category:3069
307

== Number ==
$307$ (three hundred and seven) is:
:The $63$rd prime number
:The larger of the $2$nd pair of primes whose prime gap is $14$:
::$307 - 293 = 14$
:The $3$rd non-palindromic square root after $26$, $264$ of a palindromic square:
::$307^2 = 94 \, 249$
== Also see ==
* 
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Category:Specific Numbers
Category:307
3071

== Number ==
$3071$ (three thousand and seventy-one) is:
:$37 \times 83$
:The $10$th Thabit number after $(2)$, $5$, $11$, $23$, $47$, $95$, $191$, $383$, $767$, $1535$:
::$3071 = 3 \times 2^{10} - 1$ 
== Also see ==
* 
Category:Specific Numbers
Category:3071
3078

== Number ==
$3078$ (three thousand and seventy-eight) is:
:$2 \times 3^4 \times 19$
:The $18$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $126$, $196$, $288$, $405$, $550$, $726$, $936$, $1183$, $1470$, $1800$, $2176$, $2601$:
::$3078 = \ds \sum_{k \mathop = 1}^{18} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {18^2 \paren {18 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:3078
308

== Number ==
$308$ (three hundred and eight) is:
:$2^2 \times 7 \times 11$
:The $7$th heptagonal pyramidal number after $1$, $8$, $26$, $60$, $115$, $196$:
::$308 = 1 + 7 + 18 + 34 + 55 + 81 + 112 = \dfrac {7 \paren {7 + 1} \paren {5 \times 7 - 2} } 6$
:The $51$st nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 308$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:308
3081

== Number ==
$3081$ (three thousand and eighty-one) is:
:$3 \times 13 \times 79$
:The $78$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $2211$, $2278$, $2346$, $2415$, $2485$, $2556$, $2628$, $2701$, $2775$, $2850$, $2926$, $3003$:
::$3081 = \ds \sum_{k \mathop = 1}^{78} k = \dfrac {78 \times \paren {78 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:3081
31

== Number ==
$31$ (thirty-one) is:
:The $11$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$
:The $1$st of the $2$ known numbers (with $8191$) expressible as the sum of successive powers starting from $1$ in in $2$ different ways:
::$31 = 1 + 5 + 5^2 = 1 + 2 + 2^2 + 2^3 + 2^4$
:The $2$nd of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 1^2 + 29 = 31$ ()
:The $2$nd prime number after $3$ that can be found starting from the beginning of the decimal expansion of $\pi$ (pi):
::$3 \cdotp 1 (4159 \, 26535 \, 897 \ldots)$
:The $3$rd emirp after $13$, $17$
:The $3$rd Mersenne number and the $3$rd Mersenne prime after $3$, $7$:
::$31 = 2^5 - 1$
:The $3$rd Euclid number after $3$, $7$:
::$31 = p_3\# + 1 = 2 \times 3 \times 5 + 1$
:The $4$th Euclid prime after $2$, $3$, $7$:
::$31 = p_3\# + 1 = 2 \times 3 \times 5 + 1$
:The $4$th after $2$, $3$, $19$ of the $4$ known primes $p$ such that $\dfrac {p^p - 1} {p - 1}$ is itself prime:
::$\dfrac {31^{31} - 1} {31 - 1} = 568 \, 972 \, 471 \, 024 \, 107 \, 865 \, 287 \, 021 \, 434 \, 301 \, 977 \, 158 \, 534 \, 824 \, 481$
:The larger of the $5$th pair of twin primes, with $29$
:The $6$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$
:The $7$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$
:The $8$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$
:The $8$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$:
::$31 \to 3^2 + 1^2 = 9 + 1 = 10 \to 1^2 + 0^2 = 1$
:The index of the $8$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$:
::$M_{31} = 2^{31} - 1 = 2 \, 147 \, 483 \, 647$
:The $9$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $\ldots$
:The index of the $9$th Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$, $17$, $19$ which asserted to be prime
:The $16$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $\ldots$
:The $16$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $\ldots$
:The $20$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$, $16$, $19$, $20$, $21$, $24$, $25$, $26$, $29$, $30$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $21$st integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{31} = 2 \, 147 \, 483 \, 648$
:The $23$rd (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $20$, $21$, $24$, $25$, $26$, $27$, $30$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* Square of Reversal of Small-Digit Number
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== Sources ==
* : $5$
* : $31$
* : $5$
* : $31$
Category:Specific Numbers
Category:31
31,695,652,275

