nyuuzyou/wikis huggingface dataset:
678,901

== Number ==
$678 \, 901$ (six hundred and seventy-eight thousand, nine hundred and one) is:
:The $54 \, 973$rd prime number
:The $10$th prime number after $2$, $3$, $5$, $7$, $23$, $67$, $89$, $4567$, $78 \, 901$ consisting of a string of consecutive ascending digits (allowing $0$ to follow $9$)
== Also see ==
* 
* 
Category:Specific Numbers
Category:678,901
6786

== Number ==
$6786$ (six thousand, seevn hundred and eighty-six) is:
:$2 \times 3^2 \times 13 \times 29$
:The $116$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5995$, $6105$, $6216$, $6328$, $6441$, $6555$, $6670$:
::$6786 = \ds \sum_{k \mathop = 1}^{116} k = \dfrac {116 \times \paren {116 + 1} } 2$
:The $2$nd of the $7$th pair of triangular numbers whose sum and difference are also both triangular:
::$5460 = T_{104}$, $6786 = T_{116}$, $6786 + 5460 = T_{156}$, $6786 - 5460 = T_{51}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6786
679

== Number ==
$679$ (six hundred and seventy-nine) is:
:$7 \times 97$
:The smallest number having a multiplicative persistence of $5$.
:The $32$nd number whose divisor sum is square:
::$\map {\sigma_1} {679} = 784 = 28^2$
== Also see ==
* 
* 
== Sources ==
* : $679$
* : $679$
Category:Specific Numbers
Category:679
68

== Number ==
$68$ (sixty-eight) is:
:$2^2 \times 17$
:The $7$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$:
::$\nexists m \in \Z_{>0}: \map \phi m = 68$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $12$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$:
::$68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
:The $26$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $\ldots$
:The largest positive even integer that cannot be expressed as the sum of $2$ odd positive composite integers in at least $2$ different ways.
== Also see ==
* Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways
* 
* 
* 
== Sources ==
* : $68$
Category:Specific Numbers
Category:68
680

== Number ==
$680$ (six hundred and eighty) is:
:$2^3 \times 5 \times 17$
:The $2$nd tetrahedral number after $20$ which is the sum of two tetrahedral numbers:
::$680 = 120 + 560$
:The $15$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$, $220$, $286$, $364$, $455$, $560$:
::$680 = \ds \sum_{k \mathop = 1}^{15} \frac {k \paren {k + 1} } 2 = \dfrac {15 \paren {15 + 1} \paren {15 + 2} } 6$
== Also see ==
* 
* 
== Sources ==
* : $680$
* : $680$
Category:Specific Numbers
Category:680
6816

== Number ==
$6816$ (six thousand, eight hundred and sixteen) is:
:$2^5 \times 3 \times 71$
:The $48$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $5985$, $6256$, $6533$:
::$6816 = \ds \sum_{k \mathop = 1}^{48} \paren {6 k - 5} = 48 \paren {3 \times 48 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:6816
682

== Number ==
$682$ (six hundred and eighty-two) is:
:$2 \times 11 \times 31$
:The $5$th of the $3$-digit integers $m$ which need the largest number of reverse-and-add process iterations ($23$) before reaching a palindromic number:
::$682, 968, 1837, \ldots, 8713200023178$
:The $8$th positive integer $n$ after $157$, $262$, $367$, $412$, $472$, $487$, $487$ for which $\map \phi {2 n + 1} < \map \phi {2 n}$:
::$\map \phi {2 \times 682 + 1} = 576, \map \phi {2 \times 682} = 600$
== Also see ==
* 
* 
Category:Specific Numbers
Category:682
6825

== Number ==
$6825$ (six thousand, eight hundred and twenty-five) is:
:$3 \times 5^2 \times 7 \times 13$
:The $13$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$, $4095$, $4725$, $5355$, $5775$, $5985$, $6435$, $6615$:
::$\map {\sigma_1} {6825} - 6825 = 7063 > 6825$
== Also see ==
* 
Category:Specific Numbers
Category:6825
683

