nyuuzyou/wikis huggingface dataset:
584

== Number ==
$584$ (five hundred and eighty-four) is:
:$2^3 \times 73$
:The $48$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $472$, $474$, $498$, $516$, $518$, $520$, $530$, $540$, $552$, $556$, $562$, $576$
== Also see ==
* 
Category:Specific Numbers
Category:584
585

== Number ==
$585$ (five hundred and eighty-five) is:
:$3^2 \times 5 \times 13$
:The $10$th integer after $0$, $1$, $3$, $5$, $7$, $9$, $33$, $99$, $313$ which is palindromic in both decimal and binary:
::$585_{10} = 1 \, 001 \, 001 \, 001_2$
== Also see ==
* 
Category:Specific Numbers
Category:585
587

== Number ==
$587$ (five hundred and eighty-seven) is:
:The $107$th prime number
:The $1$st of a sequence of $11$ primes formed by tripling and adding $16$.
:The larger of the $9$th pair of primes whose prime gap is $10$:
::$587 - 577 = 10$
:The $20$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$, $467$, $479$, $503$, $563$:
::$587 = 2 \times 293 + 1$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $587$
* : $587$
Category:Specific Numbers
Category:587
588

== Number ==
$588$ (five hundred and eighty-eight) is:
:$2^2 \times 3 \times 7^2$
:The $26$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $346$, $355$, $378$, $382$, $391$, $438$, $454$, $483$, $517$, $526$, $535$, $562$, $576$:
::$5 + 8 + 8 = 2 + 2 + 3 + 7 + 7 = 21$
== Also see ==
* 
Category:Specific Numbers
Category:588
5886

== Number ==
$5886$ (five thousand, eight hundred and eighty-six) is:
:$2 \times 3^3 \times 109$
:The $108$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5253$, $5356$, $5460$, $5565$, $5671$, $5778$:
::$5886 = \ds \sum_{k \mathop = 1}^{108} k = \dfrac {108 \times \paren {108 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:5886
589,786

== Number ==
$589 \, 786$ (five hundred and eighty-nine thousand, seven hundred and eighty-six) is:
:$2 \times 294 \, 893$
:The $9$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:589,786
59

== Number ==
$59$ (fifty-nine) is:
:The $17$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$
:One of the prime factors of one of the terms of one of the $2$ representations of $635 \, 318 \, 657$ as the sum of $2$ fourth powers:
::$635 \, 318 \, 657 = 59^4 + 158^4$
:The upper end of the $4$th record-breaking gap between twin primes:
::$59 - 43 = 16$
:The $6$th safe prime after $5$, $7$, $11$, $23$, $47$:
::$59 = 2 \times 29 + 1$
:The smaller of the $7$th pair of twin primes, with $61$
:The $7$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$
:The $9$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$, $53$ such that the Mersenne number $2^p - 1$ is composite
:The $10$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$
:The $11$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$ such that $\ds m = \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime
:The $27$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $\ldots$
:The $34$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $44$, $45$, $46$, $49$, $50$, $54$, $55$ which cannot be expressed as the sum of distinct pentagonal numbers.
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:59
59,049

== Number ==
$59 \, 049$ (fifty-nine thousand and forty-nine) is:
:$3^{10}$
:The $243$rd square number:
::$59 \, 049 = 243 \times 243$
:The $9$th fifth power after $1$, $32$, $243$, $1024$, $3125$, $7776$, $16 \, 807$, $32 \, 768$:
::$59 \, 049 = 9 \times 9 \times 9 \times 9 \times 9$
:The $10$th power of $3$ after $(1)$, $3$, $9$, $27$, $81$, $243$, $729$, $2187$, $6561$, $19 \, 683$:
::$59 \, 049 = 3^{10}$
:The $5$th power of $9$ after $(1)$, $9$, $81$, $729$, $6561$:
::$59 \, 049 = 9^5$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:59,049
59,403

