nyuuzyou/wikis huggingface dataset:
5040 is Product of Consecutive Numbers in Two Ways

== Theorem ==
:$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$
== Proof ==
Follows from Factorial as Product of Two Factorials:
:$10! = 6! \times 7!$
and so:
:$\dfrac {10!} {6!} = 10 \times 9 \times 8 \times 7 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Hence the result.
== Sources ==
* : $5040$
Category:Factorials
Category:5040
5041

== Number ==
$5041$ (five thousand and forty-one) is:
:$71^2$
:The $3$rd square number after $25$, $121$, and last known, of the form $n! + 1$:
::$5041 = 7! + 1 = 71^2$
:where $!$ denotes the factorial function
:The $71$st square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $4096$, $4225$, $4356$, $4489$, $4624$, $4761$, $4900$
::$5041 = 71 \times 71$
== Also see ==
* Brocard's Problem
* 
Category:Specific Numbers
Category:5041
505

== Number ==
$505$ (five hundred and five) is:
:$5 \times 101$
:The magic constant of a magic square of order $10$, after $1$, $(5)$, $15$, $34$, $65$, $111$, $175$, $260$, $369$:
::$505 = \ds \dfrac 1 {10} \sum_{k \mathop = 1}^{10^2} k = \dfrac {10 \paren {10^2 + 1} } 2$
:The $21$st positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$, $65$, $85$, $89$, $101$, $125$, $145$, $149$, $245$, $265$, $365$, $385$, $485$ such that the Fibonacci number $F_n$ ends in $n$
== Also see ==
* 
* 
Category:Specific Numbers
Category:505
5050

== Number ==
$5050$ (five thousand and fifty) is:
:$2 \times 5^2 \times 101$
:The total of all the entries in a magic square of order $10$, after $1$, $(10)$, $45$, $136$, $325$, $666$, $1225$, $2080$, $3321$:
::$5050 = \ds \sum_{k \mathop = 1}^{10^2} k = \dfrac {10^2 \paren {10^2 + 1} } 2$
:The $13$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$, $703$, $999$, $2223$, $2728$, $4879$, $4950$:
::$5050^2 = 25 \, 502 \, 500 \to 2550 + 2500 = 5050$
:The $100$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4095$, $4186$, $4278$, $4371$, $4465$, $4560$, $4656$, $4753$, $4851$, $4950$:
::$5050 = \ds \sum_{k \mathop = 1}^{100} k = \dfrac {100 \times \paren {100 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:5050
506

== Number ==
$506$ (five hundred and six) is:
:$2 \times 11 \times 23$
:The $11$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $91$, $140$, $204$, $285$, $385$:
::$506 = \ds \sum_{k \mathop = 1}^{11} k^2 = \dfrac {11 \paren {11 + 1} \paren {2 \times 11 + 1} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:506
509

== Number ==
$509$ (five hundred and nine) is:
:The $97$th prime number
:The $8$th obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$
:The $26$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $233$, $239$, $251$, $281$, $293$, $359$, $419$, $431$, $443$, $491$:
::$2 \times 509 + 1 = 1019$, which is prime.
:The $37$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $383$, $389$, $419$, $433$, $461$, $487$, $491$, $499$, $503$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:509
509,033,161

== Number ==
$509 \, 033 \, 161$ is:
:$7 \times 13 \times 19 \times 37 \times 73 \times 109$
:The $472$nd Carmichael number, and one which is the product of $2$ Carmichael numbers:
::$\forall a \in \Z: a \perp 509 \, 033 \, 161: a^{509 \, 033 \, 160} \equiv 1 \pmod {509 \, 033 \, 161}$
::$509 \, 033 \, 161 = 1729 \times 294 \, 409$:
:::$\forall a \in \Z: a \perp 1729: a^{1728} \equiv 1 \pmod {1729}$
:::$\forall a \in \Z: a \perp 294 \, 409: a^{294 \, 408} \equiv 1 \pmod {294 \, 409}$
== Also see ==
* 
== Sources ==
* : $509,033,161$
Category:Specific Numbers
Category:509,033,161
509,203

