nyuuzyou/wikis huggingface dataset:
9649

== Number ==
$9649$ (nine thousand, six hundred and forty-nine) is:
:The $1192$nd prime number
:The $19$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$, $991$, $6469$, $6949$, $9001$, $9049$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9649
967

== Number ==
$967$ (nine hundred and sixty-seven) is:
:The $163$rd prime number
:The larger of the $7$th pair of primes whose prime gap is $14$:
::$967 - 953 = 14$
:The $31$st emirp after $13$, $17$, $31$, $37$, $\ldots$, $733$, $739$, $743$, $751$, $761$, $769$, $907$, $937$, $941$, $953$
:The $52$nd left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $743$, $773$, $797$, $823$, $853$, $883$, $937$, $947$, $953$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:967
9689

== Number ==
$9689$ (nine thousand, six hundred and eighty-nine) is:
:The $1196$th prime number
:The index of the $21$st Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$, $107$, $127$, $521$, $607$, $1279$, $2203$, $2281$, $3217$, $4253$, $4423$:
::$M_{9689} = 2^{9689} - 1 \approx 4 \cdotp 782 \times 10^{2916}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9689
969

== Number ==
$969$ (nine hundred and sixty-nine) is:
:$3 \times 17 \times 19$
:The $17$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$, $220$, $286$, $364$, $455$, $560$, $680$, $816$:
::$969 = \ds \sum_{k \mathop = 1}^{17} \frac {k \paren {k + 1} } 2 = \dfrac {17 \paren {17 + 1} \paren {17 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:969
97

== Number ==
$97$ (ninety-seven) is:
:The $25$th prime number
:The larger of the $1$st pair of primes whose prime gap is $8$:
::$97 - 89 = 8$
:The $6$th Proth prime after $3$, $5$, $13$, $17$, $41$:
::$97 = 3 \times 2^5 + 1$
:The $8$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$
:The $9$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$:
::$\dfrac 1 {97} = 0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$
:The $13$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$, $73$, $79$
:The $15$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$, $53$, $59$, $67$, $71$, $73$, $79$, $83$ such that the Mersenne number $2^p - 1$ is composite
:The $15$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$
:The $19$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$, $70$, $79$, $82$, $86$, $91$, $94$:
::$97 \to 9^2 + 7^2 = 81 + 49 = 130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$
:The $35$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $77$, $83$, $89$, $97$, $\ldots$
== Also see ==
* Reciprocal of 97
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:97
97,946

== Number ==
$97 \, 946$ (ninety-seven thousand, nine hundred and forty-six) is:
:$2 \times 48 \, 973$
:The $22$nd term of a sociable chain of order $28$ whose smallest term is $14 \, 316$:
::$14 \, 316$, $19 \, 116$, $31 \, 704$, $47 \, 616$, $83 \, 328$, $177 \, 792$, $295 \, 488$, $629 \, 072$, $589 \, 786$, $294 \, 896$, $358 \, 336$, $418 \, 904$, $366 \, 556$, $274 \, 924$, $275 \, 444$, $243 \, 760$, $376 \, 736$, $381 \, 028$, $285 \, 778$, $152 \, 990$, $122 \, 410$, $97 \, 946$, $48 \, 976$, $45 \, 946$, $22 \, 976$, $22 \, 744$, $19 \, 916$, $17 \, 716$, $14 \, 316$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:97,946
970

== Number ==
$970$ (nine hundred and seventy) is:
:$2 \times 5 \times 97$
:The $20$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$, $235$, $286$, $342$, $403$, $469$, $540$, $616$, $697$, $783$, $874$:
::$970 = \ds \sum_{k \mathop = 1}^{20} \paren {5 k - 4} = \dfrac {20 \paren {5 \times 20 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:970
971

== Number ==
$971$ (nine hundred and seventy-one) is:
:The $164$th prime number
:The $31$st emirp after $13$, $17$, $31$, $37$, $\ldots$, $733$, $739$, $743$, $751$, $761$, $769$, $907$, $937$, $941$, $953$, $967$
:The $58$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $823$, $857$, $863$, $887$, $937$, $941$, $953$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:971
9730

== Number ==
$9730$ (nine thousand, seven hundred and thirty) is:
:$2 \times 5 \times 7 \times 139$
:The $70$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $8128$, $8385$, $8646$, $8911$, $9180$, $9453$:
::$9730 = \ds \sum_{k \mathop = 1}^{70} \paren {4 k - 3} = 70 \paren {2 \times 70 - 1}$
:The $139$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $8911$, $9045$, $9180$, $9316$, $9453$, $9591$:
::$9730 = \ds \sum_{k \mathop = 1}^{139} k = \dfrac {139 \times \paren {139 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9730
974

== Number ==
$974$ (nine hundred and seventy-four) is:
:$2 \times 487$
:The $17$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
::$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$, $38$, $94$, $166$, $324$, $379$, $469$, $546$, $974$
== Also see ==
* 
Category:Specific Numbers
Category:974
977

== Number ==
$977$ (nine hundred and seventy-seven) is:
:The $165$th prime number
:The $6$th Stern number after $1$, $3$, $17$, $137$, $227$
:The $6$th Stern prime after $1$, $3$, $17$, $137$, $227$
:The $15$th balanced prime after $5$, $53$, $157$, $173$, $211$, $257$, $263$, $373$, $563$, $593$, $607$, $653$, $733$, $947$:
::$977 = \dfrac {971 + 983} 2$
:The $18$th obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$, $509$, $599$, $701$, $757$, $757$, $809$, $877$, $905$, $907$, $959$
:The $44$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $727$, $733$, $757$, $773$, $787$, $797$, $811$, $877$, $881$, $883$, $887$, $911$, $919$, $929$
:The $59$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $823$, $857$, $863$, $887$, $937$, $941$, $953$, $971$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:977
979

