nyuuzyou/wikis huggingface dataset:
9,412, 343,607, 359,262, 946,971, 172,136, 294,514, 357,528, 981,378, 983,082, 541,347, 532,211, 942,640, 121,301, 590,698, 634,089, 611,468, 911,681

== Number ==
$9 \, 412 \, 343 \, 607 \, 359 \, 262 \, 946 \, 971 \, 172 \, 136 \, 294 \, 514 \, 357 \, 528 \, 981 \, 378 \, 983 \, 082 \, 541 \, 347 \, 532 \, 211 \, 942 \, 640 \, 121 \, 301 \, 590 \, 698 \, 634 \, 089 \, 611 \, 468 \, 911 \, 681$ is:
== Historical Note ==
== Sources ==
* : $\paren {11^{104} + 1} / \paren {11^8 + 1}$
Category:Specific Numbers
Category:9,412, 343,607, 359,262, 946,971, 172,136, 294,514, 357,528, 981,378, 983,082, 541,347, 532,211, 942,640, 121,301, 590,698, 634,089, 611,468, 911,681
9,412, 343,607, 359,262, 946,971, 172,136, 294,514, 357,528, 981,378, 983,082, 541,347, 532,211, 942,640, 121,301, 590,698, 634,089, 611,468, 911,681/Historical Note

== Historical Note on $9 \, 412 \, 343 \, 607 \, 359 \, 262 \, 946 \, 971 \, 172 \, 136 \, 294 \, 514 \, 357 \, 528 \, 981 \, 378 \, 983 \, 082 \, 541 \, 347 \, 532 \, 211 \, 942 \, 640 \, 121 \, 301 \, 590 \, 698 \, 634 \, 089 \, 611 \, 468 \, 911 \, 681$ ==
This was the first $100$-digit number which was factorised using a general-purpose factorisation algorithm, which was not designed to exploit any special properties of the number under investigation, nor did it use probabilitistic measures.
According to in his $1997$ work , this was achieved on $12$th October $1988$ by and .
However, this has not yet been fully corroborated, and needs to be confirmed.
== Sources ==
* 
* : $\paren {11^{104} + 1} / \paren {11^8 + 1}$
Category:Historical Notes
9,412, 343,607, 359,262, 946,971, 172,136, 294,514, 357,528, 981,378, 983,082, 541,347, 532,211, 942,640, 121,301, 590,698, 634,089, 611,468, 911,681/Mistake

== Source Work ==
:
:The Dictionary
::$\paren {11^{104} + 1} / \paren {11^8 + 1}$
== Mistake ==
:Equal to $86,759,222,313,428,390,812,218,077,095,850,708,048,977$ $\times$ $1,084,881,048,536,374,706,129,613,998,429,729,484,098,346,115,257,905,772,116,753$.
Incorrect.
The second factor is in fact:
:$108 \, 488 \, 104 \, 853 \, 637 \, 470 \, 612 \, 961 \, 399 \, 842 \, 972 \, 948 \, 409 \, 834 \, 611 \, 525 \, 790 \, 577 \, 216 \, 753$
 cites for this result, so it is possible that this is where the mistake originates.
== Sources ==
* 
* : $\paren {11^{104} + 1} / \paren {11^8 + 1}$
Category:Mistakes/Curious and Interesting Numbers
9,696,996

== Number ==
$9 \, 696 \, 996$ (nine million, six hundred and ninety-six thousand, nine hundred and ninety-six) is:
:$2^2 \times 3^4 \times 173^2$
:The $23$rd square after $1$, $4$, $9$, $16$, $25$, $\ldots$, $441$, $484$, $676$, $1444$, $7744$, $11 \, 881$, $29 \, 929$, $44 \, 944$, $55 \, 225$, $69 \, 696$ which has no more than $2$ distinct digits and does not end in $0$:
::$9 \, 696 \, 996 = 3114^2$
:The $3114$th square number:
::$9 \, 696 \, 996 = 3114 \times 3114$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9,696,996
9,800,817