== Number ==
$509 \, 203$ (five hundred and nine thousand, two hundred and three) is:
:The $42 \, 228$th prime number
:The $1$st of the integers known to be a Riesel number:
::$k \times 2^{509 \, 203} - 1$ is composite for all $k$
== Also see ==
* 
* 
== Sources ==
* : $509,203$
Category:Specific Numbers
Category:509,203
51

== Number ==
$51$ (fifty-one) is:
:$3 \times 17$
:The $3$rd term of the $1$st $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
::$\tuple {49, 50, 51, 52, 53}$
:The $6$th pentagonal number after $1$, $5$, $12$, $22$, $35$:
::$51 = 1 + 4 + 7 + 10 + 13 + 16 = \dfrac {6 \paren {3 \times 6 - 1} } 2$
:The $9$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $24$, $25$, $49$:
::$51^3 = 132 \, 6 \mathbf {51}$
:The $11$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$, $40$:
::$51 = \dfrac {6 \paren {3 \times 6 - 1} } 2$
:The $14$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $\ldots$
:The $18$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$, $35$, $38$, $39$, $46$, $49$:
::$51 = 3 \times 17$
:The $19$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $\ldots$
:The $23$rd after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$, $32$, $35$, $26$, $39$, $48$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.
:The $30$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{51} = 2 \, 251 \, 799 \, 813 \, 685 \, 248$
:Suggested by in his $1997$ book as being the smallest uninteresting number, which fact makes it intrinsically interesting.
== Also see ==
* Interesting Number Paradox
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $51$
Category:Specific Numbers
Category:51
51,001,180,160

== Number ==
$51 \, 001 \, 180 \, 160$ is:
:$2^{14} \times 5 \times 7 \times 19 \times 31 \times 151$
:The $6$th triperfect number after $120$, $672$, $523 \, 776$, $459 \, 818 \, 240$, $1 \, 476 \, 304 \, 896$:
::$\map {\sigma_1} {51 \, 001 \, 180 \, 160} = 153 \, 003 \, 540 \, 480 = 3 \times 51 \, 001 \, 180 \, 160$
== Also see ==
* 
Category:Specific Numbers
Category:51,001,180,160
510

== Number ==
$510$ (five hundred and ten) is:
:$2 \times 3 \times 5 \times 17$
:The index of the seventh power which provides the closest known approximation to a power of $10$:
::$7^{510} \approx 1 \cdotp 00000 \, 09377 \, 76536 \ldots \times 10^{431}$
:The $13$th positive integer after $200$, $202$, $204$, $205$, $206$, $208$, $320$, $322$, $324$, $325$, $326$, $328$ that cannot be made into a prime number by changing just $1$ digit
:The $27$th number whose divisor sum is square:
:
== Also see ==
* Approximation to Power of 7 by Power of 10
* 
* 
== Sources ==
* : $510$
Category:Specific Numbers
Category:510
510,510

== Number ==
$510 \, 510$ (five hundred and ten thousand, five hundred and ten) is:
:$2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$
:Can be expressed as the product of $4$ distinct consecutive Fibonacci numbers:
::$510 \, 510 = 13 \times 21 \times 34 \times 55$
:The $5$th primorial after $2$, $6$, $30$, $210$, and the largest known, which can be expressed as the product of consecutive integers:
::$510 \, 510 = 17 \# = 714 \times 715$
:The $7$th primorial after $1$, $2$, $6$, $30$, $210$, $2310$, $30 \, 030$ (counting $1$ as the zeroth):
::$510 \, 510 = p_7 \# = 17 \# = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$
:Hence the smallest positive integer with $7$ distinct prime factors.
== Also see ==
* 510,510 is Product of 4 Consecutive Fibonacci Numbers
* 
* 
== Sources ==
* : $510,510$
* : $510,510$
Category:Specific Numbers
Category:510,510
510,510 is Product of 4 Consecutive Fibonacci Numbers

