nyuuzyou/wikis huggingface dataset:
86,243/Historical Note

== Historical Note on $86 \, 243$ ==
The Mersenne number $M_{86 \, 243} = 2^{86 \, 243} - 1$ was demonstrated to be a Mersenne prime on $25$ September $1982$ by , using a CRAY-1 supercomputer.
Working at $150$ megaflops, it took $1$ hour, $3$ minutes and $22$ seconds for this calculation to be performed, but several months of preliminary work to establish that this number was indeed likely to be prime.
According to , writing in in $1986$, a mere $150$ megaflops was already old hat by then, and the more recent models available at that time could work at anything up to $1000$ megaflops.
In a final sentence of gosh-wowery, his jaw drops to the floor to report that the Cray-3 was expected to reach the order of $10$ gigaflops.
However, that machine was never commercially available, and the line was discontinued.
== Sources ==
* 
* : $2^{86243} - 1$
* : $2^{86,243} - 1$
Category:Historical Notes
86,243/Mistake

== Source Work ==
:
:The Dictionary
::$2^{86243} - 1$
:
:The Dictionary
::$2^{86,243} - 1$
== Mistake ==
:... the $28$th Mersenne prime, hunted down by on his trusty CRAY-1 in $1983$.
The date he actually found this was in fact $25$ September $1982$.
However, he delayed announcing this result until after he had officially received permission to actually begin the search (which, at that point, had already been under way for some time).
== Sources ==
* : $2^{86243} - 1$
* : $2^{86,243} - 1$
Category:Mistakes/Curious and Interesting Numbers
86,453

== Number ==
$86 \, 453$ (eighty-six thousand, four hundred and fifty-three) is:
:The $8407$th prime number
:The index of the $7$th (probable) repunit prime after $R_2$, $R_{19}$, $R_{23}$, $R_{317}$, $R_{1031}$, $R_{49 \, 081}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:86,453
861

== Number ==
$861$ (eight hundred and sixty-one) is:
:$3 \times 7 \times 41$
:The $1$st term of the $2$nd sequence of $3$ consecutive triangular numbers all of which are sphenic:
:The $21$st hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $378$, $435$, $496$, $561$, $630$, $703$, $780$:
::$861 = \ds \sum_{k \mathop = 1}^{21} \paren {4 k - 3} = 21 \paren {2 \times 21 - 1}$
:The $41$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$, $780$, $820$:
::$861 = \ds \sum_{k \mathop = 1}^{41} k = \dfrac {41 \times \paren {41 + 1} } 2$
:The $43$rd Smith number after $4$, $22$, $27$, $58$, $\ldots$, $648$, $654$, $663$, $666$, $690$, $706$, $728$, $729$, $762$, $778$, $825$, $852$:
::$8 + 6 + 1 = 3 + 7 + 4 + 1 = 15$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:861
863

== Number ==
$863$ (eight hundred and sixty-three) is:
:The $150$th prime number
:The smaller of the $6$th pair of primes whose prime gap is $14$:
::$877 - 863 = 14$
:The $23$rd safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$, $467$, $479$, $503$, $563$, $587$, $719$, $839$:
::$863 = 2 \times 431 + 1$
:The $53$rd long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $701$, $709$, $727$, $743$, $811$, $821$, $823$, $857$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:863
8637

== Number ==
$8637$ (eight thousand, six hundred and thirty-seven) is:
:$3 \times 2879$
:The $3$rd term, with $8517$ and $8577$, of the $2$nd arithmetic sequence of $3$ integers with common difference of $60$ all of which can be expressed as the sum of $4$ fourth powers:
::$8637^4 = 3450^4 + 3695^4 + 5780^4 + 8012^4$
== Also see ==
* 
== Sources ==
* : $8373$
Category:Specific Numbers
Category:8637
8640

== Number ==
$8640$ (eight thousand, six hundred and forty) is:
:$2^6 \times 3^3 \times 5$
:The $54$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $5985$, $6256$, $6533$, $6816$, $7105$, $7400$, $7701$, $8008$, $8321$:
::$8640 = \ds \sum_{k \mathop = 1}^{54} \paren {6 k - 5} = 54 \paren {3 \times 54 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:8640
8645

