nyuuzyou/wikis huggingface dataset:
40 times Heptagonal Numbers plus 9 gives Squares of Numbers ending in 7

== Theorem ==
Consider the heptagonal numbers:
:$\ds H_n = \sum_{k \mathop = 1}^n \paren {5 k - 4}$
Let $S_n$ be the sequence defined as:
:$\forall n \in \Z_{>1}: S_n = 40 \times H_n + 9$
Then $S_n$ consists of the squares of all the positive integers which end in a $7$:
:$49, 289, 729, 1369, 2209, 3249, 4489, 5929, 7569, \ldots$
that is:
:$7^2, 17^2, 27^2, 37^2, 47^2, 57^2, 67^2, 77^2, 87^2, 97^2, \ldots$
== Proof ==
For $n \in \Z_{>1}$, we have:
The positive integers of the form $10 \paren {n - 1} + 7$ for $n \in \Z_{>1}$ are precisely the positive integers which end in a $7$:
:$7, 17, 27, 37, 47, \ldots$
Category:Heptagonal Numbers
Category:Square Numbers
41

== Number ==
$41$ (forty-one) is:
:The $13$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$
:The $1$st prime number which is not the difference between a power of $2$ and a power of $3$.
:The $4$th integer after $2$, $5$, $17$ at which the prime number race between primes of the form $4 n + 1$ and $4 n - 1$ are tied.
:The $5$th Proth prime after $3$, $5$, $13$, $17$:
::$41 = 5 \times 2^3 + 1$
:The $5$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$3$, $5$, $11$, $13$, $41$
:The $5$th prime $p$ after $11$, $23$, $29$, $37$ such that the Mersenne number $2^p - 1$ is composite
:The smallest positive integer the decimal expansion of whose reciprocal has a period of $5$:
::$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$
:The smaller of the $6$th pair of twin primes, with $43$
:The $6$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$ such that the Fibonacci number $F_n$ ends in $n$
:The $6$th integer after $7$, $13$, $19$, $35$, $38$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$41^2 = 1681$; $16 = 4^2$, $81 = 9^2$
:The $6$th and largest lucky numbers of Euler after $2$, $3$, $5$, $11$, $17$:
::$n^2 + n + 41$ is prime for $0 \le n < 39$.
:The $7$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$:
::$2 \times 41 + 1 = 83$, which is prime.
:The $7$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$
:The $8$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$
:The $10$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime
:The $20$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $\ldots$
== Also see ==
* Smallest Prime Number not Difference between Power of 2 and Power of 3
* Cyclic Permutations of 5-Digit Multiples of 41
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== Sources ==
* : $41$
* : $41$
Category:Specific Numbers
Category:41
41,041

== Number ==
$41 \, 041$ (forty-one thousand and forty-one) is:
:$7 \times 11 \times 13 \times 41$
:The $1$st Carmichael number with $4$ prime factors:
::$41 \, 041 = 7 \times 11 \times 13 \times 41$
:The $11$th Carmichael number after $561$, $1105$, $1729$, $2465$, $2821$, $6601$, $8911$, $10 \, 585$, $15 \, 841$, $29 \, 341$:
::$\forall a \in \Z: a \perp 41 \, 041: a^{41 \, 040} \equiv 1 \pmod {41 \, 041}$
:The $286$th triangular number:
::$41 \, 041 = \ds \sum_{k \mathop = 1}^{286} k = \dfrac {286 \times \paren {286 + 1} } 2$
== Also see ==
* Carmichael Number with 4 Prime Factors
* 
* 
== Sources ==
* : $41,041$
Category:Specific Numbers
Category:41,041
41,328

== Number ==
$41 \, 328$ (forty-one thousand, three hundred and twenty-eight) is:
:$2^4 \times 3^2 \times 7 \times 41$
:The $144$th hexagonal number:
::$41 \, 328 = \ds \sum_{k \mathop = 1}^{144} \paren {4 k - 3} = 144 \paren {2 \times 144 - 1}$
:The $287$th triangular number:
::$41 \, 328 = \ds \sum_{k \mathop = 1}^{287} k = \dfrac {287 \times \paren {287 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:41,328
41,616