== Number ==
$59 \, 403$ (fifty-nine thousand, four hundred and three) is:
:$3 \times 19 \, 801$
:The smaller of the $4$th pandigital pair, with $76 \, 182$, of integers each have squares which are themselves pandigital:
::$59 \, 403^2 = 3 \, 528 \, 716 \, 409$, $76 \, 182^2 = 5 \, 803 \, 697 \, 124$
== Also see ==
* 
== Sources ==
* : $57,321$
Category:Specific Numbers
Category:59,403
59,554, 936,495, 441,481, 044,788, 091,271, 148,664, 944,796, 300,859, 243,635, 311,219, 048,448

== Number ==
$59 \, 554 \, 936 \, 495 \, 441 \, 481 \, 044 \, 788 \, 091 \, 271 \, 148 \, 664 \, 944 \, 796 \, 300 \, 859 \, 243 \, 635 \, 311 \, 219 \, 048 \, 448$ is:
:$2^{47} \times 9 \, 288 \, 811 \, 670 \, 405 \, 087 \times 145 \, 135 \, 534 \, 866 \, 431 \times 313 \, 887 \, 523 \, 966 \, 328 \, 699 \, 903$
:The smaller of an amicable pair, with $59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472$:
== Also see ==
* 
== Sources ==
* : $2^4 \times 7 \times 9,288,811,670,405,087 \times 145,135,534,866,431 \times 313,887,523,966,328,699,903$
Category:Specific Numbers
Category:59,554, 936,495, 441,481, 044,788, 091,271, 148,664, 944,796, 300,859, 243,635, 311,219, 048,448
59,554, 936,495, 441,891, 385,123, 332,422, 108,719, 776,971, 992,921, 810,832, 072,976, 105,472

== Number ==
$59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472$ is:
:$2^{47} \times 9 \, 288 \, 811 \, 670 \, 405 \, 087 \times 45 \, 556 \, 233 \, 678 \, 753 \, 109 \, 045 \, 286 \, 896 \, 851 \, 222 \, 527$
:The larger of an amicable pair, with $59 \, 554 \, 936 \, 495 \, 441 \, 481 \, 044 \, 788 \, 091 \, 271 \, 148 \, 664 \, 944 \, 796 \, 300 \, 859 \, 243 \, 635 \, 311 \, 219 \, 048 \, 448$:
== Also see ==
* 
== Sources ==
* : $2^4 \times 7 \times 9,288,811,670,405,087 \times 145,135,534,866,431 \times 313,887,523,966,328,699,903$
Category:Specific Numbers
Category:59,554, 936,495, 441,891, 385,123, 332,422, 108,719, 776,971, 992,921, 810,832, 072,976, 105,472
59,649,589,127,497,217

== Number ==
$59 \, 649 \, 589 \, 127 \, 497 \, 217$ is:
:The $n$th prime, where $n$ remains to be determined
:One of the two prime factors of the $7$th Fermat number $2^{\left({2^7}\right)} + 1$, the other being $5 \, 704 \, 689 \, 200 \, 685 \, 129 \, 054 \, 721$:
::$59 \, 649 \, 589 \, 127 \, 497 \, 217 = 116 \, 503 \, 103 \, 764 \, 643 \times 2^9 + 1$
== Also see ==
* 
Category:Specific Numbers
Category:59,649,589,127,497,217
59,961

== Number ==
$59 \, 961$ (fifty-nine thousand, nine hundred and sixty-one) is:
:$3 \times 11 \times 23 \times 79$
:The $25$th and largest known integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$, $105$, $160$, $661$, $2653$, $3069$, $3943$, $4053$, $4998$, $8275$, $9158$, $11 \, 164$, $43 \, 592$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$ is prime
== Also see ==
* 
Category:Specific Numbers
Category:59,961
59,994

== Number ==
$59 \, 994$ (fifty-nine thousand, nine hundred and ninety-four) is:
:$2 \times 3^3 \times 11 \times 101$
:One of the elements of the $2$-cycle to which Kaprekar's process on some $5$-digit numbers converges:
::$59 \, 994 \to 53 \, 955 \to 59 \, 994$
== Also see ==
* 
Category:Specific Numbers
Category:59,994
590