== Number ==
$683$ (six hundred and eighty-three) is:
:The $124$th prime number
:The smaller of the $8$th pair of primes whose prime gap is $8$:
::$691 - 683 = 8$
:The $31$st Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $233$, $239$, $251$, $281$, $293$, $359$, $419$, $431$, $443$, $491$, $509$, $593$, $641$, $653$, $659$:
::$2 \times 683 + 1 = 1367$, which is prime.
:The $42$nd left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $443$, $467$, $523$, $547$, $613$, $617$, $643$, $647$, $653$, $673$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:683
6859

== Number ==
$6859$ (six thousand, eight hundred and fifty-nine) is:
:$19^3$
:The $5$th cube after $6^3$, $9^3$, $12^3$, $18^3$ which can be expressed as the sum of $3$ positive cubes
:The $19$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$, $1331$, $1728$, $2197$, $2744$, $3375$, $4096$, $4913$, $5832$:
::$6859 = 19 \times 19 \times 19$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6859
6875

== Number ==
$6875$ (six thousand, eight hundred and seventy-five) is:
:$5^4 \times 11$
:The $11$th tri-automorphic number after $2$, $5$, $7$, $67$, $75$, $92$, $667$, $792$, $875$, $6667$:
::$6875 \times 3 = 141 \, 79 \mathbf {6 \, 875}$
== Also see ==
* 
Category:Specific Numbers
Category:6875
6889

== Number ==
$6889$ (six thousand, eight hundred and eighty-nine) is:
:$83^2$
:The $83$rd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $6084$, $6241$, $6400$, $6561$, $6724$
::$6889 = 83 \times 83$
== Also see ==
* 
Category:Specific Numbers
Category:6889
69

== Number ==
$69$ (sixty-nine) is:
:$3 \times 23$
:The only integer whose square and cube use each of the digits from $0$ to $9$ exactly once each:
::$69^2 = 4761$, $69^3 = 328 \, 509$
:The $4$th after $21$, $29$, $61$ of the $5$ $2$-digit positive integers which can occur as a $5$-fold repetition at the end of a square number
:The $20$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$, $36$, $38$, $47$, $48$, $53$, $57$, $62$:
::$69 = 16 + 53$
:The $24$th semiprime:
::$69 = 3 \times 23$
:The $27$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $69$, $\ldots$
:The $30$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $\ldots$
== Also see ==
* Number whose Square and Cube use all Digits Once
* 
* 
* 
* 
* 
== Sources ==
* : $69$
* : $69$
Category:Specific Numbers
Category:69
69,696

== Number ==
$69 \, 696$ (sixty-nine thousand, six hundred and ninety-six) is:
:$2^6 \times 3^2 \times 11^2$
:The $2$nd palindromic square with a non-palindromic square root after $676$:
::$69 \, 696 = 264^2$
:The $22$nd square after $1$, $4$, $9$, $16$, $25$, $\ldots$, $441$, $484$, $676$, $1444$, $7744$, $11 \, 881$, $29 \, 929$, $44 \, 944$, $55 \, 225$ which has no more than $2$ distinct digits and does not end in $0$:
::$69 \, 696 = 264^2$
:The $264$th square number:
::$69 \, 696 = 264 \times 264$
== Also see ==
* 
* 
* 
== Sources ==
* : $69,696$
* : $69,696$
Category:Specific Numbers
Category:69,696
69,720,375,229,712,477,164,533,808,935,312,303,556,800

== Number ==
$69 \, 720 \, 375 \, 229 \, 712 \, 477 \, 164 \, 533 \, 808 \, 935 \, 312 \, 303 \, 556 \, 800$ is:
:$2^6 \times 3^4 \times 5^2 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 31 \times 37 \times 41 \times 43 \times 47 \times 53 \times 59 \times 61 \times 67 \times 71 \times 73 \times 79 \times 83 \times 89 \times 97$
:The smallest positive integer which is divisible by each of the integers from $1$ to $100$
== Also see ==
* Smallest Integer Divisible by All Numbers from 1 to 100
== Sources ==
* : $69,720,375,229,712,477,164,533,808,935,312,303,556,800$
Category:Specific Numbers
Category:69,720,375,229,712,477,164,533,808,935,312,303,556,800
690