== Number ==
$41,616$ (forty-one thousand, six hundred and sixteen) is:
:$2^4 \times 3^2 \times 17^2$
:The $4$th number after $1$, $36$, $1225$ to be both square and triangular:
::$41 \, 616 = 204^2 = \dfrac {288 \times \paren {288 + 1} } 2$
:The $204$th square number:
::$41 \, 616 = 204 \times 204$
:The $288$th triangular number:
::$41 \, 616 = \ds \sum_{k \mathop = 1}^{288} k = \dfrac {288 \times \paren {288 + 1} } 2$
== Also see ==
* 
* 
* 
== Sources ==
* : $1225$
* : $1225$
Category:Specific Numbers
Category:41,616
410

== Number ==
$410$ (four hundred and ten) is:
:$2 \times 5 \times 41$
:The $46$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $290$, $310$, $318$, $322$, $345$, $354$, $357$, $366$, $370$, $374$, $385$, $399$, $402$, $406$:
::$410 = 2 \times 5 \times 41$
:The $20$th integer after $7$, $13$, $19$, $35$, $38$, $41$, $57$, $65$, $70$, $125$, $130$, $190$, $205$, $223$, $253$, $285$, $305$, $350$, $380$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$410^2 = 168 \, 100$; $16 = 4^2$, $8100 = 90^2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:410
4104

== Number ==
$4104$ (four thousand, one hundred and four) is:
:$2^3 \times 3^3 \times 19$
:The $2$nd positive integer which can be expressed as the sum of $2$ cubes in $3$ different ways:
::$4104 = 16^3 + 2^3 = 15^3 + 9^3 = \paren {-12}^3 + 18^3$
:The $2$nd taxicab number after $1729$: a positive integer which can be expressed as the sum of $2$ positive cubes in $2$ different ways:
::$4104 = 15^3 + 9^3 = 16^3 + 2^3$
== Also see ==
* 
* 
== Sources ==
* : $4104$
Category:Specific Numbers
Category:4104
411

== Number ==
$411$ (four hundred and eleven) is:
:$3 \times 137$
:The $24$th inconsummate number after $62$, $63$, $65$, $75$, $\ldots$, $371$, $372$, $377$, $381$, $383$, $386$, $387$, $395$:
::$\nexists n \in \Z_{>0}: n = 411 \times s_{10} \left({n}\right)$
== Sources ==
* 
Category:Specific Numbers
Category:411
412

== Number ==
$412$ (four hundred and twelve) is:
:$2^2 \times 103$
:The $4$th positive integer $n$ after $157$, $262$, $367$ for which $\map \phi {2 n + 1} < \map \phi {2 n}$:
::$\map \phi {2 \times 412 + 1} = 400$, $\map \phi {2 \times 412} = 408$
:The $42$nd noncototient after $10$, $26$, $34$, $50$, $\ldots$, $344$, $346$, $362$, $366$, $372$, $386$, $394$, $404$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 412$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
== Sources ==
* : $157$
Category:Specific Numbers
Category:412
4140

== Number ==
$4140$ (four thousand, one hundred and forty) is:
:$2^2 \times 3^2 \times 5 \times 23$
:The $8$th Bell number after $(1)$, $1$, $2$, $5$, $15$, $52$, $203$, $877$
== Also see ==
* 
Category:Specific Numbers
Category:4140
4141

== Number ==
$4141$ (four thousand, one hundred and forty-one) is:
:$41 \times 101$
:The $41$st heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $3010$, $3186$, $3367$, $3744$, $3940$:
::$4141 = \ds \sum_{k \mathop = 1}^{41} \paren {5 k - 4} = \dfrac {41 \paren {5 \times 41 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:4141
415,074,643

== Number ==
$415 \, 074 \, 643$ (four hundred and fifteen million, seventy-four thousand, six hundred and forty-three) is:
:The $22 \, 096 \, 548$th prime number
:The $5$th (and largest known) prime number after $2$, $5$, $71$, $369 \, 119$ which divides the sum of all smaller primes:
::$10 \, 736 \, 033 \times 415 \, 074 \, 643 = 4 \, 456 \, 255 \, 064 \, 711 \, 219 = 2 + 3 + 5 + \cdots + 415 \, 074 \, 631$
== Also see ==
* 
* 
Category:Specific Numbers
Category:415,074,643
4150