== Number ==
$9 \, 800 \, 817$ (nine million, eight hundred thousand, eight hundred and seventeen) is:
:$3 \times 13^3 \times 1487$
:The $23$rd pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $8208$, $9474$, $54 \, 748$, $92 \, 727$, $93 \, 084$, $548 \, 834$, $1 \, 741 \, 725$, $4 \, 210 \, 818$:
== Also see ==
* 
Category:Specific Numbers
Category:9,800,817
9,814,072,356

== Number ==
$9 \, 814 \, 072 \, 356$ is:
:$2^2 \times 3^2 \times 11^2 \times 19^2 \times 79^2$
:The $99 \, 066$th square number:
::$9 \, 814 \, 072 \, 356 = 99 \, 066^2$
:The largest pandigital square
== Also see ==
* Largest Pandigital Square
* 
== Sources ==
* : $9,814,072,356$
* : $9,814,072,356$
Category:Specific Numbers
Category:9,814,072,356
9,843,019

== Number ==
$9 \, 843 \, 019$ (nine million, eight hundred and forty-three thousand and nineteen) is:
:The $654 \, 926$th prime number
:The $1$st of the smallest $5$ consecutive primes in arithmetic sequence:
::$9 \, 843 \, 019 = 0 \times 30 + 9 \, 843 \, 019$
== Also see ==
* 
* 
== Sources ==
* : $9,843,019$
Category:Specific Numbers
Category:9,843,019
9,843,049

== Number ==
$9 \, 843 \, 049$ (nine million, eight hundred and forty-three thousand and forty-nine) is:
:The $654 \, 927$th prime number
:The $2$nd of the smallest $5$ consecutive primes in arithmetic sequence:
::$9 \, 843 \, 049 = 1 \times 30 + 9 \, 843 \, 019$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9,843,049
9,843,079

== Number ==
$9 \, 843 \, 079$ (nine million, eight hundred and forty-three thousand and seventy-nine) is:
:The $654 \, 928$th prime number
:The $3$rd of the smallest $5$ consecutive primes in arithmetic sequence:
::$9 \, 843 \, 079 = 2 \times 30 + 9 \, 843 \, 019$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9,843,079
9,843,109

== Number ==
$9 \, 843 \, 109$ (nine million, eight hundred and forty-three thousand, one hundred and nine) is:
:The $654 \, 929$th prime number
:The $4$th of the smallest $5$ consecutive primes in arithmetic sequence:
::$9 \, 843 \, 109 = 3 \times 30 + 9 \, 843 \, 019$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9,843,109
9,843,139

== Number ==
$9 \, 843 \, 139$ (nine million, eight hundred and forty-three thousand, one hundred and thirty-nine) is:
:The $654 \, 930$th prime number
:The $5$th of the smallest $5$ consecutive primes in arithmetic sequence:
::$9 \, 843 \, 139 = 4 \times 30 + 9 \, 843 \, 019$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9,843,139
9,876,543,210

== Number ==
$9 \, 876 \, 543 \, 210$ is:
:$2 \times 3^2 \times 5 \times 17^2 \times 379 \, 721$
:Has the property:
::$9 \, 876 \, 543 \, 210 - 0 \, 123 \, 456 \, 789 = 9 \, 753 \, 086 \, 421$
== Also see ==
* Pandigital Properties of 9,876,543,210
== Sources ==
* : $9,876,543,210$
* : $9,876,543,210$
Category:Specific Numbers
Category:9,876,543,210
9,926,315

== Number ==
$9 \, 926 \, 315$ (nine million, nine hundred and twenty-six thousand, three hundred and fifteen) is:
:$5 \times 7 \times 283,609$
:The $24$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $8208$, $9474$, $54 \, 748$, $92 \, 727$, $93 \, 084$, $548 \, 834$, $1 \, 741 \, 725$, $4 \, 210 \, 818$, $9 \, 800 \, 817$:
== Also see ==
* 
Category:Specific Numbers
Category:9,926,315
9,999,999/Historical Note

== Historical Note on $9 \, 999 \, 999$ ==
When invented what are now known as Napierian logarithms, he assigned the number $10 \, 000 \, 000$ the logarithm $0$, and the number $9 \, 999 \, 999$ the logarithm $1$.
Then, by repeatedly multiplying by $\dfrac {9 \, 999 \, 999} {10 \, 000 \, 000}$, he built the sequence of numbers that had the logarithm $2, 3, 4, \ldots$
== Sources ==
* : $9,999,999$
* : $9,999,999$
Category:Historical Notes
90