== Number ==
$690$ (six hundred and ninety) is:
:$2 \times 3 \times 5 \times 23$
:The $9$th positive integer after $1$, $7$, $102$, $110$, $142$, $159$, $187$, $381$ the sum of whose divisors is a cube:
::$\map {\sigma_1} {690} = 1728 = 12^3$
:The $35$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $576$, $588$, $627$, $634$, $636$, $645$, $648$, $654$, $663$, $666$:
::$6 + 9 + 0 = 2 + 3 + 5 + 2 + 3 = 15$
== Also see ==
* 
* 
Category:Specific Numbers
Category:690
6903

== Number ==
$6903$ (six thousand, nine hundred and three) is:
:$3^2 \times 13 \times 59$
:The $59$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $5565$, $5778$, $5995$, $6216$, $6441$, $6670$:
::$6903 = \ds \sum_{k \mathop = 1}^{59} \paren {4 k - 3} = 59 \paren {2 \times 59 - 1}$
:The $117$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5995$, $6105$, $6216$, $6328$, $6441$, $6555$, $6670$, $6786$:
::$6903 = \ds \sum_{k \mathop = 1}^{117} k = \dfrac {117 \times \paren {117 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6903
691

== Number ==
$691$ (six hundred and ninety-one) is:
:The $125$th prime number.
:The larger of the $8$th pair of primes whose prime gap is $8$:
::$691 - 683 = 8$
:The smaller of the $11$th pair of primes whose prime gap is $10$:
::$701 - 691 = 10$
:The $7$th candidate Lychrel number.
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:691
6917

== Number ==
$6917$ (six thousand, nine hundred and seventeen) is:
:The $890$th prime number
:The $21$st integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
::$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$, $38$, $94$, $166$, $324$, $379$, $469$, $546$, $974$, $1963$, $3507$, $3610$, $6917$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6917
6924

== Number ==
$6924$ (six thousand, nine hundred and twenty-four) is:
:$2^2 \times 3 \times 577$
:The magic constant of a magic square of order $24$, after $1$, $(5)$, $15$, $34$, $\ldots$, $2465$, $2925$, $3439$, $4010$, $4641$, $5335$, $6095$:
::$\ds 6924 = \dfrac 1 {24} \sum_{k \mathop = 1}^{24^2} k = \dfrac {24 \paren {24^2 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:6924
693

== Number ==
$693$ (six hundred and ninety three) is:
:$3^2 \times 7 \times 11$
:One of the legs of the Pythagorean triangle $693-1924-2045$, whose area is $666 \, 666$.
== Also see ==
* Pythagorean Triangles whose Areas are Repdigit Numbers
Category:Specific Numbers
Category:693
6930

== Number ==
$6930$ (six thousand, nine hundred and thirty) is:
:$2 \times 3^2 \times 5 \times 7 \times 11$
:The $27$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $\ldots$, $1015$, $1240$, $1496$, $1785$, $2109$, $2470$, $2870$, $3311$, $3795$, $4324$, $4900$, $5525$, $6201$:
::$6930 = \ds \sum_{k \mathop = 1}^{27} k^2 = \dfrac {27 \paren {27 + 1} \paren {2 \times 27 + 1} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:6930
6949

== Number ==
$6949$ (six thousand, nine hundred and forty-nine) is:
:The $893$rd prime number
:The $16$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$, $991$, $6469$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6949
697

== Number ==
$697$ (six hundred and ninety-seven) is:
:$17 \times 41$
:The $17$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$, $235$, $286$, $342$, $403$, $469$, $540$, $616$:
::$697 = \ds \sum_{k \mathop = 1}^{17} \paren {5 k - 4} = \dfrac {17 \paren {5 \times 17 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:697
698,896

== Number ==
$698 \, 896$ (sixty hundred and ninety-eight thousand, eight hundred and ninety-six) is:
:$2^4 \times 11^2 \times 19^2$
:The $836$th square number:
::$698 \, 896 = 836 \times 836$
:The $4$th palindromic square with a non-palindromic square root after $676$, $69 \, 696$, $94 \, 249$:
::$698 \, 896 = 836^2$
:The $1$st palindromic square with an even number of digits with a non-palindromic square root:
::$698 \, 896 = 836^2$
== Also see ==
* 
* 
* 
== Sources ==
* : $698,896$
* : $698,896$
Category:Specific Numbers
Category:698,896
6999