== Number ==
$4150$ (four thousand, one hundred and fifty) is:
:$2 \times 5^2 \times 83$
:The smallest positive integer which can be expressed as the sum of the $5$th powers of its digits:
::$4150 = 4^5 + 1^5 + 5^5 + 0^5$
== Also see ==
* Perfect Digital Invariant/Examples/4150
== Sources ==
* : $4150$
Category:Specific Numbers
Category:4150
416

== Number ==
$416$ (four hundred and sixteen) is:
:$2^5 \times 13$
:The $19$th positive integer after $64$, $96$, $128$, $144$, $\ldots$, $320$, $324$, $336$, $352$, $360$, $384$, $400$ with $6$ or more prime factors:
::$416 = 2 \times 2 \times 2 \times 2 \times 2 \times 13$
:The $25$th inconsummate number after $62$, $63$, $65$, $75$, $\ldots$, $371$, $372$, $377$, $381$, $383$, $386$, $387$, $395$, $411$:
::$\nexists n \in \Z_{>0}: n = 416 \map {\times s_{10} } n$
== Also see ==
* 
* 
Category:Specific Numbers
Category:416
418

== Number ==
$418$ (four hundred and eighteen) is:
:$2 \times 11 \times 19$
:The $7$th integer after $1$, $14$, $30$, $105$, $248$, $264$ whose divisor sum divided by its Euler $\phi$ value is a square:
::$\dfrac {\map {\sigma_1} {418} } {\map \phi {418} } = \dfrac {720} {180} = 4 = 2^2$
:The $47$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $385$, $399$, $402$, $406$, $410$:
::$418 = 2 \times 11 \times 19$
== Also see ==
* 
* 
Category:Specific Numbers
Category:418
418,904

== Number ==
$418 \, 904$ (four hundred and eighteen thousand, nine hundred and four) is:
:$2^3 \times 52 \, 363$
:The $12$th 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:418,904
4181

== Number ==
$4181$ (four thousand, one hundred and eighty-one) is:
:$37 \times 113$
:The first composite Fibonacci number with a prime index:
::$4181 = 37 \times 113$
:The $19$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$, $34$, $55$, $89$, $144$, $233$, $377$, $610$, $987$, $1597$, $2584$:
::$4181 = 1597 + 2584$
== Also see ==
* 
* 
== Sources ==
* : $4181$
* : $4181$
Category:Specific Numbers
Category:4181
4186

== Number ==
$4186$ (four thousand, one hundred and eighty-six) is:
:$2 \times 7 \times 13 \times 23$
:The second of the $5$th pair of triangular numbers whose sum and difference are also both triangular:
::$1485 = T_{54}$, $4186 = T_{91}$, $1485 + 4186 = T_{106}$, $4186 - 1485 = T_{73}$
:The $46$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $2145$, $2278$, $2415$, $2556$, $2701$, $2850$, $3003$, $3160$, $3321$, $3486$, $3655$, $3828$, $4005$:
::$4186 = \ds \sum_{k \mathop = 1}^{46} \paren {4 k - 3} = 46 \paren {2 \times 46 - 1}$
:The $91$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $3655$, $3741$, $3828$, $3916$, $4005$, $4095$:
::$4186 = \ds \sum_{k \mathop = 1}^{91} k = \dfrac {91 \times \paren {91 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:4186
419

== Number ==
$419$ (four hundred and nineteen) is:
:The $81$st prime number
:The larger of the $6$th pair of primes whose prime gap is $10$:
::$419 - 409 = 10$
:The upper end of the $7$th record-breaking gap between twin primes:
::$419 - 349 = 70$
:The smaller of the $22$nd pair of twin primes:
::$421 - 419 = 2$
:The $22$nd Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$, $131$, $173$, $179$, $191$, $233$, $239$, $251$, $281$, $293$, $359$:
::$2 \times 419 + 1 = 839$, which is prime.
:The $30$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $269$, $313$, $337$, $367$, $379$, $383$, $389$
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:419
41 is Smallest Number whose Period of Reciprocal is 5

