nyuuzyou/wikis huggingface dataset:
7957

== Number ==
$7957$ (seven thousand, nine hundred and fifty-seven) is:
:$73 \times 109$
:The $19$th Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$, $2465$, $2701$, $2821$, $3277$, $4033$, $4369$, $4371$, $4681$, $5461$, $6601$:
::$2^{7957} \equiv 2 \pmod {7957}$: $7957 = 73 \times 109$
== Also see ==
* 
Category:Specific Numbers
Category:7957
797

== Number ==
$797$ (seven hundred and ninety-seven) is:
:The $139$th prime number
:The $12$th two-sided prime after $2$, $3$, $5$, $7$, $23$, $37$, $53$, $73$, $313$, $317$, $373$:
::$797$ is prime; $79$, $7$ are prime; $97$, $7$ are prime
:The larger of the $13$th pair of primes whose prime gap is $10$:
::$797 - 787 = 10$
:The $18$th palindromic prime:
::$2$, $3$, $5$, $7$, $11$, $101$, $131$, $151$, $181$, $191$, $313$, $353$, $373$, $383$, $727$, $757$, $787$, $797$, $\ldots$
:The $27$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $\ldots$, $317$, $373$, $379$, $593$, $599$, $719$, $733$, $739$
:The $35$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $353$, $373$, $383$, $433$, $443$, $449$, $499$, $557$, $577$, $599$, $661$, $677$, $727$, $733$, $757$, $773$, $787$
:The $45$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $443$, $467$, $523$, $547$, $613$, $617$, $643$, $647$, $653$, $673$, $683$, $743$, $773$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:797
798,644

== Number ==
$798 \, 644$ (seven hundred and ninety-eight thousand, six hundred and forty-four) is:
:$2^2 \times 7 \times 11 \times 2593$
:The $2$nd positive integer after $836$ whose square is a palindromic number with an even number of digits:
::$798 \, 644^2 = 637 \, 832 \, 238 \, 736$ ($12$ digits)
== Also see ==
* 
== Sources ==
* : $798,644$
* : $836$
* : $798,644$
Category:Specific Numbers
Category:798,644
7998

== Number ==
$7998$ (seven thousand, nine hundred and ninety-eight) is:
:$2 \times 3 \times 31 \times 43$
:The $2$nd after $6999$ of the $4$-digit integers $m$ which need the largest number of reverse-and-add process iterations ($20$) before reaching a palindromic number:
::$7998$, $16995$, $76956$, $\ldots$, $16668488486661$
== Also see ==
* 
== Sources ==
* : $6999$
* : $6999$
Category:Specific Numbers
Category:7998
8,212,890,625

== Number ==
$8 \, 212 \, 890 \, 625$ is:
:$5^{10} \times 29^2$
:The $90 \, 625$th square number:
::$8 \, 212 \, 890 \, 625 = 90 \, 625 \times 90 \, 625$
:The $19$th automorphic number after $1$, $5$, $6$, $25$, $76$, $376$, $625$, $9376$, $\ldots$, $12 \, 890 \, 625$, $87 \, 109 \, 376$, $212 \, 890 \, 625$, $787 \, 109 \, 376$, $1 \, 787 \, 109 \, 376$:
::$8 \, 212 \, 890 \, 625^2 = 67 \, 451 \, 572 \, 41\mathbf {8 \, 212 \, 890 \, 625}$
:and the $2$nd of the only $2$ such with $10$ digits.
== Also see ==
* 
* 
== Sources ==
* : $1,787,109,376$
* : $8,212,890,625$
* : $1,787,109,376$
Category:Specific Numbers
Category:8,212,890,625
8,388,607

== Number ==
$8 \, 388 \, 607$ (eight million, three hundred and eighty-eight thousand, six hundred and seven) is:
:$47 \times 178 \, 481$
:The $2$nd Mersenne number after $2047$ which is composite:
::$8 \, 388 \, 607 = 2^{23} - 1 = 47 \times 178 \, 481$
:The $9$th Mersenne number after $3$, $7$, $31$, $127$, $2047$, $8191$, $131 \, 071$, $524 \, 287$:
::$8 \, 388 \, 607 = 2^{23} - 1$
== Historical Note ==
== Also see ==
* 
* 
Category:Specific Numbers
Category:8,388,607
8,388,608