== Number ==
$590$ (five hundred and ninety) is:
:$2 \times 5 \times 59$
:The $20$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$, $176$, $210$, $247$, $287$, $330$, $330$, $376$, $425$, $477$, $532$:
::$590 = \ds \sum_{k \mathop = 1}^{20} \paren {3 k - 2} = \dfrac {20 \paren {3 \times 20 - 1} } 2$
:The $39$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$, $532$, $551$:
::$590 = \ds \sum_{k \mathop = 1}^{20} \paren {3 k - 2} = \dfrac {20 \paren {3 \times 20 - 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:590
5906

== Number ==
$5906$ (five thousand, nine hundred and six) is:
:$2 \times 2953$
:The $1$st integer which can be expressed as the sum of two rational $4$th powers, but not two integer $4$th powers
== Also see ==
* Sum of Two Rational 4th Powers but not Two Integer 4th Powers
== Sources ==
* : $5906$
Category:Specific Numbers
Category:5906
5913

== Number ==
$5913$ (five thousand, nine hundred and thirteen) is:
:$3^4 \times 73$
:The sum of the first $7$ factorials:
::$5913 = 1! + 2! + 3! + 4! + 5! + 6! + 7!$
== Also see ==
* 
== Sources ==
* : $5913$
* : $5913$
Category:Specific Numbers
Category:5913
5929

== Number ==
$5929$ (five thousand, nine hundred and twenty-nine) is:
:$7^2 \times 11^2$
:The $49$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $3744$, $3940$, $4141$, $4347$, $4558$, $4774$, $4995$, $5221$, $5452$, $5688$:
::$5929 = \ds \sum_{k \mathop = 1}^{49} \paren {5 k - 4} = \dfrac {49 \paren {5 \times 49 - 3} } 2$
:The $77$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $5041$, $5184$, $5329$, $5476$, $5625$, $5776$
::$5929 = 77 \times 77$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5929
593

== Number ==
$593$ (five hundred and ninety-three) is:
:The $108$th prime number
:The largest positive integer that cannot be expressed in the form $23 x + 28 y$ for positive integers $x$ and $y$.
:The $10$th balanced prime after $5$, $53$, $157$, $173$, $211$, $257$, $263$, $373$, $563$:
::$593 = \dfrac {587 + 599} 2$
:The $22$nd right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$, $233$, $239$, $293$, $311$, $313$, $317$, $373$, $379$
:The $27$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $233$, $239$, $251$, $281$, $293$, $359$, $419$, $431$, $443$, $491$, $509$:
::$2 \times 593 + 1 = 1187$, which is prime.
:The $41$st long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $491$, $499$, $503$, $509$, $541$, $571$, $577$
== Also see ==
* Largest Number not Expressible as Sum of Multiples of 23 and 28
* 
* 
* 
* 
* 
== Sources ==
* : $593$
* : $593$
Category:Specific Numbers
Category:593
594

== Number ==
$594$ (five hundred and ninety-four) is:
:$2 \times 3^3 \times 11$
:The $7$th positive integer after $1$, $24$, $26$, $87$, $168$, $388$ whose Euler $\phi$ value is equal to the product of its digits: 
::$\map \phi {594} = 180 = 5 \times 9 \times 4$
== Also see ==
* 
Category:Specific Numbers
Category:594
5941

== Number ==
$5941$ (five thousand, nine hundred and forty-one) is:
:$13 \times 457$
:The $45$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $4219$, $4447$, $4681$, $4921$, $5167$, $5419$, $5677$:
::$5941 = \ds 1 + \sum_{k \mathop = 1}^{45 - 1} 6 k = 45^3 - 44^3$
== Also see ==
* 
Category:Specific Numbers
Category:5941
595

== Number ==
$595$ (five hundred and ninety-five) is:
:$5 \times 7 \times 17$
:The $8$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$.
:The $34$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$:
::$595 = \ds \sum_{k \mathop = 1}^{34} k = \dfrac {34 \times \paren {34 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:595
598

== Number ==
$598$ (five hundred and ninety-eight) is:
:$2 \times 13 \times 23$
:The $15$th number after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $89$, $135$, $175$, $518$ which is the sum of the increasing powers of its digits taken in order:
::$5^1 + 9^2 + 8^3 = 598$
== Also see ==
* 
== Sources ==
* : $135$
* : $135$
Category:Specific Numbers
Category:598
5984