== Theorem ==
$41$ is the first positive integer the decimal expansion of whose reciprocal has a period of $5$:
:$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$
== Proof ==
From Reciprocal of $41$:
Counting the digits, it is seen that this has a period of recurrence of $5$.
It remains to be shown that $41$ is the smallest positive integer which has this property.
Category:41
Category:Examples of Reciprocals
42

__NOTOC__
== Number ==
$42$ (forty-two) is:
:$2 \times 3 \times 7$
:The $2$nd sphenic number after $30$:
::$42 = 2 \times 3 \times 7$
:The magic constant of a magic cube of order $3$, after $1$, $(9)$:
::$42 = \ds \dfrac 1 {3^2} \sum_{k \mathop = 1}^{3^3} k = \dfrac {3 \paren {3^3 + 1} } 2$
:The $4$th and largest positive integer after $1$, $3$, $14$ of which the product of its Euler $\phi$ function and its divisor count equals its divisor sum:
::$\map \phi {42} \map {\sigma_0} {42} = 12 \times 8 = 96 = \map {\sigma_1} {42}$
:The $5$th Catalan number after $(1)$, $1$, $2$, $5$, $14$:
::$42 = \dfrac 1 {5 + 1} \dbinom {2 \times 5} 5 = \dfrac 1 6 \times 252$
:The $8$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$ such that no factorial of an integer can end with $n$ zeroes.
:The $8$th abundant number after $12$, $18$, $20$, $24$, $30$, $36$, $40$:
::$1 + 2 + 3 + 6 + 7 + 14 + 21 = 54 > 42$
:The $10$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$:
::$42 = 7 + 14 + 21$
:The number of integer partitions for $10$:
::$\map p {10} = 42$
:The $15$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$:
::$\map {\sigma_1} {42} = 96$
:The $17$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$, $16$, $18$, $24$, $30$, $36$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
:The $25$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$, $32$, $33$, $37$, $38$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $29$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $27$, $30$, $31$, $32$, $35$, $36$, $37$, $38$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Historical Note ==
== Also see ==
* 
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== Sources ==
* : $42$
* : $42$
Category:Specific Numbers
Category:42
42,209

== Number ==
$42 \, 209$ (forty-two thousand, two hundred and nine) is:
:The $4413$th prime number
:The $19$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$, $1021$, $2657$, $3229$, $4547$, $4787$, $11 \, 549$, $13 \, 649$, $18 \, 523$, $23 \, 801$, $24 \, 029$, $42 \, 209$
== Also see ==
* 
* 
Category:Specific Numbers
Category:42,209
42,549,416

== Number ==
$42 \, 549 \, 416$ is:
:$2^3 \times 7 \times 13 \times 211 \times 277$
:The smallest positive integer which can be expressed as the sum of $2$ cubes in $4$ different ways:
== Also see ==
* Smallest Integer which is Sum of 2 Cubes in 4 Ways
== Sources ==
* : $42,549,416$
Category:Specific Numbers
Category:42,549,416
420

== Number ==
$420$ (four hundred and twenty) is:
:$2^2 \times 3 \times 5 \times 7$
:The $17$th integer $n$ after $1$, $3$, $15$, $30$, $35$, $56$, $70$, $78$, $105$, $140$, $168$, $190$, $210$, $248$, $264$, $357$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} {420} = 24$, $\map \phi {420} = 96$, $\map {\sigma_1} {420} = 1344$
:The $34$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $120$, $144$, $168$, $180$, $210$, $216$, $240$, $288$, $300$, $336$, $360$:
::$\map {\sigma_1} {420} = 1344$
== Also see ==
* 
* 
Category:Specific Numbers
Category:420
4200