== Number ==
$8 \, 388 \, 608$ (eight million, three hundred and eighty-eight thousand, six hundred and eight) is:
:$2^{23}$
:The $23$rd power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $\ldots$, $131 \, 072$, $262 \, 144$, $524 \, 288$, $1 \, 048 \, 576$, $2 \, 097 \, 152$, $4 \, 194 \, 304$:
::$8 \, 388 \, 608 = 2^{23}$
:The $24$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $\ldots$, $65 \, 536$, $131 \, 072$, $262 \, 144$, $524 \, 288$, $1 \, 048 \, 576$, $2 \, 097 \, 152$, $4 \, 194 \, 304$:
::$\map {\sigma_1} {8 \, 388 \, 608} = 16 \, 777 \, 215 = 2 \times 8 \, 388 \, 608 - 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:8,388,608
8,549,176,320

== Number ==
$8 \, 549 \, 176 \, 320$ is:
:$2^{10} \times 3^3 \times 5 \times 61 \, 843$
:The pandigital integer formed from the digits from $0$ to $9$ arranged in alphabetical order
== Also see ==
* Pandigital Integer Formed by Digits in Alphabetical Order
== Sources ==
* : $8,549,176,320$
Category:Specific Numbers
Category:8,549,176,320
8,616,460,799

== Number ==
$8 \, 616 \, 460 \, 799$ (eight billion, six hundred and sixteen million, four hundred and sixty thousand, seven hundred and ninety-nine) is:
:$89 \, 681 \times 96 \, 079$
:Known as Jevons' number.
== Also see ==
* Definition:Jevons' Number
== Historical Note ==
== Sources ==
* 
Category:Specific Numbers
Category:8,616,460,799
8,981,279

== Number ==
$8 \, 981 \, 279$ (eight million, nine hundred and eighty-one thousand, two hundred and seventy-nine) is:
:The $601 \, 286$th prime number
:The $4$th term of the smallest Cunningham chain of the first kind of length $7$:
::$1 \, 122 \, 659$, $2 \, 245 \, 319$, $4 \, 490 \, 639$, $8 \, 981 \, 279$, $17 \, 962 \, 559$, $35 \, 925 \, 119$, $71 \, 850 \, 239$
== Also see ==
* 
* 
Category:Specific Numbers
Category:8,981,279
80

== Number ==
$80$ (eighty) is:
:$2 \times 4 \times 5$
:The $18$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$, $66$, $72$, $78$:
::$80 = 2 + 8 + 10 + 20 + 40$
:The $42$nd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $50$, $54$, $55$, $59$, $60$, $61$, $65$, $66$, $67$, $72$, $77$ which cannot be expressed as the sum of distinct pentagonal numbers.
== Also see ==
* 
* 
Category:Specific Numbers
Category:80
8000

== Number ==
$8000$ (eight thousand) is:
:$2^6 \times 5^3$
:The $6$th cube after $6^3$, $9^3$, $12^3$, $18^3$, $19^3$ which can be expressed as the sum of $3$ positive cubes
:The $20$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$, $1331$, $1728$, $2197$, $2744$, $3375$, $4096$, $4913$, $5832$, $6859$:
::$8000 = 20 \times 20 \times 20$
:$8000 = 20^3 = 11^3 + 12^3 + 13^3 + 14^3$
== Also see ==
* Cube of 20 is Sum of Sequence of 4 Consecutive Cubes
* 
* 
== Sources ==
* : $8000$
* : $8000$
Category:Specific Numbers
Category:8000
8001

== Number ==
$8001$ (eight thousand and one) is:
:$3^2 \times 7 \times 127$
:The $5$th triangular number after $28$, $91$, $351$, $2926$ which is the sum of $2$ cubes:
::$8001 = 1 + 8000 = 1^3 + 20^3$
:The $126$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $6786$, $6903$, $7021$, $7140$, $7260$, $7381$, $7503$, $7626$, $7750$, $7875$:
::$8001 = \ds \sum_{k \mathop = 1}^{126} k = \dfrac {126 \times \paren {126 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:8001
8008

== Number ==
$8008$ (eight thousand and eight) is:
:$2^3 \times 7 \times 11 \times 13$
:The $52$nd octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $5985$, $6256$, $6533$, $6816$, $7105$, $7400$, $7701$:
::$8008 = \ds \sum_{k \mathop = 1}^{52} \paren {6 k - 5} = 52 \paren {3 \times 52 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:8008
802,359,150,003,121,605,557,551,380,867,519,560,344,356,971