== Number ==
$90$ (ninety) is:
:$2 \times 3^2 \times 5$
:The $3$rd unitary perfect number after $6$, $60$:
::$90 = 1 + 2 + 5 + 9 + 10 + 18 + 45$
:The $4$th element of the $1$st set of $4$ positive integers which form an arithmetic sequence which all have the same Euler $\phi$ value:
::$\map \phi {72} = \map \phi {78} = \map \phi {84} = \map \phi {90} = 24$
:The smallest positive integer which can be expressed as the sum of $2$ odd primes in $9$ ways.
:The $11$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$, $68$, $74$, $76$, $86$:
::$\nexists m \in \Z_{>0}: \map \phi m = 90$
:where $\map \phi n$ denotes the Euler $\phi$ function
:The $20$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$:
::$\map {\sigma_1} {90} = 234$
:The $20$th of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
::$1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$, $16$, $18$, $24$, $30$, $36$, $42$, $48$, $60$, $90$, $\ldots$
:The $21$st semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$, $66$, $72$, $78$, $80$, $84$, $88$:
::$90 = 15 + 30 + 45$
:The $35$th and last of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $69$, $74$, $75$, $79$, $81$, $82$, $87$, $88$, $90$
:The $48$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $77$, $78$, $79$, $84$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
Category:Specific Numbers
Category:90
90,625

== Number ==
$90 \, 625$ (ninety thousand, six hundred and twenty-five) is:
:$5^5 \times 29$
:The $9$th automorphic number after $1$, $5$, $6$, $25$, $76$, $376$, $625$, $9376$:
::$90 \, 625^2 = 8 \, 212 \, 8 \mathbf{90 \, 625}$
:and the only automorphic number with $5$ digits which does not begin with a zero
== Also see ==
* Automorphic Numbers with 5 Digits
* 
== Sources ==
* : $90,625$
* : $90,625$
Category:Specific Numbers
Category:90,625
900

== Number ==
$900$ (nine hundred) is:
:$2^2 \times 3^2 \times 5^2$
:The $11$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $324$, $400$, $576$, $784$ to be the divisor sum value of some (strictly) positive integer:
::$900 = \map {\sigma_1} {472}$
:The $30$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $625$, $676$, $729$, $784$, $841$:
::$900 = 30 \times 30$
== Also see ==
* 
* 
Category:Specific Numbers
Category:900
9001

== Number ==
$9001$ (nine thousand and one) is:
:The $1118$th prime number
:The $17$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$, $991$, $6469$, $6949$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9001
9010

== Number ==
$9010$ (nine thousand and ten) is:
:$2 \times 5 \times 17 \times 53$
:$9010^2 = 25^2 + 26^2 + \cdots + 624^2$
== Also see ==
* Sums of Consecutive Sequences of Squares that equal Squares
== Sources ==
* : $9010$
Category:Specific Numbers
Category:9010
9025

== Number ==
$9025$ (nine thousand and twenty-five) is:
:$5^2 \times 19^2$
:The $95$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $7396$, $7569$, $7744$, $7921$, $8100$, $8281$, $8464$, $8649$, $8836$
::$9025 = 95 \times 95$
== Also see ==
* 
Category:Specific Numbers
Category:9025
903

== Number ==
$903$ (nine hundred and three) is:
:$3 \times 7 \times 43$
:The $42$nd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$, $780$, $820$, $861$:
::$903 = \ds \sum_{k \mathop = 1}^{42} k = \dfrac {42 \times \paren {42 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:903
904

== Number ==
$904$ (nine hundred and four) is:
:$2^3 \times 113$
:The $15$th even integer after $2$, $4$, $94$, $96$, $98$, $400$, $402$, $404$, $514$, $516$, $518$, $784$, $786$, $788$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
== Also see ==
* 
Category:Specific Numbers
Category:904
9045