== Number ==
$4200$ (four thousand two hundred) is:
:$2^3 \times 3 \times 5^2 \times 7$
:The $20$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $126$, $196$, $288$, $405$, $550$, $726$, $936$, $1183$, $1470$, $1800$, $2176$, $2601$, $3078$, $3610$:
::$4200 = \ds \sum_{k \mathop = 1}^{20} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {20^2 \paren {20 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:4200
4204

== Number ==
$4204$ (four thousand, two hundred and four) is:
:$2^2 \times 1051$
:The $33$rd even integer after $2$, $4$, $94$, $96$, $\ldots$, $1148$, $1264$, $1266$, $1268$, $1354$, $1356$, $1358$, $3244$, $3246$, $3248$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
== Also see ==
* 
Category:Specific Numbers
Category:4204
4206

== Number ==
$4206$ (four thousand, two hundred and six) is:
:$2 \times 3 \times 701$
:The $34$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $1148$, $1264$, $1266$, $1268$, $1354$, $1356$, $1358$, $3244$, $3246$, $3248$, $4204$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
== Also see ==
* 
Category:Specific Numbers
Category:4206
4208

== Number ==
$4208$ (four thousand, two hundred and eight) is:
:$2^4 \times 263$
:The $35$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $1148$, $1264$, $1266$, $1268$, $1354$, $1356$, $1358$, $3244$, $3246$, $3248$, $4204$, $4206$, and conjectured to be the last, that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
== Also see ==
* 
::Conjectured to be the last.
Category:Specific Numbers
Category:4208
421

== Number ==
$421$ (four hundred and twenty-one) is:
:The $82$nd prime number
:The smaller of the $7$th pair of primes whose prime gap is $10$:
::$431 - 421 = 10$
:The $15$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 14^2 + 29 = 421$ ()
:The larger of the $22$nd pair of twin primes:
::$421 - 419 = 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:421
4219

== Number ==
$4219$ (four thousand, two hundred and nineteen) is:
:The $578$th prime number
:The $38$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $2791$, $2977$, $3169$, $3367$, $3571$, $3781$, $3997$:
::$4219 = \ds 1 + \sum_{k \mathop = 1}^{38 - 1} 6 k = 38^3 - 37^3$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4219
422,481

== Number ==
$422 \, 481$ (four hundred and twenty-two thousand, four hundred and eighty-one) is:
:$3 \times 140 \, 827$
:The $1$st positive integer whose fourth power can be expressed as the sum of $3$ fourth powers, thus refuting the Euler Quartic Conjecture:
::$422 \, 481^4 = 95 \, 800^4 + 217 \, 519^4 + 414 \, 560^4$
== Also see ==
* 
== Sources ==
* : $20,615,673$
Category:Specific Numbers
Category:422,481
4225

== Number ==
$4225$ (four thousand, two hundred and twenty-five) is:
:$5^2 \times 13^2$
:The $8$th square after $49$, $169$, $361$, $1225$, $1444$, $1681$, $3249$ whose decimal representation can be split into two parts which are each themselves square:
::$4225 = 65^2$; $4 = 2^2$, $225 = 15^2$
:The $65$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $3249$, $3364$, $3481$, $3600$, $3721$, $3844$, $3969$, $4096$:
::$4225 = 65 \times 65$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4225
425

== Number ==
$425$ (four hundred and twenty-five) is:
:$5^2 \times 17$
:The $2$nd positive integer after $325$ which can be expressed as the sum of two square numbers in $3$ distinct ways:
::$425 = 20^2 + 5^2 = 19^2 + 8^2 = 16^2 + 13^2$
:The $17$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$, $176$, $210$, $247$, $287$, $330$, $376$:
::$425 = \ds \sum_{k \mathop = 1}^{17} \paren {3 k - 2} = \dfrac {17 \paren {3 \times 17 - 1} } 2$
:The $33$rd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $222$, $247$, $260$, $287$, $301$, $330$, $345$, $376$, $392$:
::$425 = \ds \sum_{k \mathop = 1}^{17} \paren {3 k - 2} = \dfrac {17 \paren {3 \times 17 - 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:425
4253