== Number ==
$6999$ (six thousand, nine hundred and ninety-nine) is:
:$3 \times 2333$
:The $1$st of the $4$-digit integers $m$ which need the largest number of reverse-and-add process iterations ($20$) before reaching a palindromic number:
::$6999$, $16995$, $76956$, $\ldots$, $16668488486661$
== Also see ==
* 
== Sources ==
* : $6999$
* : $6999$
Category:Specific Numbers
Category:6999
6 Pints from 7 Pints and 5 Pints

== Classic Problem ==
:You have $12$ pints of wine in a barrel and you wish to divide it into $6$ pints for a friend and $6$ pints for yourself,
:but you only have containers holding $7$ pints and $5$ pints.
:How can you succeed?
== Solution ==
Fill the $5$ pint jug from the barrel, and empty its contents into the $7$ pint jug.
Fill the $5$ pint jug from the barrel again, and fill the $7$ pint jug from it.
There will then be $3$ pints in the $5$ pint jug.
Empty the $7$ pint jug into the barrel.
Pour the $3$ pints from the $5$ pint jug into the $7$ pint jug, which now contains just those $3$ pints.
Fill the $5$ pint jug from the barrel, and fill the $7$ pint jug from it.
This takes $4$ pints, so the $5$ pint jug now contains $1$ pint.
Empty the $7$ pint jug into the barrel, and pour the $1$ pint from the $5$ pint jug into the $7$ pint jug.
Fill the $5$ pint jug from the barrel, and pour the contents into the $7$ pint jug.
The $7$ pint jug now contains $6$ pints, and there are $6$ pints remaining in the barrel.
== Sources ==
* 
* : Rational Amusements for Winter Evenings: $151$
Category:Liquid Pouring Problems
7,109,376

== Number ==
$7 \, 109 \, 376$ (seven million, one hundred and nine thousand, three hundred and seventy-six) is:
:$2^8 \times 3 \times 9257$
:The $13$th automorphic number after $1$, $5$, $6$, $25$, $76$, $376$, $625$, $9376$, $90 \, 625$, $109 \, 376$, $890 \, 625$, $2 \, 890 \, 625$:
::$7 \, 109 \, 376^2 = 50 \, 543 \, 22 \mathbf {7 \, 109 \, 376}$
== Also see ==
* 
Category:Specific Numbers
Category:7,109,376
7,529,536

== Number ==
$7 \, 529 \, 536$ (seven million, five hundred and twenty-nine thousand, five hundred and thirty-six) is:
:$2^6 \times 7^6$
:The $6$th power of $14$ after $(1)$, $14$, $196$, $2744$, $38 \, 416$, $537 \, 824$:
::$7 \, 529 \, 536 = 14^6$
:The $14$th sixth power after $1$, $64$, $729$, $4096$, $15 \, 625$, $46 \, 656$, $117 \, 649$, $262 \, 144$, $531 \, 441$, $1 \, 000 \, 000$, $1 \, 771 \, 561$, $2 \, 985 \, 984$, $4 \, 826 \, 809$:
::$7 \, 529 \, 536 = 14 \times 14 \times 14 \times 14 \times 14 \times 14$
:The $196$th cube number:
::$7 \, 529 \, 536 = 196 \times 196 \times 196$
:The $2744$th square number:
::$7 \, 529 \, 536 = 2744 \times 2744$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:7,529,536
7,683,984

== Number ==
$7 \, 683 \, 984$ is:
:$2^4 \times 3^4 \times 7^2 \times 11^2$
:The $2772$nd square number:
::$7 \, 683 \, 984 = 2772 \times 2772$
:An EPORN which is not a multiple of $10$:
::$7 \, 683 \, 984 = 2772 \times 2772 = 1584 \times 4851 = \paren {2^2 \times 3^2 \times 7 \times 11} \times \paren {2^2 \times 3^2 \times 7 \times 11} = \paren {2^4 \times 3^2 \times 11} \times \paren {3^2 \times 7^2 \times 11}$
== Also see ==
* EPORN/Examples/7,683,984
* 
== Sources ==
* : $63,504$
Category:Specific Numbers
Category:7,683,984
7,845,601