== Number ==
$8645$ (eight thousand, six hundred and forty-five) is:
:$5 \times 7 \times 13 \times 19$
:The smaller of the $3$rd pair of consecutive positive integers which are each the product of exactly $4$ distinct prime numbers:
::$8645 = 5 \times 7 \times 13 \times 19$, $8646 = 2 \times 3 \times 11 \times 131$
== Also see ==
* 
Category:Specific Numbers
Category:8645
8646

== Number ==
$8646$ (eight thousand, six hundred and forty-six) is:
:$2 \times 3 \times 11 \times 131$
:The larger of the $3$rd pair of consecutive positive integers which are each the product of exactly $4$ distinct prime numbers:
::$8645 = 5 \times 7 \times 13 \times 19$, $8646 = 2 \times 3 \times 11 \times 131$
:The $66$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $6903$, $7140$, $7381$, $7626$, $7875$, $8128$, $8385$:
::$8646 = \ds \sum_{k \mathop = 1}^{66} \paren {4 k - 3} = 66 \paren {2 \times 66 - 1}$
:The $131$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$, $8128$, $8256$, $8385$, $8515$:
::$8646 = \ds \sum_{k \mathop = 1}^{131} k = \dfrac {131 \times \paren {131 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:8646
8649

== Number ==
$8649$ (eight thousand, six hundred and forty-nine) is:
:$3^2 \times 31^2$
:The $93$rd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $7396$, $7569$, $7744$, $7921$, $8100$, $8281$, $8464$
::$8649 = 93 \times 93$
== Also see ==
* 
Category:Specific Numbers
Category:8649
8672

== Number ==
$8672$ (eight thousand, six hundred and seventy-two) is:
:$2^5 \times 271$
:The $7$th and last integer $m$ after $0$, $1$, $2$, $8$, $24$, $260$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$:
::$8672^2 = \dbinom {767} 0 + \dbinom {767} 1 + \dbinom {767} 2 + \dbinom {767} 3$
== Also see ==
* 
Category:Specific Numbers
Category:8672
87

== Number ==
$87$ (eighty-seven) is:
::$3 \times 29$
:The $2$nd positive integer after $1$ whose divisor sum of its Euler $\phi$ value equals its divisor sum:
::$\map {\sigma_1} {\map \phi {87} } = \map {\sigma_1} {56} = 120 = \map {\sigma_1} {87}$
:The $4$th after $4$, $13$, $38$ in the sequence formed by adding the squares of the first $n$ primes:
::$87 = \ds \sum_{i \mathop = 1}^4 {p_i}^2 = 2^2 + 3^2 + 5^2 + 7^2$
:The $4$th positive integer after $1$, $24$, $26$ whose Euler $\phi$ value is equal to the product of its digits: 
::$\map \phi {87} = 56 = 8 \times 7$
:The $20$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $75$, $79$, $87$, $\ldots$
:The $29$th semiprime:
::$87 = 3 \times 29$
:The $33$rd of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $69$, $74$, $75$, $79$, $81$, $82$, $87$, $\ldots$
== Also see ==
* 
* 
* 
* 
* 
* 
== Sources ==
* : $87$
Category:Specific Numbers
Category:87
87,109,376

== Number ==
$87 \, 109 \, 376$ (eighty-seven million, one hundred and nine thousand, three hundred and seventy-six) is:
:$2^8 \times 19 \times 17 \, 909$
:The $15$th automorphic number after $1$, $5$, $6$, $25$, $76$, $376$, $625$, $9376$, $90 \, 625$, $109 \, 376$, $890 \, 625$, $2 \, 890 \, 625$, $7 \, 109 \, 376$, $12 \, 890 \, 625$:
::$87 \, 109 \, 376^2 = 7 \, 588 \, 043 \, 3 \mathbf {87 \, 109 \, 376}$
== Also see ==
* 
Category:Specific Numbers
Category:87,109,376
87,360

== Number ==
$87 \, 360$ (eighty-seven thousand, three hundred and sixty) is:
:$2^6 \times 3 \times 5 \times 7 \times 13$
:The $4$th unitary perfect number after $6$, $60$, $90$:
::$87 \, 360 = \paren {2^6 + 1} \paren {3 + 1} \paren {5 + 1} \paren {7 + 1} \paren {14 + 1} - 87 \, 360$
== Also see ==
* 
== Sources ==
* : $87,360$
Category:Specific Numbers
Category:87,360
87,531