== Theorem ==
$510 \, 510$ can be expressed as the product of $4$ distinct consecutive Fibonacci numbers:
:$510 \, 510 = 13 \times 21 \times 34 \times 55$
and is also the $7$th primorial:
:$510 \, 510 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$
== Proof ==
By observation:
== Historical Note ==
== Sources ==
* : $510,510$
* : $510,510$
Category:Fibonacci Numbers
Category:Primorials
Category:510,510
510,510 is Product of 4 Consecutive Fibonacci Numbers/Historical Note

== Historical Note on $510 \, 510$ is Product of $4$ Consecutive Fibonacci Numbers ==
 report in his $1997$ work that this result can be found in an article by in volume $12$, but this has not been corroborated.
== Sources ==
* : $510,510$
* : $510,510$
Category:Historical Notes
511/Historical Note

== Historical Note on $511$ ==
$511$ was the $1$st number of the form $2^n - 1$, where $n$ is odd, that was demonstrated to be composite.
This was done by in $1636$, who demonstrated that $2^9 - 1 = 511 = 7 \times 73$.
== Sources ==
* : Preface
Category:Historical Notes
512

== Number ==
$512$ (five hundred and twelve) is:
:$2^9$
:In binary:
::$1 \, 000 \, 000 \, 000$
:In octal:
::$1 \, 000$
:The $3$rd cube after $0$, $1$ whose digits add up to its root:
::$512 = 8^3$, while $5 + 1 + 2 = 8$
:The $3$rd power of $8$ after $(1)$, $8$, $64$:
::$512 = 8^3$
:The $8$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$:
::$512 = 8 \times 8 \times 8$
:The $9$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$:
::$512 = 2^9$
:The $10$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$:
::$\map {\sigma_1} {512} = 1023 = 2 \times 512 - 1$
:The $10$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$, $480$ with $7$ or more prime factors:
::$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times \, 2 \times 2}$
:The index (after $2$, $3$, $6$, $30$, $75$, $81$, $115$, $123$, $249$, $362$, $384$, $462$) of the $13$th Woodall prime:
::$512 \times 2^{512} - 1$
:The $14$th positive integer after $200$, $202$, $204$, $205$, $206$, $208$, $320$, $322$, $324$, $325$, $326$, $328$, $510$ that cannot be made into a prime number by changing just $1$ digit
:The $25$th positive integer after $64$, $96$, $128$, $144$, $\ldots$, $384$, $400$, $416$, $432$, $448$, $480$, $504$ with $6$ or more prime factors:
::$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times \, 2 \times 2 \times 2}$
:The $39$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $289$, $324$, $343$, $361$, $392$, $400$, $432$, $441$, $484$, $500$:
::$512 = 2^9$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $512$
* : $512$
Category:Specific Numbers
Category:512
514

== Number ==
$514$ (five hundred and fourteen) is:
:$2 \times 257$
:The $9$th even integer after $2$, $4$, $94$, $96$, $98$, $400$, $402$, $404$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
:The $15$th positive integer after $200$, $202$, $204$, $205$, $206$, $208$, $320$, $322$, $324$, $325$, $326$, $328$, $510$, $512$ that cannot be made into a prime number by changing just $1$ digit
== Also see ==
* 
* 
Category:Specific Numbers
Category:514
5151

== Number ==
$5151$ (five thousand, one hundred and fifty-one) is:
:$51 \times 101$
:The $7$th hexamorphic number after $1$, $45$, $66$, $1225$, $1326$, $4950$:
::$5151 = H_{51}$
:The $51$st hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $4005$, $4186$, $4371$, $4560$, $4950$:
::$5151 = \ds \sum_{k \mathop = 1}^{51} \paren {4 k - 3} = 51 \paren {2 \times 51 - 1}$
:The $101$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4560$, $4656$, $4753$, $4851$, $4950$, $5050$:
::$5151 = \ds \sum_{k \mathop = 1}^{101} k = \dfrac {101 \times \paren {101 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:5151
516

== Number ==
$516$ (five hundred and sixteen) is:
:$2^2 \times 3 \times 43$
:The $10$th even integer after $2$, $4$, $94$, $96$, $98$, $400$, $402$, $404$, $514$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
:The $39$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $324$, $326$, $336$, $342$, $372$, $406$, $408$, $426$, $430$, $448$, $472$, $474$, $498$
== Also see ==
* 
* 
Category:Specific Numbers
Category:516
5167