== Number ==
$31 \, 695 \, 652 \, 275$ is:
:$3^3 \times 5^2 \times 11 \times 13 \times 431 \times 719$
:The smaller of the $2$nd of a set of $3$ amicable pairs whose sum is $64 \, 795 \, 852 \, 800$:
::$\map {\sigma_1} {31 \, 695 \, 652 \, 275} = \map {\sigma_1} {33 \, 100 \, 200 \, 525} = 64 \, 795 \, 852 \, 800 = 31 \, 695 \, 652 \, 275 + 33 \, 100 \, 200 \, 525$
== Also see ==
* 
Category:Specific Numbers
Category:31,695,652,275
31,704

== Number ==
$31 \, 704$ (thirty-one thousand, seven hundred and four) is:
:$2^3 \times 3 \times 1321$
:The $3$rd term of a sociable chain of order $28$ whose smallest term is $14 \, 316$:
::$14 \, 316$, $19 \, 116$, $31 \, 704$, $47 \, 616$, $83 \, 328$, $177 \, 792$, $295 \, 488$, $629 \, 072$, $589 \, 786$, $294 \, 896$, $358 \, 336$, $418 \, 904$, $366 \, 556$, $274 \, 924$, $275 \, 444$, $243 \, 760$, $376 \, 736$, $381 \, 028$, $285 \, 778$, $152\, 990$, $122 \, 410$, $97 \, 946$, $48 \, 976$, $45 \, 946$, $22 \, 976$, $22 \, 744$, $19 \, 916$, $17 \, 716$, $14 \, 316$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:31,704
310

== Number ==
$310$ (three hundred and ten) is:
:$2 \times 5 \times 31$
:The $16$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$, $115$, $119$, $170$, $210$, $214$, $217$, $265$, $282$ whose divisor sum is square:
:
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $19$ different ways
:The $31$st noncototient after $10$, $26$, $34$, $50$, $\ldots$, $244$, $260$, $266$, $268$, $274$, $290$, $292$, $298$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 310$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $33$rd sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $246$, $255$, $258$, $266$, $273$, $282$, $285$, $286$, $290$:
::$310 = 2 \times 5 \times 31$
:The $47$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$, $302$:
::$310 \to 3^2 + 1^2 + 0^2 = 9 + 1 + 0 = 10 \to 1^2 + 0^2 = 1$
== Also see ==
* 
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Category:Specific Numbers
Category:310
311

== Number ==
$311$ (three hundred and eleven) is:
:The $64$th prime number
:The $14$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$, $227$, $229$, $233$, $277$
:The $16$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$, $157$, $167$, $179$, $199$
:The $17$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$, $233$, $239$, $293$
:The $17$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $113$, $131$, $199$
:The smaller of the $20$th pair of twin primes, with $313$
== Also see ==
* 
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Category:Specific Numbers
Category:311
312

== Number ==
$312$ (three hundred and twelve) is:
:$2^3 \times 3 \times 13$
:The $26$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $135$, $144$, $175$, $212$, $216$, $224$:
::$312 = 52 \times 6 = 52 \times \left({3 \times 1 \times 2}\right)$
== Also see ==
* 
Category:Specific Numbers
Category:312
3125

== Number ==
$3125$ (three thousand, one hundred and twenty-five) is:
:$5^5$
:The $5$th fifth power after $1$, $32$, $243$, $1024$:
::$3125 = 5 \times 5 \times 5 \times 5 \times 5$
:The $5$th power of $5$ after $(1)$, $5$, $25$, $125$, $625$:
::$3125 = 5^5$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3125
3126

== Number ==
$3126$ (three thousand, one hundred and twenty-six) is:
:$2 \times 3 \times 521$
:The $5$th Sierpiński number of the first kind after $2$, $5$, $28$, $257$:
::$3126 = 5^5 + 1$
== Also see ==
* 
Category:Specific Numbers
Category:3126
313

== Number ==
$313$ (three hundred and thirteen) is:
:The $65$th prime number
:The $4$th prime number after $3$, $5$, $7$ which is palindromic in both decimal and binary:
::$313_{10} = 100 \, 111 \, 001_2$
:The lower end of the $6$th record-breaking gap between twin primes:
::$347 - 313 = 34$
:The $9$th integer after $0$, $1$, $3$, $5$, $7$, $9$, $33$, $99$ which is palindromic in both decimal and binary:
::$313_{10} = 100 \, 111 \, 001_2$
:The $9$th two-sided prime after $2$, $3$, $5$, $7$, $23$, $37$, $53$, $73$:
::$313$ is prime; $31$, $3$ are prime; $13$, $3$ are prime
:The $11$th palindromic prime:
::$2$, $3$, $5$, $7$, $11$, $101$, $131$, $151$, $181$, $191$, $313$, $\ldots$
:The $15$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$, $227$, $229$, $233$, $277$, $311$
:The $18$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$, $233$, $239$, $293$, $311$
:The larger of the $20$th pair of twin primes, with $311$
:The $23$rd left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$, $113$, $137$, $167$, $173$, $197$, $223$, $283$
:The $24$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $181$, $193$, $223$, $229$, $233$, $257$, $263$, $269$
:The $48$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$, $302$, $310$:
::$313 \to 3^2 + 1^2 + 3^2 = 9 + 1 + 9 = 19 \to 1^2 + 9^2 = 1 + 81 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
== Also see ==
* 
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== Sources ==
* : $313$
Category:Specific Numbers
Category:313
3136