== Number ==
$5984$ (five thousand, nine hundred and eighty-four) is:
:$2^5 \times 11 \times 17$
:The $32$nd tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $816$, $969$, $1140$, $1330$, $1540$, $1771$, $2024$, $2300$, $2600$, $2925$, $3276$, $3276$, $4060$, $4495$, $4960$, $5456$:
::$5984 = \ds \sum_{k \mathop = 1}^{32} \frac {k \paren {k + 1} } 2 = \dfrac {32 \paren {32 + 1} \paren {32 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:5984
5985

== Number ==
$5985$ (five thousand, nine hundred and eighty-five) is:
:$3^2 \times 5 \times 7 \times 19$
:The $10$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$, $4095$, $4725$, $5355$, $5775$:
::$\map {\sigma_1} {5985} - 5985 = 6495 > 5985$
:The $45$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $3816$, $4033$, $4256$, $4485$, $4720$, $4961$, $5208$, $5461$, $5720$:
::$5985 = \ds \sum_{k \mathop = 1}^{45} \paren {6 k - 5} = 45 \paren {3 \times 45 - 2}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5985
599

== Number ==
$599$ (five hundred and ninety-nine) is:
:The $109$th prime number
:The $9$th obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$, $509$
:The $23$rd right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$, $233$, $239$, $293$, $311$, $313$, $317$, $373$, $379$, $593$
:The $27$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $353$, $373$, $383$, $433$, $443$, $449$, $499$, $557$, $577$
:The smaller of the $27$th pair of twin primes:
::$601 - 599 = 2$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:599
5993

== Number ==
$5993$ (five thousand, nine hundred and ninety-three) is:
:$13 \times 461$
:The $10$th (and largest known) Stern number after $1$, $3$, $17$, $137$, $227$, $977$, $1187$, $1493$, $5777$
:The second and greater of the only two known composite Stern numbers after $5777$
== Also see ==
* 
== Sources ==
* : $5777$
* : $5777$
Category:Specific Numbers
Category:5993
5995

== Number ==
$5995$ (five thousand, nine hundred and ninety-six) is:
:$5 \times 11 \times 109$
:The $11$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$, $666$, $3003$
:The $55$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $4005$, $4186$, $4371$, $4560$, $4950$, $5151$, $5356$, $5565$, $5778$:
::$5995 = \ds \sum_{k \mathop = 1}^{55} \paren {4 k - 3} = 55 \paren {2 \times 55 - 1}$
:The $109$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5253$, $5356$, $5460$, $5565$, $5671$, $5778$, $5886$:
::$5995 = \ds \sum_{k \mathop = 1}^{109} k = \dfrac {109 \times \paren {109 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:5995
5^x + 12^y equals 13^z has Unique Solution

== Theorem ==
The Diophantine equation:
:$5^x + 12^y = 13^z$
has exactly one solution in (strictly) positive integers:
:$5^2 + 12^2 = 13^2$
== Proof ==
== Sources ==
* 
* 
* : $2$
Category:Number Theory
5 Numbers such that Sum of any 3 is Square

== Theorem ==
This set of $5$ integers has the property that the sum of any $3$ of them is square:
== Proof ==
Taking the $\dbinom 5 3 = 10$ subsets of $3$ integers at a time:
== Sources ==
* 
* : $26,072,323,311,568,661,931$
Category:Square Numbers
5th Cyclotomic Ring has no Elements with Field Norm of 2 or 3

== Theorem ==
Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring.
There are no elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ whose field norm is either $2$ or $3$.
== Proof ==
Let $\map N z$ denote the field norm of $z \in \Z \sqbrk {i \sqrt 5}$.
Let $z = x + i y$.
Then:
But Square Root of Prime is Irrational so $x \notin \Z$.
Similarly for $\map N z = 3$.
== Sources ==
* : Chapter $9$: Rings: Exercise $19 \ \text {(ii)}$
Category:Cyclotomic Rings
5th Cyclotomic Ring is not a Unique Factorization Domain