== Number ==
$802 \, 359 \, 150 \, 003 \, 121 \, 605 \, 557 \, 551 \, 380 \, 867 \, 519 \, 560 \, 344 \, 356 \, 971$ is:
:A prime number
:The $1$st term of a prime quadruple:
::$802 \, 359 \, 150 \, 003 \, 121 \, 605 \, 557 \, 551 \, 380 \, 867 \, 519 \, 560 \, 344 \, 356 \, 971$
::$802 \, 359 \, 150 \, 003 \, 121 \, 605 \, 557 \, 551 \, 380 \, 867 \, 519 \, 560 \, 344 \, 356 \, 973$
::$802 \, 359 \, 150 \, 003 \, 121 \, 605 \, 557 \, 551 \, 380 \, 867 \, 519 \, 560 \, 344 \, 356 \, 977$
::$802 \, 359 \, 150 \, 003 \, 121 \, 605 \, 557 \, 551 \, 380 \, 867 \, 519 \, 560 \, 344 \, 356 \, 979$
== Also see ==
* 
== Historical Note ==
== Sources ==
* : $802,359,150,003,121,605,557,551,380,867,519,560,344,356,971$
* : $802,359,150,003,121,605,557,551,380,867,519,560,344,356,971$
Category:Specific Numbers
Category:802,359,150,003,121,605,557,551,380,867,519,560,344,356,971
802,359,150,003,121,605,557,551,380,867,519,560,344,356,971/Historical Note

== Historical Note on $802,359,150,003,121,605,557,551,380,867,519,560,344,356,971$ ==
According to in his of $1986$, this factoid appears in Volume $14$ issue $3$ of , but this has not been corroborated.
== Sources ==
* : $802,359,150,003,121,605,557,551,380,867,519,560,344,356,971$
* : $802,359,150,003,121,605,557,551,380,867,519,560,344,356,971$
Category:Specific Numbers
Category:802,359,150,003,121,605,557,551,380,867,519,560,344,356,971
8042

== Number ==
$8042$ (eight thousand and forty-two) is:
:$2 \times 4021$
:The largest number (probably) that cannot be expressed as the sum of fewer than $8$ cubes
== Also see ==
* Largest Number Not Expressible as Sum of Fewer than 8 Cubes
== Sources ==
* : $8042$
Category:Specific Numbers
Category:8042
805

== Number ==
$805$ (eight hundred and five) is:
:$5 \times 7 \times 23$
:The $23$rd second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $222$, $260$, $301$, $345$, $392$, $442$, $495$, $551$, $610$, $672$, $737$:
::$805 = \dfrac {23 \paren {3 \times 23 + 1} } 2$
:The $46$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$, $532$, $551$, $590$, $610$, $651$, $672$, $715$, $737$, $782$:
::$805 = \dfrac {23 \paren {3 \times 23 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:805
806

== Number ==
$806$ (eight hundred and six) is:
: $2 \times 13 \times 31$
:The $3$rd positive integer after $504$, $756$ which can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other:
::$806 = 13 \times 62 = 26 \times 31$
== Also see ==
* 
Category:Specific Numbers
Category:806
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

== Number ==
$808 \, 017 \, 424 \, 794 \, 512 \, 875 \, 886 \, 459 \, 904 \, 961 \, 710 \, 757 \, 005 \, 754 \, 368 \, 000 \, 000 \, 000$ is:
:$2^{46} \times 3^{20} \times 5^9 \times 7^6 \times 11^2 \times 13^3 \times 17 \times 19 \times 23 \times 29 \times 31 \times 41 \times 47 \times 59 \times 71$
:The order of the Fischer-Griess Monster $\mathrm M$
== Also see ==
* 196,883
* Order of Fischer-Griess Monster
== Sources ==
* : $808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000$
Category:Specific Numbers
Category:808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
8085

== Number ==
$8085$ (eight thousand and eighty-five) is:
:$3 \times 5 \times 7^2 \times 11$
:The $17$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$, $4095$, $4725$, $5355$, $5775$, $5985$, $6435$, $6615$, $6825$, $7245$, $7425$, $7875$:
::$\map {\sigma_1} {8085} - 8085 = 8331 > 8085$
== Also see ==
* 
Category:Specific Numbers
Category:8085
809