== Number ==
$87,531$ (eighty-seven thousand, five hundred and thirty-one) is:
:$3 \times 163 \times 179$
:With $9642$, forms the largest integer that can be obtained by multiplying $2$ integers which between them use all the digits from $1$ to $9$
::$9642 \times 87 \, 531 = 843 \, 973 \, 902$
== Also see ==
* Largest Product of Pandigital Factors
== Sources ==
* : $9642$
* : $9642$
Category:Specific Numbers
Category:87,531
87,539,319

== Number ==
$87 \, 539 \, 319$ is:
:$3^3 \times 7 \times 31 \times 67 \times 223$
:The $3$rd Hardy-Ramanujan number after $2$, $1729$: the smallest positive integer which can be expressed as the sum of $2$ cubes in $3$ different ways:
== Also see ==
* 
== Sources ==
* : $87,539,319$
* : $87,539,319$
Category:Specific Numbers
Category:87,539,319
870

== Number ==
$870$ (eight hundred and seventy) is:
:$2 \times 3 \times 5 \times 29$
:The magic constant of a magic square of order $12$, after $1$, $(5)$, $15$, $34$, $65$, $111$, $175$, $260$, $369$, $505$, $671$:
::$870 = \ds \dfrac 1 {12} \sum_{k \mathop = 1}^{12^2} k = \dfrac {12 \paren {12^2 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:870
8712

== Number ==
$8712$ (eight thousand, seven hundred and twelve) is:
:$2^3 \times 3^2 \times 11^2$
:The $1$st of only $2$ numbers with $4$ digits or fewer which is a multiple of its reversal:
::$8712 = 4 \times 2178$
== Also see ==
* 
== Sources ==
* : $8712$
* : $8712$
Category:Specific Numbers
Category:8712
872

== Number ==
$872$ (eight hundred and seventy-two) is:
:$2^3 \times 109$
:The $17$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$, $154$, $320$, $340$, $399$, $427$, $872$
== Also see ==
* 
Category:Specific Numbers
Category:872
872,894

== Number ==
$872\,894$ (eight hundred and seventy-two thousand, eight hundred and ninety-four) is:
:$2 \times 11^2 \times 3607$
:The $1$st of a sequence of $5$ consecutive integers which cannot be made prime by changing $1$ digit
== Also see ==
* 
== Sources ==
* : $872,894$
Category:Specific Numbers
Category:872,894
873

== Number ==
$873$ (eight hundred and seventy-three) is:
:$3^2 \times 97$
:The sum of the first $6$ factorials:
::$873 = 1! + 2! + 3! + 4! + 5! + 6!$
== Also see ==
* 
== Sources ==
* : $873$
* : $873$
Category:Specific Numbers
Category:873
874

== Number ==
$874$ (eight hundred and seventy-four) is:
:$2 \times 19 \times 23$
:The $19$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$, $235$, $286$, $342$, $403$, $469$, $540$, $616$, $697$, $783$:
::$874 = \ds \sum_{k \mathop = 1}^{19} \paren {5 k - 4} = \dfrac {19 \paren {5 \times 19 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:874
875

== Number ==
$875$ (eight hundred and seventy-five) is:
:$5^3 \times 7$
:The $9$th tri-automorphic number after $2$, $5$, $7$, $67$, $75$, $92$, $667$, $792$:
::$875^2 \times 3 = 2 \, 296 \, \mathbf {875}$
:The $24$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $\ldots$, $501$, $624$, $625$, $749$, $751$:
::$875^3 = 669 \, 921 \, \mathbf {875}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:875
876

== Number ==
$876$ (eight hundred and seventy-six) is:
:$2^2 \times 3 \times 73$
:The $24$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $222$, $260$, $301$, $345$, $392$, $442$, $495$, $551$, $610$, $672$, $737$, $805$:
::$876 = \dfrac {24 \paren {3 \times 24 + 1} } 2$
:The $48$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$, $532$, $551$, $590$, $610$, $651$, $672$, $715$, $737$, $782$, $805$, $852$:
::$876 = \dfrac {24 \paren {3 \times 24 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:876
877