== Number ==
$9045$ (nine thousand and forty-five) is:
:$3^3 \times 5 \times 67$
:The $134$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$, $8128$, $8256$, $8385$, $8515$, $8646$, $8778$, $8911$:
::$9045 = \ds \sum_{k \mathop = 1}^{134} k = \dfrac {134 \times \paren {134 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:9045
9049

== Number ==
$9049$ (nine thousand and forty-nine) is:
:The $1125$th prime number
:The $18$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$, $991$, $6469$, $6949$, $9001$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9049
905

== Number ==
$905$ (nine hundred and five) is:
:$5 \times 181$
:The $15$th obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$, $509$, $599$, $701$, $757$, $757$, $809$, $877$
== Also see ==
* 
Category:Specific Numbers
Category:905
906

== Number ==
$906$ (nine hundred and six) is:
:$2 \times 3 \times 151$
:The $16$th even integer after $2$, $4$, $94$, $96$, $98$, $400$, $402$, $404$, $514$, $516$, $518$, $784$, $786$, $788$, $904$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
== Also see ==
* 
Category:Specific Numbers
Category:906
907

== Number ==
$907$ (nine hundred and seven) is:
:The $155$th prime number
:The $16$th obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$, $509$, $599$, $701$, $757$, $757$, $809$, $877$, $905$
:The $28$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $311$, $337$, $347$, $359$, $389$, $709$, $733$, $739$, $743$, $751$, $761$, $769$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:907
9079

== Number ==
$9079$ (nine thousand and seventy-nine) is:
:$7 \times 1297$
:The $7$th integer after $91$, $428$, $573$, $727$, $846$, $7810$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive integers:
::$9079^2 = 82 \, 428 \, 241$
:The $2$nd integer after $91$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive decreasing integers:
::$9079^2 = 82 \, 428 \, 241$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9079
908

== Number ==
$908$ (nine hundred and eight) is:
:$2^2 \times 227$
:The $17$th even integer after $2$, $4$, $94$, $96$, $98$, $400$, $402$, $404$, $514$, $516$, $518$, $784$, $786$, $788$, $904$, $906$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
== Also see ==
* 
Category:Specific Numbers
Category:908
909,091

== Number ==
$909 \, 091$ (nine hundred and nine thousand and ninety-one) is:
:The $71 \, 954$th prime number
:The $8$th unique period prime after $3$, $11$, $37$, $101$, $9091$, $9901$, $333 \, 667$: its period is $14$:
::$\dfrac 1 {909 \, 091} = 0 \cdotp \dot 00000 \, 10999 \, 998 \dot 9$
== Also see ==
* Reciprocal of 909,091
* 
* 
Category:Specific Numbers
Category:909,091
9091

== Number ==
$9091$ (nine thousand and ninety-one) is:
:The $1128$th prime number
:The $5$th unique period prime after $3$, $11$, $37$, $101$: its period is $10$:
::$\dfrac 1 {9091} = 0 \cdotp \dot 00010 \, 9998 \dot 9$
== Also see ==
* Reciprocal of 9091
* 
* 
Category:Specific Numbers
Category:9091
91

== Number ==
$91$ (ninety-one) is:
:$7 \times 13$
:The $1$st integer whose square has a decimal representation consisting of the concatenation of $2$ consecutive integers:
::$91^2 = 8281$
:The $1$st integer whose square has a decimal representation consisting of the concatenation of $2$ consecutive decreasing integers:
::$91^2 = 8281$
:The $1$st Fermat pseudoprime to base $3$:
::$3^{91} \equiv 3 \pmod {91}$
:The $2$nd triangular number after $28$ which is the sum of $2$ cubes:
::$91 = 27 + 64 = 3^3 + 4^3$
:The $2$nd of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
::$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$
:The $3$rd Fermat pseudoprime to base $4$ after $15$, $85$:
::$4^{91} \equiv 4 \pmod {91}$
:The $3$rd after $1$, $55$ of the $4$ square pyramidal numbers which are also triangular.
:The $5$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $85$, $91$, $\ldots$
:The $6$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$:
::$91 = 1 + 6 + 12 + 18 + 24 + 30 = 6^3 - 5^3$
:The $6$th square pyramidal number after $1$, $5$, $14$, $30$, $55$:
::$91 = 1 + 4 + 9 + 16 + 25 + 36 = \dfrac {6 \paren {6 + 1} \paren {2 \times 6 + 1} } 6$
:The $7$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$:
::$91 = 1 + 5 + 9 + 13 + 17 + 21 + 25 = 7 \paren {2 \times 7 - 1}$
:The $8$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $\ldots$
:The $13$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$, $78$:
::$91 = \ds \sum_{k \mathop = 1}^{13} k = \dfrac {13 \times \paren {13 + 1} } 2$
:The $17$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$, $70$, $79$, $82$, $86$:
::$91 \to 9^2 + 1^2 = 81 + 1 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
:The $17$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$, $61$, $67$, $73$, $79$, $85$ such that no factorial of an integer can end with $n$ zeroes
:The $30$th semiprime:
::$91 = 7 \times 13$
:The $49$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $77$, $78$, $79$, $84$, $90$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* 91 is Smallest Fermat Pseudoprime Base 3
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $91$
* : $91$
Category:Specific Numbers
Category:91
910