== Number ==
$809$ (eight hundred and nine) is:
:The $140$th prime number.
:The upper end of the $8$th record-breaking gap between twin primes:
::$809 - 661 = 148$
:The $12$th obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$, $509$, $599$, $701$, $757$
:The smaller of the $31$st pair of twin primes:
::$811 - 809 = 2$
:The $35$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $431$, $443$, $491$, $509$, $593$, $641$, $653$, $659$, $683$, $719$, $743$, $761$:
::$2 \times 809 + 1 = 1619$, which is prime.
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:809
81

== Number ==
$81$ (eighty-one) is:
:$3^4$
:The $1$st square number which is also heptagonal:
::$81 = \dfrac {6 \paren {5 \times 6 - 3} } 2 = 9^2$
:The $2$nd power of $9$ after $(1)$, $9$:
::$81 = 9^2$
:The $2$nd, after $1$, of the $4$ harshad numbers which can each be expressed as the product of the sum of its digits and the reversal of the sum of its digits:
::$81 = 9 \times 9 = 9 \times \paren {1 + 8}$
:The $3$rd and last number after $0$, $1$ whose square root equals the sum of its digits:
::$\sqrt {81} = 9 = 8 + 1$
:The $3$rd fourth power after $1$, $16$:
::$81 = 3 \times 3 \times 3 \times 3$
:The $4$th power of $3$ after $(1)$, $3$, $9$, $27$:
::$81 = 3^4$
:The $6$th heptagonal number after $1$, $7$, $18$, $34$, $55$:
::$81 = 1 + 7 + 11 + 16 + 21 + 26 = \dfrac {6 \paren {5 \times 6 - 3} } 2$
:The $6$th number after $1$, $3$, $22$, $66$, $70$, and $2$nd square number after $1$, whose divisor sum is square:
:
:The index (after $2$, $3$, $6$, $30$, $75$) of the $6$th Woodall prime:
::$81 \times 2^{81} - 1$
:The $9$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$:
::$81 = 9 \times 9$
:The $9$th square after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$ which has no more than $2$ distinct digits and does not end in $0$:
::$81 = 9^2$
:The smallest positive integer the decimal expansion of whose reciprocal has a period of $9$:
::$\dfrac 1 {81} = 0 \cdotp \dot 01234 \, 567 \dot 9$
:The $13$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$
:The $31$st of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $69$, $74$, $75$, $79$, $81$, $\ldots$
:The $35$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{81} = 2 \, 417 \, 851 \, 639 \, 229 \, 258 \, 349 \, 412 \, 352$
:The $43$rd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $50$, $54$, $55$, $59$, $60$, $61$, $65$, $66$, $67$, $72$, $77$, $80$ which cannot be expressed as the sum of distinct pentagonal numbers
:One of the cycle of $5$ numbers to which Kaprekar's process on $2$-digit numbers converges:
::$81 \to 63 \to 27 \to 45 \to 09 \to 81$
:A harshad number:
::$81 = 9 \times 9 = 9 \times \left({1 + 8}\right)$
== Also see ==
* Square Numbers whose Divisor Sum is Square
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $81$
* : $81$
Category:Specific Numbers
Category:81
81,081

== Number ==
$81 \, 081$ (eighty-one thousand and eighty-one) is:
:$3^4 \times 7 \times 11 \times 13$
:The $1$st odd abundant number which does not have $5$ as a prime factor:
::$\map {\sigma_1} {81 \, 081} - 81 \, 081 = 81 \, 543 > 81 \, 081$
== Also see ==
* 
== Sources ==
* : $81,081$
Category:Specific Numbers
Category:81,081
8100

== Number ==
$8100$ (eight thousand, one hundred) is:
:$2^2 \times 3^4 \times 5^2$
:The $34$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $5184$, $5776$, $6084$, $6400$, $7056$, $7744$ to be the divisor sum value of some (strictly) positive integer:
::$8100 = \map {\sigma_1} {3560}$
:The $90$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $6084$, $6241$, $6400$, $6561$, $6724$, $6889$, $7056$, $7225$, $7396$, $7569$, $7744$, $7921$
::$8100 = 90 \times 90$
== Also see ==
* 
* 
Category:Specific Numbers
Category:8100
8101

== Number ==
$8101$ (eight thousand, one hundred and one) is:
:The $1019$th prime number
:The $18$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$, $257$, $401$, $577$, $677$, $1297$, $1601$, $2917$, $3137$, $4357$, $5477$, $7057$:
::$8101 = 90^2 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:8101
811