== Number ==
$5167$ (five thousand, one hundred and sixty-seven) is:
:The $688$th prime number
:The $42$nd centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $2791$, $2977$, $3169$, $3367$, $3571$, $3781$, $3997$, $4219$, $4447$, $4681$, $4921$:
::$5167 = \ds 1 + \sum_{k \mathop = 1}^{42 - 1} 6 k = 42^3 - 41^3$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5167
517

== Number ==
$517$ (five hundred and seventeen) is:
:$11 \times 47$
:The $21$st Smith number after $4$, $22$, $27$, $58$, $\ldots$, $346$, $355$, $378$, $382$, $391$, $438$, $454$, $483$:
::$5 + 1 + 7 = 1 + 1 + 4 + 7 = 13$
:The $28$th number whose divisor sum is square:
:
== Also see ==
* 
* 
Category:Specific Numbers
Category:517
517,482

== Number ==
$517 \, 482$ (five hundred and seventeen thousand, four hundred and eighty-two) is:
:$2 \times 3^3 \times 7 \times 37^2$
:Equal to $666 \times 777$, and an anagram of $142 \, 857$
== Sources ==
* : $147,852$
* : $147,852$
Category:Specific Numbers
Category:517,482
517,842/Mistake

== Source Work ==
:
:The Dictionary
::$147,852$
== Mistake ==
:The digits $147852$ in various orders ... occur in several other products also. For example, $666 \times 777 = 517,842$ and $333 \times 777 = 258,741$.
== Correction ==
$666 \times 777 = 517,482$ not $517,842$.
This is a misprint that does not occur in of $1986$.
== Sources ==
* : $147,852$
Category:Mistakes/Curious and Interesting Numbers
518

== Number ==
$518$ (five hundred and eighteen) is:
:$2 \times 7 \times 37$
:The $4$th integer after $0$, $1$, $2$ such that its double factorial plus $1$ is prime:
::$518!! + 1$ is prime
:The $11$th even integer after $2$, $4$, $94$, $96$, $98$, $400$, $402$, $404$, $514$, $516$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
:The $14$th number after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $89$, $135$, $175$ which is the sum of the increasing powers of its digits taken in order:
::$5^1 + 1^2 + 8^3 = 518$
:The $40$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $324$, $326$, $336$, $342$, $372$, $406$, $408$, $426$, $430$, $448$, $472$, $474$, $498$, $516$
:The $49$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $386$, $394$, $404$, $412$, $436$, $466$, $470$, $474$, $482$, $490$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 518$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
* 
* 
* 
== Sources ==
* : $135$
* : $135$
* : $518$
Category:Specific Numbers
Category:518
5184

== Number ==
$5184$ (five thousand, one hundred and eighty-four) is:
:$2^6 \times 3^4$
:The $28$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $1936$, $2304$, $2704$, $2916$, $3136$, $3600$, $3844$, $4096$, $4356$, $4624$, $4900$ to be the divisor sum value of some (strictly) positive integer:
::$5184 = \map {\sigma_1} {2130}$
:The $72$nd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $4096$, $4225$, $4356$, $4489$, $4624$, $4761$, $4900$, $5041$
::$5184 = 72 \times 72$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5184
5186

== Number ==
$5186$ (five thousand, one hundred and eighty-six) is:
:$2 \times 2593$
:The $1$st term of the $1$st triplet of consecutive integers with equal Euler $\phi$ values:
::$\map \phi {5186} = \map \phi {5187} = \map \phi {5188} = 2592 = 2^5 \times 3^4$
== Also see ==
* Euler Phi Function of 5186
* Consecutive Triples with Same Euler Phi value: $5186$, $5187$ and $5188$
== Sources ==
* : $5186$
* : $5186$
Category:Specific Numbers
Category:5186
5187