== Number ==
$3136$ (three thousand, one hundred and thirty-six) is:
:$2^6 \times 7^2$
:The $21$st square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $1936$, $2304$, $2704$, $2916$ to be the divisor sum value of some (strictly) positive integer:
::$3136 = \map {\sigma_1} {1092} = \map {\sigma_1} {1612} = \map {\sigma_1} {2037} = \map {\sigma_1} {2899} = \map {\sigma_1} {3007}$
:The $46$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $2209$, $2304$, $2500$, $2809$, $2986$, $3136$, $\ldots$
:The $56$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1936$, $2025$, $2116$, $2209$, $2304$, $2401$, $2500$, $2601$, $2704$, $2809$, $2916$, $3025$:
::$3136 = 56 \times 56$
== Also see ==
* 
* 
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Category:Specific Numbers
Category:3136
3137

== Number ==
$3137$ (three thousand, one hundred and thirty-seven) is:
:The $446$th prime number
:The $13$th two-sided prime after $2$, $3$, $5$, $7$, $23$, $37$, $53$, $73$, $313$, $317$, $373$, $797$:
::$3137$ is prime; $313$, $31$, $3$ are prime; $137$, $37$, $7$ are prime
:The $14$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$, $257$, $401$, $577$, $677$, $1297$, $1601$, $2917$:
::$3137 = 56^2 + 1$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:3137
314

== Number ==
$314$ (three hundred and fourteen) is:
:$2 \times 157$
:The $52$nd nontotient:
::$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 314$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
== Also see ==
* 
Category:Specific Numbers
Category:314
314,159

== Number ==
$314 \, 159$ (three hundred and fourteen thousand, one hundred and fifty-nine) is:
:The $27 \, 131$st prime number
:The $3$rd prime number after $3$, $31$ that can be found starting from the beginning of the decimal expansion of $\pi$ (pi):
::$3 \cdotp 14159 \, (26535 \, 897 \ldots)$
== Also see ==
* 
* 
== Sources ==
* : $314,159$
Category:Specific Numbers
Category:314,159
315

== Number ==
$315$ (three hundred and fifteen) is:
:$3^2 \times 5 \times 7$
:The $1$st of the $3$rd ordered triple of consecutive integers after $\tuple {105, 106, 107}$ and $\tuple {165, 166, 167}$ that have Euler $\phi$ values which are strictly increasing:
::$\map \phi {315} = 144$, $\map \phi {316} = 156$, $\map \phi {317} = 316$
:The magic constant of a magic cube of order $5$, after $1$, $(9)$, $42$, $130$:
::$315 = \ds \dfrac 1 {5^2} \sum_{k \mathop = 1}^{5^3} k = \dfrac {5 \paren {5^3 + 1} } 2$
:The $27$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $135$, $144$, $175$, $212$, $216$, $224$, $312$:
::$315 = 21 \times 15 = 21 \times \paren {3 \times 1 \times 5}$
== Also see ==
* 
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Category:Specific Numbers
Category:315
316

== Number ==
$316$ (three hundred and sixteen) is:
:$2^2 \times 79$
:The $2$nd of the $3$rd ordered triple of consecutive integers after $\tuple {105, 106, 107}$ and $\tuple {165, 166, 167}$ that have Euler $\phi$ values which are strictly increasing:
::$\map \phi {315} = 144$, $\map \phi {316} = 156$, $\map \phi {317} = 316$
== Also see ==
* 
Category:Specific Numbers
Category:316
3160

== Number ==
$3160$ (three thousand, one hundred and sixty) is:
:$2^3 \times 5 \times 79$
:The $40$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $2145$, $2278$, $2415$, $2556$, $2701$, $2850$, $3003$:
::$3160 = \ds \sum_{k \mathop = 1}^{40} \paren {4 k - 3} = 40 \paren {2 \times 40 - 1}$
:The $79$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $2701$, $2775$, $2850$, $2926$, $3003$, $3081$:
::$3160 = \ds \sum_{k \mathop = 1}^{79} k = \dfrac {79 \times \paren {79 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3160
3169