== Number ==
$811$ (eight hundred and eleven) is:
:The $141$st prime number.
:The smaller of the $14$th pair of primes whose prime gap is $10$:
::$821 - 811 = 10$
:The larger of the $31$st pair of twin primes:
::$811 - 809 = 2$
:The $36$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $727$, $733$, $757$, $773$, $787$, $797$
:The $49$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $701$, $709$, $727$, $743$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:811
8125

== Number ==
$8125$ (eight thousand, one hundred and twenty-five) is:
:$5^4 \times 13$
:The $25$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $\ldots$, $1470$, $1800$, $2176$, $2601$, $3078$, $3610$, $4200$, $4851$, $5566$, $6348$, $7200$:
::$8125 = \ds \sum_{k \mathop = 1}^{25} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {25^2 \paren {25 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:8125
8128

== Number ==
$8128$ (eight thousand, one hundred and twenty-eight) is:
:$2^6 \times 127$
:The $4$th perfect number after $6$, $28$, $496$:
::$8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = \map {\sigma_1} {8128} - 8128 = 64 \times 127 = 2^{7 - 1} \paren {2^7 - 1}$
:The $64$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $6903$, $7140$, $7381$, $7626$, $7875$:
::$8128 = \ds \sum_{k \mathop = 1}^{64} \paren {4 k - 3} = 64 \paren {2 \times 64 - 1}$
:The $127$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$:
::$8128 = \ds \sum_{k \mathop = 1}^{127} k = \dfrac {127 \times \paren {127 + 1} } 2$
== Also see ==
* 
* 
* 
== Sources ==
* : $127$
* : $8128$
* : $127$
* : $8128$
Category:Specific Numbers
Category:8128
816

== Number ==
$816$ (eight hundred and sixteen) is:
:$2^4 \times 3 \times 17$
:The $16$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$, $220$, $286$, $364$, $455$, $560$, $680$:
::$816 = \ds \sum_{k \mathop = 1}^{16} \frac {k \paren {k + 1} } 2 = \dfrac {16 \paren {16 + 1} \paren {16 + 2} } 6$
:The $34$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $216$, $224$, $312$, $315$, $384$, $432$, $612$, $624$, $672$, $735$:
::$816 = 17 \times 48 = 17 \times \paren {8 \times 1 \times 6}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:816
817

== Number ==
$817$ (eight hundred and seventeen) is:
:$19 \times 43$
:The $17$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$, $331$, $397$, $469$, $547$, $631$, $721$:
::$817 = \ds 1 + \sum_{k \mathop = 1}^{17 - 1} 6 k = 17^3 - 16^3$
== Also see ==
* 
Category:Specific Numbers
Category:817
819

== Number ==
$819$ (eight hundred and nineteen) is:
:$3^2 \times 7 \times 13$
:The $13$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $91$, $140$, $204$, $285$, $385$, $506$, $650$:
::$819 = \ds \sum_{k \mathop = 1}^{13} k^2 = \dfrac {13 \paren {13 + 1} \paren {2 \times 13 + 1} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:819
8191

== Number ==
$8191$ (eight thousand, one hundred and ninety-one) is:
:The $1028$th prime number
:The $2$nd of the $2$ known numbers (with $31$) expressible as the sum of successive powers starting from $1$ in in $2$ different ways:
::$8191 = 1 + 90 + 90^2 = 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10} + 2^{11} + 2^{12}$
:The $5$th Mersenne prime after $3$, $7$, $31$, $127$:
::$8191 = 2^{13} - 1$
:The $6$th Mersenne number after $3$, $7$, $31$, $127$, $2047$:
::$8191 = 2^{13} - 1$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $8191$
* : $2^{8191} - 1$
* : $8191$
* : $2^{8191} - 1$
Category:Specific Numbers
Category:8191
8192

== Number ==
$8192$ (eight thousand, one hundred and ninety-two) is:
:$2^{13}$
:The $13$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$, $4096$:
::$8192 = 2^{13}$
:The $14$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$, $4096$:
::$\map {\sigma_1} {8192} = 16 \, 383 = 2 \times 8192 - 1$
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Category:Specific Numbers
Category:8192
82

== Number ==
$82$ (eighty-two) is:
:$2 \times 41$
:The $15$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$, $70$, $79$:
::$82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
:The $26$th semiprime:
::$82 = 2 \times 41$
:The $32$nd of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $69$, $74$, $75$, $79$, $81$, $82$, $\ldots$
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Category:Specific Numbers
Category:82
82,428,241