== Number ==
$5187$ (five thousand, one hundred and eighty-seven) is:
:$3 \times 7 \times 13 \times 19$
:The $2$nd term of the $1$st triplet of consecutive integers with equal Euler $\phi$ values:
::$\map \phi {5186} = \map \phi {5187} = \map \phi {5188} = 2592 = 2^5 \times 3^4$
== Also see ==
* Euler Phi Function of 5187
* Consecutive Triples with Same Euler Phi value: $5186$, $5187$ and $5188$
== Sources ==
* : $5186$
* : $5186$
Category:Specific Numbers
Category:5187
5188

== Number ==
$5188$ (five thousand, one hundred and eighty-seven) is:
:$2^2 \times 1297$
:The $3$rd term of the $1$st triplet of consecutive integers with equal Euler $\phi$ values:
::$\map \phi {5186} = \map \phi {5187} = \map \phi {5188} = 2592 = 2^5 \times 3^4$
== Also see ==
* Euler Phi Function of 5188
* Consecutive Triples with Same Euler Phi value: $5186$, $5187$ and $5188$
== Sources ==
* : $5186$
* : $5186$
Category:Specific Numbers
Category:5188
519,650,114,814,905,002,347

== Number ==
$519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347$ is:
:$3 \times 41 \times 281 \times 7759 \times 1 \, 937 \, 732 \, 593 \, 591$
:The $5$th of $5$ integers which have the property that the sum of any $3$ of them is square:
== Also see ==
* 
Category:Specific Numbers
Category:519,650,114,814,905,002,347
52

== Number ==
$52$ (fifty-two) is:
:$2^2 \times 13$
:The $4$th term of the $1$st $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
::$\tuple {49, 50, 51, 52, 53}$
:The $3$rd untouchable number after $2$, $5$
:The $5$th Bell number after $(1)$, $1$, $2$, $5$, $15$
:The $5$th noncototient after $10$, $26$, $34$, $50$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 52$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The length of God's Algorithm for Sam Loyd's Fifteen Puzzle.
== Also see ==
* Length of God's Algorithm for Sam Loyd's Fifteen Puzzle
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $52$
* : $52$
Category:Specific Numbers
Category:52
52/Historical Note

== Historical Note on 52 (Fifty-Two) ==
There are two particular cultural significances for the number $52$:
:There are $52$ weeks (plus $1$ or $2$ days) in a year, divided into $4$ seasons of $13$ weeks each.
:There are $52$ cards in a standard deck divided into $4$ suits of $13$ cards each.
== Sources ==
* : $52$
* : $52$
Category:Historical Notes
520

== Number ==
$520$ (five hundred and twenty) is:
:$2^3 \times 5 \times 13$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $28$ different ways
:The $41$st untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $324$, $326$, $336$, $342$, $372$, $406$, $408$, $426$, $430$, $448$, $472$, $474$, $498$, $516$, $518$
:The $50$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $470$, $474$, $482$, $490$, $518$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 520$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:520
5208

== Number ==
$5208$ (five thousand, two hundred and eight) is:
:$2^3 \times 3 \times 7 \times 31$
:The $42$nd octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $3008$, $3201$, $3400$, $3605$, $3816$, $4033$, $4256$, $4485$, $4720$, $4961$:
::$5208 = \ds \sum_{k \mathop = 1}^{42} \paren {6 k - 5} = 42 \paren {3 \times 42 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:5208
521

== Number ==
$521$ (five hundred and twenty-one) is:
:The $98$th prime number
:The $8$th Lucas prime after $2$, $3$, $7$, $11$, $29$, $47$, $199$.
:The $13$th Lucas number after $(2)$, $1$, $3$, $4$, $7$, $11$, $18$, $29$, $47$, $76$, $123$, $199$, $322$:
::$521 = 199 + 322$
:The index of the $13$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$, $107$, $127$:
::$M_{521} = 2^{521} - 1 \approx 6 \cdotp 865 \times 10^{156}$
:The smaller of the $25$th pair of twin primes:
::$523 - 521 = 2$
== Also see ==
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $2^{521} - 1$
* : $521$
* : $2^{521} - 1$
Category:Specific Numbers
Category:521
521/Historical Note