== Number ==
$7 \, 845 \, 601$ (seven million, eight hundred and forty-five thousand, six hundred and one) is:
:$2801^2$
:The $7$th and last after $1 \, 048 \, 576$, $1 \, 056 \, 784$, $1 \, 085 \, 764$, $5 \, 740 \, 816$, $5 \, 764 \, 801$, $6 \, 754 \, 801$ of a sequence of $7$ numbers which are square and anagrams of each other
:The $2801$st square number:
::$7 \, 845 \, 601 = 2801 \times 2801$
== Also see ==
* 
* 
Category:Specific Numbers
Category:7,845,601
7,906,276

== Number ==
$7 \, 906 \, 276$ (seven million, nine hundred and six thousand, two hundred and seventy-six) is:
:$2^2 \times 7 \times 41 \times 71 \times 97$
:The $4$th integer after $1$, $210$, $40 \, 755$ to be both pentagonal and triangular
:The $2296$th pentagonal number:
::$7 \, 906 \, 276 = \ds \sum_{k \mathop = 1}^{2296} \paren {3 k - 2} = \dfrac {2296 \paren {3 \times 2296 - 1} } 2$
:The $3976$th triangular number:
::$7 \, 906 \, 276 = \ds \sum_{k \mathop = 1}^{3976} k = \dfrac {3976 \times \paren {3976 + 1} } 2$
:The $4951$st generalized pentagonal number:
::$7 \, 906 \, 276 = \ds \sum_{k \mathop = 1}^{2296} \paren {3 k - 2} = \dfrac {2296 \paren {3 \times 2296 - 1} } 2$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:7,906,276
70

== Number ==
$70$ (seventy) is:
:$2 \times 5 \times 7$
:The $1$st weird number:
::$\map {\sigma_1} {70} - 70 = 74$: its aliquot parts are $1$, $2$, $5$, $7$, $10$, $14$, $35$, from which $70$ cannot be made.
:The $2$nd primitive abundant number after $20$:
::$1 + 2 + 5 + 7 + 10 + 14 + 35 = 74 > 70$
:The $4$th sphenic number after $30$, $42$, $66$:
::$70 = 2 \times 5 \times 7$
:The $4$th central binomial coefficient after $2$, $6$, $20$:
::$70 = \dbinom {2 \times 4} 4 := \dfrac {8!} {\paren {4!}^2}$
:The $5$th pentatope number after $1$, $5$, $15$, $35$:
::$70 = 1 + 4 + 10 + 20 + 35 = \dfrac {5 \paren {5 + 1} \paren {5 + 2} \paren {5 + 3} } {24}$
:The $5$th number after $1$, $3$, $22$, $66$ whose divisor sum is square:
:
:The $7$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$:
::$70 = 1 + 4 + 7 + 10 + 13 + 16 + 19 = \dfrac {7 \paren {3 \times 7 - 1} } 2$
:The $7$th integer $n$ after $1$, $3$, $15$, $30$, $35$, $56$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} {70} = 8$, $\map \phi {70} = 24$, $\map {\sigma_1} {70} = 144$
:The $9$th integer after $7$, $13$, $19$, $35$, $38$, $41$, $57$, $65$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$70^2 = 4900$; $4 = 2^2$, $900 = 30^2$
:The $13$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$, $40$, $51$, $57$:
::$70 = \dfrac {7 \paren {3 \times 7 - 1} } 2$
:The $13$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$:
::$70 \to 7^2 + 0^2 = 49 \to 4^2 + 9^2 = 16 + 81 = 97 \to 9^2 + 7^2 = 81 + 49 = 130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $70$
* : $70$
Category:Specific Numbers
Category:70
701