== Number ==
$910$ (nine hundred and ten) is:
:$2 \times 5 \times 7 \times 13$
:The $18$th primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$, $304$, $350$, $368$, $464$, $490$, $496$, $550$, $572$, $650$, $748$, $770$:
::$910 = 1 + 2 + 5 + 7 + 10 + 13 + 26 + 35 + 65 + 70 + 91 + 130 + 455$
== Also see ==
* 
Category:Specific Numbers
Category:910
9108

== Number ==
$9108$ (nine thousand, one hundred and eight) is:
:$2^2 \times 3^2 \times 11 \times 23$
: The $7$th integer after $954$, $1980$, $2961$, $3870$, $5823$, $7641$ whose digits are unchanged when subtracting its reversal:
::$9108 - 8019 = 1089$
== Also see ==
* 
Category:Specific Numbers
Category:9108
911

== Number ==
$911$ (nine hundred and eleven) is:
:The $156$th prime number
:The smaller of the $13$th pair of primes whose prime gap is $8$:
::$919 - 911 = 8$
:The $22$nd of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 21^2 + 29 = 911$ ()
:The $36$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $431$, $443$, $491$, $509$, $593$, $641$, $653$, $659$, $683$, $719$, $743$, $761$, $809$:
::$2 \times 911 + 1 = 1823$, which is prime.
:The $41$st near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $727$, $733$, $757$, $773$, $787$, $797$, $811$, $877$, $881$, $883$, $887$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:911
912,985,153

== Number ==
$912 \, 985 \, 153$ is:
:$17 \times 53 \, 705 \, 009$
:The $31$st pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $88 \, 593 \, 477$, $146 \, 511 \, 208$, $472 \, 335 \, 975$, $534 \, 494 \, 836$:
== Also see ==
* 
Category:Specific Numbers
Category:912,985,153
9126

== Number ==
$9126$ (nine thousand, one hundred and twenty-six) is:
:$2 \times 3^3 \times 13^2$
:The $26$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $\ldots$, $1470$, $1800$, $2176$, $2601$, $3078$, $3610$, $4200$, $4851$, $5566$, $6348$, $7200$, $8125$:
::$9126 = \ds \sum_{k \mathop = 1}^{26} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {26^2 \paren {26 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:9126
913

== Number ==
$913$ (nine hundred and thirteen) is:
:$11 \times 83$
:The $45$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $648$, $654$, $663$, $666$, $690$, $706$, $728$, $729$, $762$, $778$, $825$, $852$, $895$:
::$9 + 1 + 3 = 1 + 1 + 8 + 3 = 13$
== Also see ==
* 
Category:Specific Numbers
Category:913
9139

== Number ==
$9139$ (nine thousand, one hundred and thirty-nine) is:
:$13 \times 19 \times 37$
:The $37$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $3276$, $3276$, $4060$, $4495$, $4960$, $5456$, $5984$, $6545$, $7140$, $7770$, $8436$:
::$9139 = \ds \sum_{k \mathop = 1}^{37} \frac {k \paren {k + 1} } 2 = \dfrac {37 \paren {37 + 1} \paren {37 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:9139
915