== Theorem ==
Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring.
Then $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ is not a unique factorization domain.
The following elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ are irreducible:
:$2$
:$3$
:$1 + i \sqrt 5$
:$1 - i \sqrt 5$
== Proof ==
By definition, a unique factorization domain $D$ is an integral domain with the properties that:
:For all $x \in D$ such that $x$ is non-zero and not a unit of $D$:
::$(1): \quad x$ possesses a complete factorization in $D$
::$(2): \quad$ Any two complete factorizations of $x$ are equivalent.
A complete factorization is a tidy factorization 
:$x = u \circ y_1 \circ y_2 \circ \cdots \circ y_n$
such that:
:$u$ is a unit of $D$
:all of $y_1, y_2, \ldots, y_n$ are irreducible.
From Units of 5th Cyclotomic Ring, the only units of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ are $1$ and $-1$.
From Irreducible Elements of 5th Cyclotomic Ring, all of the elements of the set $S$, where:
:$S := \set {2, 3, 1 + i \sqrt 5, 1 - i \sqrt 5}$
are irreducible in $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$
Then we have:
So there are two tidy factorizations of $6$ in $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ which are not equivalent.
Hence the result.
== Sources ==
* : Chapter $9$: Rings: Exercise $19 \ \text {(iv)}$
Category:Cyclotomic Rings
Category:Unique Factorization Domains
6,172,882,716

== Number ==
$6 \, 172 \, 882 \, 716$ is:
:$2^2 \times 3 \times 7 \times 11 \times 13 \times 17 \times 19 \times 37 \times 43$
:The $111 \, 111$th triangular number:
::$6 \, 172 \, 882 \, 716 = \dfrac {111 \, 111 \times \left({111 \, 111 + 1}\right)} 2$
:The $10$th palindromic triangular number after $0$, $1$, $3$, $6$, $66$, $3003$, $66 \, 066$, $617 \, 716$, $3 \, 544 \, 453$ whose index is itself palindromic:
::$6 \, 172 \, 882 \, 716 = T_{111 \, 111}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6,172,882,716
6,355,441

== Number ==
$6 \, 355 \, 441$ (six million, three hundred and fifty-five thousand, four hundred and forty-one) is:
:$2521^2$
:The $4$th centered hexagonal number after $1, 169, 32 \, 761$ which is also square:
::$6 \, 355 \, 441 = 1456^3 - 1455^3 = 2521^2$
:The $1456$th centered hexagonal number:
::$6 \, 355 \, 441 = 3 \times 1456 \times \paren {1456 - 1} + 1 = 1456^3 - 1455^3$
:The $2521$st square number:
::$6 \, 355 \, 441 = 2521 \times 2521$
== Also see ==
* 
* 
* 
== Sources ==
* : $169$
Category:Specific Numbers
Category:6,355,441
6,505,941,701,960,039

== Number ==
$6 \, 505 \, 941 \, 701 \, 960 \, 039$ is:
:A prime number
:The lower of the $1$st pair of consecutive prime numbers separated by a prime gap of $864$:
::$6 \, 505 \, 941 \, 701 \, 960 \, 903 - 6 \, 505 \, 941 \, 701 \, 960 \, 039 = 864$
== Also see ==
* First Occurrence of Prime Gap of 864
* 
== Sources ==
* : $6,505,941,701,960,039$
Category:Specific Numbers
Category:6,505,941,701,960,039
6,661,661,161

== Number ==
$6 \, 661 \, 661 \, 161$ is:
:$81 \, 619^2$
:The $24$th (and largest known) square after $1$, $4$, $9$, $16$, $25$, $\ldots$, $11 \, 881$, $29 \, 929$, $44 \, 944$, $55 \, 225$, $69 \, 696$, $9 \, 696 \, 996$ which has no more than $2$ distinct digits and does not end in $0$:
::$6 \, 661 \, 661 \, 161 = 81 \, 619^2$
:The $81 \, 619$th square number:
::$6 \, 661 \, 661 \, 161 = 81 \, 619 \times 81 \, 619$
== Also see ==
* 
* 
== Sources ==
* : $6,661,661,161$
Category:Specific Numbers
Category:6,661,661,161
6,700,417