== Number ==
$82 \, 428 \, 241$ is:
:$7^2 \times 1297^2$
:The $2$nd square number after $8281$ whose digits form $2$ consecutive decreasing integers:
::$82 \, 428 \, 241 = 9079^2$
:The $7$th square number after $8281$, $183 \, 184$, $328 \, 329$, $528 \, 529$, $715 \, 716$, $60 \, 996 \, 100$ whose digits form $2$ consecutive integers:
::$82 \, 428 \, 241 = 9079^2$
:The $9079$th square number:
::$82 \, 428 \, 241 = 9079 \times 9079$
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Category:Specific Numbers
Category:82,428,241
82,962

== Number ==
$82 \, 962$ (eighty-two thousand, nine hundred and sixty-two) is:
:$2 \times 3^2 \times 11 \times 419$
:One of the elements of one of the $4$-cycles to which Kaprekar's process on some $5$-digit numbers converges:
::$82 \, 962 \to 75 \, 933 \to 63 \, 954 \to 61 \, 974 \to 82 \, 962$
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Category:Specific Numbers
Category:82,962
820

== Number ==
$820$ (eight hundred and twenty) is:
:$2^2 \times 5 \times 41$
:The $40$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$, $780$:
::$820 = \ds \sum_{k \mathop = 1}^{40} k = \dfrac {40 \times \paren {40 + 1} } 2$
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Category:Specific Numbers
Category:820
8208

== Number ==
$8208$ (eight thousand, two hundred and eight) is:
:$2^4 \times 3^3 \times 19$
:The $15$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $153$, $370$, $371$, $407$, $1634$:
::$8^4 + 2^4 + 0^4 + 8^4 = 4096 + 16 + 0 + 4096 = 8208$
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== Sources ==
* : $8208$
* : $8208$
Category:Specific Numbers
Category:8208
821

== Number ==
$821$ (eight hundred and twenty-one) is:
:The $142$nd prime number.
:The larger of the $14$th pair of primes whose prime gap is $10$:
::$821 - 811 = 10$
:The smaller of the $32$nd pair of twin primes:
::$823 - 821 = 2$
:The $50$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $701$, $709$, $727$, $743$, $811$
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Category:821
822

== Number ==
$822$ (eight hundred and twenty-two) is:
:$2 \times 3 \times 137$
:The index (after $2$, $3$, $6$, $30$, $75$, $81$, $115$, $123$, $249$, $362$, $384$, $462$, $512$, $751$) of the $15$th Woodall prime:
::$822 \times 2^{822} - 1$
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Category:Specific Numbers
Category:822
823

== Number ==
$823$ (eight hundred and twenty-three) is:
:The $143$rd prime number.
:The larger of the $32$nd pair of twin primes:
::$823 - 821 = 2$
:The $46$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $443$, $467$, $523$, $547$, $613$, $617$, $643$, $647$, $653$, $673$, $683$, $743$, $773$, $797$
:The $51$st long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $701$, $709$, $727$, $743$, $811$, $821$
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Category:823
823,543

== Number ==
$823 \, 543$ (eight hundred and twenty-three thousand, five hundred and forty-three) is:
:$7^7$
:The $7$th $7$th power after after $1$, $128$, $2187$, $16 \, 384$, $78 \, 125$, $279 \, 936$:
::$823 \, 543 = 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7$
:The $7$th power of $7$ after $(1)$, $7$, $49$, $343$, $2401$, $16 \, 807$, $117 \, 649$:
::$823 \, 543 = 7^7$
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Category:823,543
823,544

== Number ==
$823 \, 544$ (forty-six thousand, six hundred and fifty-seven) is:
:$2^3 \times 113 \times 911$
:The $7$th Sierpiński number of the first kind after $2$, $5$, $28$, $257$, $3126$, $46 \, 657$:
::$823 \, 544 = 7^7 + 1$
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Category:Specific Numbers
Category:823,544
825

== Number ==
$825$ (eight hundred and twenty-five) is:
:$3 \times 5^2 \times 11$
:The $41$st Smith number after $4$, $22$, $27$, $58$, $\ldots$, $648$, $654$, $663$, $666$, $690$, $706$, $728$, $729$, $762$, $778$:
::$8 + 2 + 5 = 3 + 5 + 5 + 1 + 1 = 15$
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Category:Specific Numbers
Category:825