== Number ==
$4253$ (four thousand, two hundred and fifty-three) is:
:The $583$rd prime number
:The index of the $19$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$, $107$, $127$, $521$, $607$, $1279$, $2203$, $2281$, $3217$:
::$M_{4253} = 2^{4253} - 1 \approx 1 \cdotp 908 \times 10^{1280}$
== Also see ==
* 
* 
== Historical Note ==
== Sources ==
* : $2^{4253} - 1$
* : $2^{4253} - 1$
Category:Specific Numbers
Category:4253
4253/Historical Note

== Historical Note on $4253$ ==
The Mersenne number $M_{4253} = 2^{4253} - 1$ was demonstrated to be a Mersenne prime by on $3$ November $1961$, using an IBM 7090, along with $M_{4423}$, whose primality was similarly demonstrated in the same computer run.
These were the first titanic primes ever discovered.
It is a moot question which one was actually discovered first.
== Sources ==
* : $2^{4253} - 1$
* : $2^{4253} - 1$
Category:Historical Notes
4256

== Number ==
$4256$ (four thousand, two hundred and fifty-six) is:
:$2^5 \times 7 \times 19$
:The $38$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $2296$, $2465$, $2640$, $2821$, $3008$, $3201$, $3400$, $3605$, $3816$, $4033$:
::$4256 = \ds \sum_{k \mathop = 1}^{38} \paren {6 k - 5} = 38 \paren {3 \times 38 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:4256
426

== Number ==
$426$ (four hundred and twenty-six) is:
:$2 \times 3 \times 71$
:The $33$rd untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $324$, $326$, $336$, $342$, $372$, $406$, $408$
:The $48$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $385$, $399$, $402$, $406$, $410$, $418$:
::$426 = 2 \times 3 \times 71$
== Also see ==
* 
* 
Category:Specific Numbers
Category:426
427

== Number ==
$427$ (four hundred and twenty-seven) is:
:$7 \times 61$
:The $16$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$, $154$, $320$, $340$, $399$, $427$
== Also see ==
* 
Category:Specific Numbers
Category:427
4278

== Number ==
$4278$ (four thousand, two hundred and seventy-eight) is:
:$2 \times 3 \times 23 \times 31$
:The $92$nd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $3655$, $3741$, $3828$, $3916$, $4005$, $4095$, $4186$:
::$4278 = \ds \sum_{k \mathop = 1}^{92} k = \dfrac {92 \times \paren {92 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:4278
428

== Number ==
$428$ (four hundred and twenty-eight) is:
:$2^2 \times 107$
:The $2$nd integer after $91$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive integers:
::$428^2 = 183 \, 184$
:The $1$st integer whose square has a decimal representation consisting of the concatenation of $2$ consecutive increasing integers:
::$428^2 = 183 \, 184$
== Also see ==
* 
* 
== Sources ==
* : $183,184$
* : $183,184$
Category:Specific Numbers
Category:428
429

== Number ==
$429$ (four hundred and twenty-nine) is:
:$3 \times 11 \times 13$
:The $7$th Catalan number after $(1)$, $1$, $2$, $5$, $14$, $42$, $132$:
::$429 = \dfrac 1 {7 + 1} \dbinom {2 \times 7} 7 = \dfrac 1 8 \times 3432$
:The $49$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $385$, $399$, $402$, $406$, $410$, $418$, $426$:
::$429 = 3 \times 11 \times 13$
== Also see ==
* 
* 
Category:Specific Numbers
Category:429
429,315,678

== Number ==
$429 \, 315 \, 678$ is:
:$2 \times 3^2 \times 11 \times 19 \times 139 \times 821$
:Prepended with zero, a pandigital integer which can be expressed as the product of a pandigital doubleton in $3$ different ways:
== Also see ==
* Pandigital Product of Pandigital Pairs in 3 Ways
== Sources ==
* : $0,429,315,678$
* : $0,429,315,678$
Category:Specific Numbers
Category:429,315,678
4297

== Number ==
$4297$ (four thousand, two hundred and ninety-seven) is:
:The $590$th prime number
:The $14$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$3$, $5$, $11$, $13$, $41$, $89$, $317$, $337$, $991$, $1873$, $2053$, $2377$, $4093$, $4297$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4297