== Number ==
$877$ (eight hundred and seventy-seven) is:
:The $151$st prime number
:The larger of the $6$th pair of primes whose prime gap is $14$:
::$877 - 863 = 14$
:The $7$th Bell number after $(1)$, $1$, $2$, $5$, $15$, $52$, $203$
:The $14$th obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$, $509$, $599$, $701$, $757$, $757$, $809$
:The $37$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $727$, $733$, $757$, $773$, $787$, $797$, $811$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:877
8778

== Number ==
$8778$ (eight thousand, seven hundred and seventy-eight) is:
:$2 \times 3 \times 7 \times 11 \times 19$
:The $2$nd of the $8$th pair of triangular numbers whose sum and difference are also both triangular:
::$7875 = T_{125}$, $8778 = T_{132}$, $8778 + 7875 = T_{182}$, $8778 - 7875 = T_{42}$
:The $12$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$, $666$, $3003$, $5995$
:The $132$nd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$, $8128$, $8256$, $8385$, $8515$, $8646$:
::$8778 = \ds \sum_{k \mathop = 1}^{132} k = \dfrac {132 \times \paren {132 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:8778
88

== Number ==
$88$ (eighty-eight) is:
:$2^3 \times 11$
:The index of the $1$st non-integer term of the $3$-Göbel sequence.
:The $3$rd primitive abundant number after $20$, $70$:
::$1 + 2 + 4 + 8 + 11 + 22 + 44 = 92 > 88$
:The $4$th untouchable number after $2$, $5$, $52$.
:The $4$th primitive semiperfect number after $6$, $20$, $28$:
::$88 = 1 + 2 + 8 + 11 + 22 + 44$
:The $20$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$, $66$, $72$, $78$, $80$, $84$
:The $34$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $69$, $74$, $75$, $79$, $81$, $82$, $87$, $88$, $\ldots$
:A repdigit whose square is two repdigits together:
::$88^2 = 7744$
== Also see ==
* Non-Integral Value of 3-Göbel Sequence
* 
* 
* 
* 
* 
== Sources ==
* : $88$
* : $43$
* : $88$
Category:Specific Numbers
Category:88
88,593,477

== Number ==
$88 \, 593 \, 477$ is:
:$3 \times 7 \times 17 \times 248 \, 161$
:The $27$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $9 \, 926 \, 315$, $24 \, 678 \, 050$, $24 \, 678 \, 051$:
== Also see ==
* 
Category:Specific Numbers
Category:88,593,477
880

== Number ==
$880$ (eight hundred and eighty) is:
:$2^4 \times 5 \times 11$
:The number of different magic squares of order $4$, up to rotation and reflection
:The $7$th and last of the $3$-digit integers $m$ which need the largest number of reverse-and-add process iterations ($23$) before reaching a palindromic number:
::$880$, $968$, $1837$, $\ldots$, $8 \, 713 \, 200 \, 023 \, 178$
:The $10$th heptagonal pyramidal number after $1$, $8$, $26$, $60$, $115$, $196$, $308$, $456$, $645$:
::$880 = 1 + 7 + 18 + 34 + 55 + 81 + 112 + 148 + 189 + 235 = \dfrac {10 \paren {10 + 1} \paren {5 \times 10 - 2} } 6$
== Also see ==
* Number of Magic Squares of Order 4
* 
* 
== Sources ==
* : $880$
* : $880$
Category:Specific Numbers
Category:880
8801

== Number ==
$8801$ (eight thousand, eight hundred and one) is:
:$13 \times 677$
:The magic constant of a magic square of order $26$, after $1$, $(5)$, $15$, $34$, $\ldots$, $2465$, $2925$, $3439$, $4010$, $4641$, $5335$, $6095$, $6924$, $7825$:
::$\ds 8801 = \dfrac 1 {26} \sum_{k \mathop = 1}^{26^2} k = \dfrac {26 \paren {26^2 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:8801
881

== Number ==
$881$ (eight hundred and eighty-one) is:
:The $152$nd prime number
:The $13$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$
:The smaller of the $35$th pair of twin primes:
::$883 - 881 = 2$
:The $38$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $727$, $733$, $757$, $773$, $787$, $797$, $811$, $877$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:881
883