== Number ==
$6 \, 700 \, 417$ is:
:The $457 \, 523$rd prime number
:One of the two prime factors of the $6$th Fermat number $2^{\paren {2^5} } + 1$, the other being $641$:
::$6 \, 700 \, 417 = 52 \, 347 \times 2^7 + 1$
== Also see ==
* 
* 
== Sources ==
* : $257$
* : $257$
Category:Specific Numbers
Category:6,700,417
6,754,801

== Number ==
$6 \, 754 \, 801$ (six million, seven hundred and fifty-four thousand, eight hundred and one) is:
:$23^2 \times 113^2$
:The $6$th after $1 \, 048 \, 576$, $1 \, 056 \, 784$, $1 \, 085 \, 764$, $5 \, 740 \, 816$, $5 \, 764 \, 801$ of a sequence of $7$ numbers which are square and anagrams of each other
:The $2599$th square number:
::$6 \, 754 \, 801 = 2599 \times 2599$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6,754,801
6,963,472,309,248

== Number ==
$6 \, 963 \, 472 \, 309 \, 248$ is:
:$2^{10} \times 3^3 \times 7 \times 13 \times 19 \times 31 \times 37 \times 127$
:The $4$th Hardy-Ramanujan number after $2$, $1729$, $87 \, 539 \, 319$: the smallest positive integer which can be expressed as the sum of $2$ cubes in $4$ different ways:
== Also see ==
* 
== Sources ==
* : $6,963,472,309,248$
Category:Specific Numbers
Category:6,963,472,309,248
6/Historical Note

== Historical Note on $6$ ==
The number $6$ was associated by the Pythagoreans with health.
It was also, like $5$, associated with marriage, as it is the product of $2$, the first female number, and $3$, the first male number.
It also stood for equilibrium, its symbol being two triangles placed base to base:
:
The number $6$ was also believed by the Pythagoreans to be perfect, as it was the sum of:
:$1$, the unity
:$2$, the principle of diversity
:$3$, the "sacred trinity".
The fact that $1 + 2 + 3 = 1 \times 2 \times 3$ added to its mystique.
:Six is a perfect number in itself ... God created all things in six days because this number is perfect. And it would remain perfect, even if the work of six days did not exist.
:::: -- as reported by Ludwig Bieler
== Sources ==
* : $6$
* : Chapter $\text {A}.2$: Pythagoras (ca. B.C.)
* : $6$
Category:Historical Notes
60

== Number ==
$60$ (sixty) is:
:$2^2 \times 3 \times 5$
:The $2$nd unitary perfect number after $6$:
::$60 = 1 + 3 + 4 + 5 + 12 + 15 + 20$
:The $4$th heptagonal pyramidal number after $1$, $8$, $26$:
::$60 = 1 + 7 + 18 + 34 = \dfrac {4 \paren {4 + 1} \paren {5 \times 4 - 2} } 6$
:The $5$th special highly composite number after $1$, $2$, $6$, $12$
:The smallest positive integer which can be expressed as the sum of $2$ odd primes in $6$ ways:
::$60 = 7 + 53 = 13 + 47 = 17 + 43 = 19 + 41 = 23 + 37 = 29 + 31$
:The $9$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$:
::$\map {\sigma_0} {60} = 12$
:The $9$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$:
::$\dfrac {\map {\sigma_1} {60} } {60} = \dfrac {168} {60} = 2 \cdotp 8$
:The $11$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$ such that no factorial of an integer can end with $n$ zeroes
:The $14$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$:
::$60 = 10 + 20 + 30$
:The $17$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$:
::$\map {\sigma_1} {60} = 168$
:The $19$th of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
::$1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$, $16$, $18$, $24$, $30$, $36$, $42$, $48$, $60$, $\ldots$
:The $23$rd positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $\ldots$
:The $35$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $44$, $45$, $46$, $49$, $50$, $54$, $55$, $59$ which cannot be expressed as the sum of distinct pentagonal numbers
:The $37$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $37$, $38$, $42$, $43$, $44$, $48$, $49$, $50$, $54$, $55$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:The base of the sexagesimal system
:The number of degrees in an internal angle of an equilateral triangle
== Also see ==
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* 
* 
* 
* 
* 
* 
* 
* 
* 
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* 
== Historical Note ==
== Sources ==
* : $60$
* : $60$
Category:Specific Numbers
Category:60