== Number ==
$915$ (nine hundred and fifteen) is:
:$3 \times 5 \times 61$
:The $46$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $648$, $654$, $663$, $666$, $690$, $706$, $728$, $729$, $762$, $778$, $825$, $852$, $895$, $913$:
::$9 + 1 + 5 = 3 + 5 + 6 + 1 = 15$
== Also see ==
* 
Category:Specific Numbers
Category:915
9158

== Number ==
$9158$ (nine thousand, one hundred and fifty-eight) is:
:$2 \times 19 \times 241$
:The $22$nd integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$, $105$, $160$, $661$, $2653$, $3069$, $3943$, $4053$, $4998$, $8275$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$ is prime
== Also see ==
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Category:Specific Numbers
Category:9158
9180

== Number ==
$9180$ (nine thousand, one hundred and eighty) is:
:$2^2 \times 3^3 \times 5 \times 17$
:The $68$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $6903$, $7140$, $7381$, $7626$, $7875$, $8128$, $8385$, $8646$, $8911$:
::$9180 = \ds \sum_{k \mathop = 1}^{68} \paren {4 k - 3} = 68 \paren {2 \times 68 - 1}$
:The $135$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$, $8128$, $8256$, $8385$, $8515$, $8646$, $8778$, $8911$, $9045$:
::$9180 = \ds \sum_{k \mathop = 1}^{135} k = \dfrac {135 \times \paren {135 + 1} } 2$
== Also see ==
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Category:Specific Numbers
Category:9180
919

== Number ==
$919$ (nine hundred and nineteen) is:
:The $157$th prime number.
:The $4$th and last of the $4$ cubic recurring digital invariants after $55$, $136$, $160$:
::$919 \to 1459 \to 919$
:The larger of the $13$th pair of primes whose prime gap is $8$:
::$919 - 911 = 8$
:The smaller of the $16$th pair of primes whose prime gap is $10$:
::$929 - 919 = 10$
:The $18$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$, $331$, $397$, $469$, $547$, $631$, $721$, $817$:
::$919 = \ds 1 + \sum_{k \mathop = 1}^{18 - 1} 6 k = 18^3 - 17^3$
:The $19$th palindromic prime:
::$2$, $3$, $5$, $7$, $11$, $101$, $131$, $151$, $181$, $191$, $313$, $353$, $373$, $383$, $727$, $757$, $787$, $797$, $919$, $\ldots$
:The $21$st permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $113$, $131$, $199$, $311$, $337$, $373$, $733$
:The $42$nd near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $727$, $733$, $757$, $773$, $787$, $797$, $811$, $877$, $881$, $883$, $887$, $911$
== Also see ==
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Category:Specific Numbers
Category:919
91 is Pseudoprime to 35 Bases less than 91

== Theorem ==
$91$ is a Fermat pseudoprime in $35$ bases less than itself:
:$3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, 90$
== Proof ==
By definition of a Fermat pseudoprime, we need to check for $a < 91$:
:$a^{90} \equiv 1 \pmod {91}$
is satisfied or not.
By Chinese Remainder Theorem, this is equivalent to checking whether:
:$a^{90} \equiv 1 \pmod 7$
and:
:$a^{90} \equiv 1 \pmod {13}$
are both satisfied.
If $a$ is a multiple of $7$ or $13$, $a^{90} \not \equiv 1 \pmod {91}$.
Therefore we consider $a$ not divisible by $7$ or $13$.
By Fermat's Little Theorem, we have:
:$a^6 \equiv 1 \pmod 7$
and thus:
:$a^{90} \equiv 1^{15} \equiv 1 \pmod 7$
Now by Fermat's Little Theorem again:
:$a^{12} \equiv 1 \pmod {13}$
and thus:
:$a^{90} \equiv a^6 \paren{1^7} \equiv a^6 \pmod {13}$
We have:
and thus $a$ must be equivalent to $1, 3, 4, 9, 10, 12 \pmod {13}$.
This gives $1$ and the $35$ bases less than $91$ listed above.
== Historical Note ==
== Sources ==
* : $91$
Category:Fermat Pseudoprimes
Category:91