== Number ==
$883$ (eight hundred and eighty-three) is:
:The $153$rd prime number
:The larger of the $35$th pair of twin primes:
::$883 - 881 = 2$
:The $39$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $727$, $733$, $757$, $773$, $787$, $797$, $811$, $877$, $881$
:The $48$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $443$, $467$, $523$, $547$, $613$, $617$, $643$, $647$, $653$, $673$, $683$, $743$, $773$, $797$, $823$, $853$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:883
8836

== Number ==
$8836$ (eight thousand, eight hundred and thirty-six) is:
:$2^2 \times 47^2$
:The $94$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $7396$, $7569$, $7744$, $7921$, $8100$, $8281$, $8464$, $8649$
::$8836 = 94 \times 94$
== Also see ==
* 
Category:Specific Numbers
Category:8836
8837

== Number ==
$8837$ (eight thousand, eight hundred and thirty-seven) is:
:The $1101$st prime number
:The $19$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$, $257$, $401$, $577$, $677$, $1297$, $1601$, $2917$, $3137$, $4357$, $5477$, $7057$, $8101$:
::$8837 = 94^2 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:8837
887

== Number ==
$887$ (eight hundred and eighty-seven) is:
:The $154$th prime number
:The $11$th candidate Lychrel number
:The $24$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$, $467$, $479$, $503$, $563$, $587$, $719$, $839$, $863$:
::$887 = 2 \times 443 + 1$
:The $40$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $727$, $733$, $757$, $773$, $787$, $797$, $811$, $877$, $881$, $883$
:The $54$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $701$, $709$, $727$, $743$, $811$, $821$, $823$, $857$, $863$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:887
8874

== Number ==
$8874$ (eight thousand, eight hundred and seventy-four) is:
:$2 \times 3^2 \times 17 \times 29$
:Its square consists of $2$ consecutive odd or even integers juxtaposed:
::$8874^2 = 78 \, 747 \, 876$
== Also see ==
* 
== Sources ==
* : $1127$
* : $1127$
Category:Specific Numbers
Category:8874
888

== Number ==
$888$ (eight hundred and eighty-eight) is:
:$2^3 \times 3 \times 37$
:The smallest multiple of $24$ whose digits add up to $24$:
::$888 = 37 \times 24$; $8 + 8 + 8 = 24$
== Sources ==
* : $888$
Category:Specific Numbers
Category:888
89

== Number ==
$89$ (eighty-nine) is:
:The $24$th prime number
:With $98$, gives the longest reverse-and-add sequence of any $2$-digit integers, of $24$ terms.
:The smaller of the $1$st pair of primes whose prime gap is $8$:
::$97 - 89 = 8$
:The $1$st term of the smallest Cunningham chain of the first kind of length $6$:
::$\left({89, 179, 359, 719, 1439, 2879}\right)$
:The $5$th Fibonacci prime after $2$, $3$, $5$, $13$
:The $6$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$3$, $5$, $11$, $13$, $41$, $89$
:The $7$th prime number after $2$, $3$, $5$, $7$, $23$, $67$ consisting of a string of consecutive ascending digits
:The $9$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$
:The $10$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$:
::$2 \times 89 + 1 = 179$, which is prime.
:The index of the $10$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$:
::$M_{89} = 2^{89} - 1 = 618 \, 970 \, 019 \, 642 \, 690 \, 137 \, 449 \, 562 \, 111$
:The $11$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$, $34$, $55$:
::$89 = 34 + 55$
:The $11$th number after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ which is the sum of the increasing powers of its digits taken in order:
::$8^1 + 9^2 = 89$
:The $11$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$, $65$, $85$ such that the Fibonacci number $F_n$ ends in $n$
:The $34$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $77$, $83$, $89$, $\ldots$
:The $45$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $61$, $65$, $66$, $67$, $72$, $77$, $80$, $81$, $84$ which cannot be expressed as the sum of distinct pentagonal numbers.
== Also see ==
* 2-Digit Numbers forming Longest Reverse-and-Add Sequence
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== Sources ==
* : $89$
* : $89$
Category:Specific Numbers
Category:89
890,625