== Number ==
$3169$ (three thousand, one hundred and sixty-nine) is:
:The $449$th prime number
:The $33$rd centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $1801$, $1951$, $2107$, $2269$, $2437$, $2611$, $2791$, $2977$:
::$3169 = \ds 1 + \sum_{k \mathop = 1}^{33 - 1} 6 k = 33^3 - 32^3$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3169
317

__NOTOC__
== Number ==
$317$ (three hundred and seventeen) is:
:The $66$th prime number
:The smaller of the $3$rd pair of primes whose prime gap is $14$:
::$331 - 317 = 14$
:The $3$rd of the $3$rd ordered triple of consecutive integers after $\tuple {105, 106, 107}$ and $\tuple {165, 166, 167}$ that have Euler $\phi$ values which are strictly increasing:
::$\map \phi {315} = 144$, $\map \phi {316} = 156$, $\map \phi {317} = 316$
:The index of the $4$th repunit prime after $R_2$, $R_{19}$, $R_{23}$
:The $7$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$3$, $5$, $11$, $13$, $41$, $89$, $317$
:The $10$th two-sided prime after $2$, $3$, $5$, $7$, $23$, $37$, $53$, $73$, $313$:
::$317$ is prime; $31$, $3$ are prime; $17$, $7$ are prime
:The $13$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 12^2 + 29 = 317$ ()
:The $19$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$, $233$, $239$, $293$, $311$, $313$
:The $24$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$, $113$, $137$, $167$, $173$, $197$, $223$, $283$, $313$
== Historical Note ==
== Also see ==
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== Sources ==
* : $11,111,111, \ldots 111,111$
* : $11,111,111, \ldots 111,111$
Category:Specific Numbers
Category:317
317/Mistake

== Source Work ==
:
:The Dictionary
::$11,111,111, \ldots 111,111$
== Mistake ==
:The $3$rd known prime repunit.
== Correction ==
The $4$th known, in fact, after $R_2$, $R_{19}$ and $R_{23}$.
This mistake does not appear in of $1986$, where it states:
:The $4$th and largest known prime repunit.
Some considerably larger ones have been discovered since.
== Sources ==
* : $11,111,111, \ldots 111,111$
Category:Mistakes/Curious and Interesting Numbers
318

== Number ==
$318$ (three hundred and eighteen) is:
:$2 \times 3 \times 53$
:The $34$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $246$, $255$, $258$, $266$, $273$, $282$, $285$, $286$, $290$, $310$:
::$318 = 2 \times 3 \times 53$
:The $53$rd nontotient:
::$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 318$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
::No further elements of this sequence will be tabulated on .
Category:Specific Numbers
Category:318
3186

== Number ==
$3186$ (three thousand, one hundred and eighty-six) is:
:$2 \times 3^3 \times 59$
:The $36$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $2205$, $2356$, $2512$, $2673$, $2839$, $3010$:
::$3186 = \ds \sum_{k \mathop = 1}^{36} \paren {5 k - 4} = \dfrac {36 \paren {5 \times 36 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:3186
319

== Number ==
$319$ (three hundred and nineteen) is:
:$11 \times 29$
:The $4$th positive integer after $79$, $159$, $239$ which cannot be expressed as the sum of fewer than $19$ fourth powers:
::$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$
:or:
::$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$
:The $12$th Smith number after $4$, $22$, $27$, $58$, $85$, $94$, $121$, $166$, $202$, $265$, $274$:
::$3 + 1 + 9 = 1 + 1 + 2 + 9 = 13$
:The $49$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$, $302$, $310$, $313$:
::$319 \to 3^2 + 1^2 + 9^2 = 9 + 1 + 81 = 91 \to 9^2 + 1^2 = 81 + 1 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
== Also see ==
* 
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== Sources ==
* : $319$
* : $319$
Category:Specific Numbers
Category:319
3197

== Number ==
$3197$ (three thousand, one hundred and ninety-seven) is:
:$23 \times 139$
:The $46$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $1962$, $2072$, $2185$, $2301$, $2420$, $2542$, $2667$, $2795$, $2926$, $3060$:
::$3197 = \dfrac {46 \paren {3 \times 46 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:3197
319 is not Expressible as Sum of Fewer than 19 Fourth Powers