== Historical Note on $521$ ==
The Mersenne number $2^{521} - 1$ was proven to be the $13$th Mersenne prime on $30$th January $1952$, on the Standards Western Automatic Computer.
The program used was written by over the Christmas period of $1951$, using an implementation of the Lucas-Lehmer Test.
This and the $14$th Mersenne prime $2^{607} - 1$ were thus discovered after a mere few hours.
== Sources ==
* : $2^{521} - 1$
* : $2^{521} - 1$
Category:Historical Notes
521/Mistake

== Source Work ==
:
:The Dictionary
::$2^{521} - 1$
:
:The Dictionary
::$2^{521} - 1$
== Mistake ==
:In a few hours on the night of $30$ January $1952$, using the SWAC computer, proved that $2^{521} - 1$ and the $183$-digit number $2^{607} - 1$ are both Mersenne primes.
This statement goes against the usual account that it was who (by writing and running a computer program) accomplished this feat.
However, it is worth pointing out that the algorithm he implemented was designed by in the first place.
== Sources ==
* : $2^{521} - 1$
* : $2^{521} - 1$
Category:Mistakes/Curious and Interesting Numbers
5221

== Number ==
$5221$ (five thousand, two hundred and twenty-one) is:
:$23 \times 227$
:The $46$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $3744$, $3940$, $4141$, $4347$, $4558$, $4774$, $4995$:
::$5221 = \ds \sum_{k \mathop = 1}^{46} \paren {5 k - 4} = \dfrac {46 \paren {5 \times 46 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:5221
523

== Number ==
$523$ (five hundred and twenty-three) is:
:The $99$th prime number
:The $1$st element of the $1$st sequence of $3$ consecutive positive integers whose Euler $\phi$ values are strictly decreasing:
::$\map \phi {523} = 522$, $\map \phi {524} = 260$, $\map \phi {525} = 240$
:The larger of the $25$th pair of twin primes:
::$523 - 521 = 2$
:The $34$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $197$, $223$, $283$, $313$, $317$, $337$, $347$, $353$, $367$, $373$, $383$, $397$, $443$, $467$
== Also see ==
* First Sequence of Three Consecutive Strictly Decreasing Euler Phi Values
* 
* 
* 
== Sources ==
* : $523$
Category:Specific Numbers
Category:523
523,776

== Number ==
$523 \, 776$ (five hundred and twenty-three thousand, seven hundred and seventy-six) is:
:$2^9 \times 3 \times 11 \times 31$
:The $3$rd triperfect number after $120$, $672$:
::$\map {\sigma_1} {523 \, 776} = 1 \, 571 \, 328 = 3 \times 523 \, 776$
== Also see ==
* 
== Sources ==
* : $523,776$
* : $523,776$
Category:Specific Numbers
Category:523,776
524,287

== Number ==
$524 \, 287$ (five hundred and twenty-four thousand, two hundred and eighty-seven) is:
:The $43 \, 390$th prime number
:The $7$th Mersenne prime after $3$, $7$, $31$, $127$, $8191$, $131 \, 071$:
::$524 \, 287 = 2^{19} - 1$
:The $8$th Mersenne number after $3$, $7$, $31$, $127$, $2047$, $8191$, $131 \, 071$:
::$524 \, 287 = 2^{19} - 1$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:524,287
524,288

== Number ==
$524 \, 288$ (five hundred and twenty-four thousand, two hundred and eighty-eight) is:
:$2^{19}$
:The $19$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $\ldots$, $16 \, 384$, $32 \, 768$, $65 \, 536$, $131 \, 072$, $262 \, 144$:
::$524 \, 288 = 2^{19}$
:The $20$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $\ldots$, $16 \, 384$, $32 \, 768$, $65 \, 536$, $131 \, 072$, $262 \, 144$:
::$\map {\sigma_1} {524 \, 288} = 1 \, 048 \, 575 = 2 \times 524 \, 288 - 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:524,288
5253

== Number ==
$5253$ (five thousand, two hundred and fifty-three) is:
:$3 \times 17 \times 103$
:The $102$nd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4560$, $4656$, $4753$, $4851$, $4950$, $5050$, $5151$:
::$5253 = \ds \sum_{k \mathop = 1}^{102} k = \dfrac {102 \times \paren {102 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:5253