== Number ==
$701$ (seven hundred and one) is:
:The $126$th prime number.
:The smaller of the $9$th pair of primes whose prime gap is $8$:
::$709 - 701 = 8$
:The $10$th obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$, $509$, $599$
:The larger of the $11$th pair of primes whose prime gap is $10$:
::$701 - 691 = 10$
:The $21$st emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$, $157$, $167$, $179$, $199$, $311$, $337$, $347$, $359$, $389$
:The $45$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $491$, $499$, $503$, $509$, $541$, $571$, $577$, $593$, $619$, $647$, $659$
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:701
7021

== Number ==
$7021$ (seven thousand and twenty-one) is:
:$7 \times 17 \times 59$
:The $118$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5995$, $6105$, $6216$, $6328$, $6441$, $6555$, $6670$, $6786$, $6903$:
::$7021 = \ds \sum_{k \mathop = 1}^{118} k = \dfrac {118 \times \paren {118 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:7021
703

== Number ==
$703$ (seven hundred and three) is:
:$19 \times 37$
:The second of the $3$rd pair of triangular numbers whose sum and difference are also both triangular:
::$378 = T_{27}$, $703 = T_{37}$, $378 + 703 = T_{46}$, $703 - 378 = T_{25}$
:The $5$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$:
::$3^{703} \equiv 3 \pmod {703}$
:The $7$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$:
::$703^2 = 494 \, 209 \to 494 + 209 = 703$
:The $9$th Fermat pseudoprime to base $4$ after $15$, $85$, $91$, $341$, $435$, $451$, $561$, $645$:
::$4^{703} \equiv 4 \pmod {703}$
:The $19$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $378$, $435$, $496$, $561$, $630$:
::$703 = \ds \sum_{k \mathop = 1}^{19} \paren {4 k - 3} = 19 \paren {2 \times 19 - 1}$
:The $37$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$, $630$, $666$:
::$703 = \ds \sum_{k \mathop = 1}^{37} k = \dfrac {37 \times \paren {37 + 1} } 2$
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Category:Specific Numbers
Category:703
704

== Number ==
$704$ (seven hundred and four) is:
:$2^6 \times 11$
:The $15$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$, $480$, $512$, $576$, $640$, $648$, $672$ with $7$ or more prime factors:
::$704 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 11$
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Category:Specific Numbers
Category:704
7056

== Number ==
$7056$ (seven thousand and fifty-six) is:
:$2^4 \times 3^2 \times 49^2$
:The $32$nd square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $5184$, $5776$, $6084$, $6400$ to be the divisor sum value of some (strictly) positive integer:
::$7056 = \map {\sigma_1} {2460}$
:The $84$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $6084$, $6241$, $6400$, $6561$, $6724$, $6889$
::$7056 = 84 \times 84$
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Category:7056
7057

== Number ==
$7057$ (seven thousand and fifty-seven) is:
:The $907$th prime number
:The $17$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$, $257$, $401$, $577$, $677$, $1297$, $1601$, $2917$, $3137$, $4357$, $5477$:
::$7057 = 84^2 + 1$
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Category:7057
706

== Number ==
$706$ (seven hundred and six) is:
:$2 \times 353$
:The $11$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $85$, $91$, $130$, $214$, $226$, $370$, $526$, $706$, $\ldots$
:The $23$rd positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $196$, $214$, $226$, $289$, $324$, $370$, $400$, $526$, $529$, $625$, $676$, $706$, $\ldots$
:The $36$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $576$, $588$, $627$, $634$, $636$, $645$, $648$, $654$, $663$, $666$, $690$:
::$7 + 0 + 6 = 2 + 3 + 5 + 3 = 13$
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Category:706
709

== Number ==
$709$ (seven hundred and nine) is:
:The $127$th prime number.
:The larger of the $9$th pair of primes whose prime gap is $8$:
::$709 - 701 = 8$
:The smaller of the $12$th pair of primes whose prime gap is $10$:
::$719 - 709 = 10$
:The $22$nd emirp after $13$, $17$, $31$, $37$, $\ldots$, $311$, $337$, $347$, $359$, $389$
:The $46$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $491$, $499$, $503$, $509$, $541$, $571$, $577$, $593$, $619$, $647$, $659$, $701$
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Category:Specific Numbers
Category:709