== Number ==
$890 \, 625$ (eight hundred and ninety thousand, six hundred and twenty-five) is:
:$3 \times 5^6 \times 19$
:The $11$th automorphic number after $1$, $5$, $6$, $25$, $76$, $376$, $625$, $9376$, $90 \, 625$, $109 \, 376$:
::$890 \, 625^2 = 793 \, 212 \, \mathbf {890 \, 625}$
== Also see ==
* 
Category:Specific Numbers
Category:890,625
8911

== Number ==
$8911$ (eight thousand, nine hundred and eleven) is:
:$7 \times 19 \times 67$
:The $7$th Carmichael number after $561$, $1105$, $1729$, $2465$, $2821$, $6601$:
::$\forall a \in \Z: a \perp 8911: a^{8910} \equiv 1 \pmod {8911}$
:The $22$nd Poulet number after $341$, $561$, $645$, $1105$, $1387$, $\ldots$, $4369$, $4371$, $4681$, $5461$, $6601$, $7957$, $8321$, $8481$:
::$2^{8911} \equiv 2 \pmod {8911}$: $8911 = 7 \times 19 \times 67$
:The $67$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $6903$, $7140$, $7381$, $7626$, $7875$, $8128$, $8385$, $8646$:
::$8911 = \ds \sum_{k \mathop = 1}^{67} \paren {4 k - 3} = 67 \paren {2 \times 67 - 1}$
:The $133$rd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$, $8128$, $8256$, $8385$, $8515$, $8646$, $8778$:
::$8911 = \ds \sum_{k \mathop = 1}^{133} k = \dfrac {133 \times \paren {133 + 1} } 2$
== Also see ==
* 
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Category:Specific Numbers
Category:8911
895

== Number ==
$895$ (eight hundred and ninety-five) is:
:$5 \times 179$
:The $7$th Woodall number after $1$, $7$, $23$, $63$, $159$, $383$:
::$895 = 7 \times 2^7 - 1$
:The $44$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $648$, $654$, $663$, $666$, $690$, $706$, $728$, $729$, $762$, $778$, $825$, $852$:
::$8 + 9 + 5 = 5 + 1 + 7 + 9 = 22$
== Also see ==
* 
* 
Category:Specific Numbers
Category:895
8965

== Number ==
$8965$ (eight thousand, nine hundred and sixty-five) is:
:$5 \times 11 \times 163$
:The $55$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $5985$, $6256$, $6533$, $6816$, $7105$, $7400$, $7701$, $8008$, $8321$, $8640$:
::$8965 = \ds \sum_{k \mathop = 1}^{55} \paren {6 k - 5} = 55 \paren {3 \times 55 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:8965
897

== Number ==
$897$ (eight hundred and ninety-seven) is:
:$3 \times 13 \times 23$
:The $8$th Cullen number after $1$, $3$, $9$, $25$, $65$, $161$, $385$:
::$897 = 7 \times 2^7 + 1$
== Also see ==
* 
Category:Specific Numbers
Category:897
8997

== Number ==
$8997$ (eight thousand, nine hundred and ninety-seven) is:
:$3 \times 2999$
:The $3$rd after $6999$, $7998$ of the $4$-digit integers $m$ which need the largest number of reverse-and-add process iterations ($20$) before reaching a palindromic number:
::$8997$, $16995$, $76956$, $\ldots$, $16668488486661$
== Also see ==
* 
Category:Specific Numbers
Category:8997
8 Mutually Non-Attacking Queens on Chessboard

== Theorem ==
On a standard chessboard, it is possible to arrange a maximum of $8$ queens so that no queen is attacking any other queen.
There are $12$ such arrangements, up to rotation and reflection.
== Proof ==
That there are only these $12$ can be proved by brute force.
$9$ queens cannot be so placed.
This is clear from the Pigeonhole Principle, which would have at least one row with $2$ queens on it, and so attacking each other.
== Sources ==
* : $8$
* : $8$
* : $12$
Category:8
Category:12
Category:Recreational Chess
9,192,631,770

== Number ==
$9 \, 192 \, 631 \, 770$ is:
:$2 \times 3^2 \times 5 \times 7^2 \times 47 \times 44351$
:The number of periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium $133$ atom at rest at $0 \ \mathrm K$ in one second.
== Also see ==
* Definition:Second of Time
== Sources ==
Category:Specific Numbers
Category:9,192,631,770