== Theorem ==
$319$ cannot be expressed as the sum of fewer than $19$ fourth powers:
:$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$
or:
:$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$
== Proof ==
First note that $5^4 = 625 > 319$.
Then note that $2 \times 4^4 = 512 > 319$.
Hence any expression of $319$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances of $1^4$.
Now we have:
and it can be seen that the first two use the least number of fourth powers, at $19$.
== Also see ==
* Hilbert-Waring Theorem
* Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers
* 159 is not Expressible as Sum of Fewer than 19 Fourth Powers
* 399 is not Expressible as Sum of Fewer than 19 Fourth Powers
== Sources ==
* : $319$
* : $319$
Category:Fourth Powers
Category:Hilbert-Waring Theorem
Category:319
31 is Smallest Prime whose Reciprocal has Odd Period

== Theorem ==
$31$ is the smallest prime number to have a decimal expansion of the reciprocal with an odd period greater than $1$:
:$\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$
== Proof ==
From Reciprocal of $31$:
Counting the digits, it is seen that this has a period of recurrence of $15$, an odd integer.
The prime numbers less than $31$ are $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$.
We investigate the reciprocal of each of these:
and it is seen that none has an odd period greater than $1$.
Hence the result.
== Sources ==
* : $31$
* : $31$
Category:31
Category:Examples of Reciprocals
32

== Number ==
$32$ (thirty-two) is:
:$2^5$
:In binary:
::$100 \, 000$
:The $2$nd fifth power after $1$:
::$32 = 2 \times 2 \times 2 \times 2 \times 2$
:The $2$nd element of the $2$nd pair of integers $m$ whose values of $m \map {\sigma_0} m$ is equal:
::$24 \times \map {\sigma_0} {24} = 192 = 32 \times \map {\sigma_0} {32}$
:The $5$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$:
::$32 = 2^5$
:The $6$th almost perfect number after $1$, $2$, $4$, $8$, $16$:
::$\map {\sigma_1} {32} = 63 = 2 \times 32 - 1$
:The $8$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$
:The $8$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
::$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$
:The $9$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$:
::$32 \to 3^2 + 2^2 = 9 + 4 = 13 \to 1^2 + 3^2 = 9 + 1 = 10 \to 1^2 + 0^2 = 1$
:The $13$th even number after $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $20$, $22$, $26$, $28$ which cannot be expressed as the sum of $2$ composite odd numbers.
:The $17$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $\ldots$
:The $18$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.
:The $21$st positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $22$nd integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{32} = 4 \, 294 \, 967 \, 296$
:The $24$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $20$, $21$, $24$, $25$, $26$, $27$, $30$, $31$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$:There exist $32$ distinct point groups.
:There exist $32$ distinct point groups.
== Also see ==
* 32 Distinct Point Groups
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $32$
* : $32$
* 
* 
Category:Specific Numbers
Category:32
32,129,958,525

== Number ==
$32 \, 129 \, 958 \, 525$ is:
:$3 \times 5^2 \times 7 \times 31 \times 41 \times 179 \times 269$
:The smaller of the $3$rd of a set of $3$ amicable pairs whose sum is $64 \, 795 \, 852 \, 800$:
::$\map {\sigma_1} {32 \, 129 \, 958 \, 525} = \map {\sigma_1} {32 \, 665 \, 894 \, 275} = 64 \, 795 \, 852 \, 800 = 32 \, 129 \, 958 \, 525 + 32 \, 665 \, 894 \, 275$
== Also see ==
* 
Category:Specific Numbers
Category:32,129,958,525
32,223

== Number ==
$32 \, 223$ (thirty-two thousand, two hundred and twenty-three) is:
:$3 \times 23 \times 467$
:The $14$th integer after $0$, $1$, $3$, $5$, $7$, $9$, $33$, $99$, $313$, $585$, $717$, $7447$, $15 \, 351$ which is palindromic in both decimal and binary:
::$32 \, 223_{10} = 111 \, 110 \, 111 \, 011 \, 111_2$
== Also see ==
* 
Category:Specific Numbers
Category:32,223
32,665,894,275

== Number ==
$32 \, 665 \, 894 \, 275$ is:
:$3 \times 5^2 \times 7 \times 31 \times 59 \times 34 \, 019$
:The larger of the $3$rd of a set of $3$ amicable pairs whose sum is $64 \, 795 \, 852 \, 800$:
::$\map {\sigma_1} {32 \, 129 \, 958 \, 525} = \map {\sigma_1} {32 \, 665 \, 894 \, 275} = 64 \, 795 \, 852 \, 800 = 32 \, 129 \, 958 \, 525 + 32 \, 665 \, 894 \, 275$
== Also see ==
* 
Category:Specific Numbers
Category:32,665,894,275