== Number ==
$979$ (nine hundred and seventy-nine) is:
:$11 \times 89$
:The $18$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $99$, $111$, $141$, $151$, $171$, $303$, $393$, $535$, $717$, $727$, $777$, $787$, $979$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:979
98

== Number ==
$98$ (ninety-eight) is:
:$2 \times 7^2$
:With $89$, gives the longest reverse-and-add sequence of any $2$-digit integers, of $24$ terms
:The $1$st even integer that cannot be expressed as the sum of $2$ prime numbers of which the smaller one is $3$, $5$ or $7$
:The $1$st of the $3$rd pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {98} = \map {\sigma_0} {99} = 6$
:The $5$th even integer after $2$, $4$, $94$, $96$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
:The $5$th after $0$, $2$, $3$, $27$ of the $6$ integers which are the middle term of a sequence of $5$ consecutive integers whose cubes add up to a square
::$96^3 + 97^3 + 98^3 + 99^3 + 100^3 = 4 \, 708 \, 900 = 2170^2$
:The $13$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$, $68$, $74$, $76$, $86$, $90$, $94$:
::$\nexists m \in \Z_{>0}: \map \phi m = 98$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $19$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$, $61$, $67$, $73$, $79$, $85$, $91$, $92$ such that no factorial of an integer can end with $n$ zeroes
== Also see ==
* 2-Digit Numbers forming Longest Reverse-and-Add Sequence
* Reciprocal of 98
* 
* 
* 
* 
* 
* 
== Sources ==
* : $98$
Category:Specific Numbers
Category:98
98,029,801

== Number ==
$98 \, 029 \, 801$ is:
:$9901^2$
:The $3$rd square number after $8281$, $82 \, 428 \, 241$ whose digits form $2$ consecutive decreasing integers:
::$98 \, 029 \, 801 = 9901^2$
:The $8$th square number after $8281$, $183 \, 184$, $328 \, 329$, $528 \, 529$, $715 \, 716$, $60 \, 996 \, 100$, $82 \, 428 \, 241$ whose digits form $2$ consecutive integers:
::$98 \, 029 \, 801 = 9901^2$
:The $9901$th square number:
::$98 \, 029 \, 801 = 9901 \times 9901$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:98,029,801
9801

== Number ==
$9801$ (nine thousand, eight hundred and one) is:
:$3^4 \times 11^2$
:The $99$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $7396$, $7569$, $7744$, $7921$, $8100$, $8281$, $8464$, $8649$, $8836$, $9025$, $9216$, $9409$, $9604$
::$9801 = 99 \times 99$
:The $2$nd of only $2$ numbers with $4$ digits or fewer which is a multiple of its reversal:
::$9801 = 9 \times 1089$
== Also see ==
* 
* 
== Sources ==
* : $9801$
* : $9801$
Category:Specific Numbers
Category:9801
9801/Mistake

== Source Work ==
:
:The Dictionary
::$9801$
:
:The Dictionary
::$9801$
== Mistake ==
:$9801 = 99^2$ and $98 + 01 = 99$, so $9801$ is a Kaprekar number.
== Correction ==
By the usual definition, this makes $99$ a Kaprekar number, not $9801$.
The squares of Kaprekar numbers are not considered to be Kaprekar numbers unless they are Kaprekar in their own right.
== Sources ==
* : $9801$
* : $9801$
Category:Mistakes/Curious and Interesting Numbers
981

== Number ==
$981$ (nine hundred and eighty-one) is:
:$3^3 \times 109$
:The common sum of the only known set of $5$ triplets of integers with the property that the sum of the integers in each triplet are equal and the product of the integers in each triplet are equal:
::$\tuple {6, 480, 495}$, $\tuple {11, 160, 810}$, $\tuple {12, 144, 825}$, $\tuple {20, 81, 880}$, $\tuple {33, 48, 900}$
== Also see ==
* Set of 5 Triplets whose Sums and Products are Equal
* 1,425,600
== Sources ==
* : $981$
* : $981$
Category:Specific Numbers
Category:981
983

== Number ==
$983$ (nine hundred and eighty-three) is:
:The $166$th prime number.
:The smaller of the $15$th pair of primes whose prime gap is $8$:
::$991 - 983 = 8$
:The $25$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$, $467$, $479$, $503$, $563$, $587$, $719$, $839$, $863$, $887$:
::$983 = 2 \times 491 + 1$
:The $32$nd emirp after $13$, $17$, $31$, $37$, $\ldots$, $733$, $739$, $743$, $751$, $761$, $769$, $907$, $937$, $941$, $953$, $967$, $971$
:The $53$rd left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $743$, $773$, $797$, $823$, $853$, $883$, $937$, $947$, $953$, $967$
:The $60$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $823$, $857$, $863$, $887$, $937$, $941$, $953$, $971$, $977$
== Also see ==
* 
* 
* 
* 
* 
* 
::
Category:Specific Numbers
Category:983
985

== Number ==
$985$ (nine hundred and eighty-five) is:
:$5 \times 197$
:The $49$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $778$, $825$, $852$, $895$, $913$, $915$, $922$, $958$:
::$9 + 8 + 5 = 5 + 1 + 9 + 7 = 22$
== Also see ==
* 
Category:Specific Numbers
Category:985
9855

== Number ==
$9855$ (nine thousand, eight hundred and fifty-five) is:
:$3^3 \times 5 \times 73$
:The magic constant of a magic square of order $27$, after $1$, $(5)$, $15$, $34$, $\ldots$, $2465$, $2925$, $3439$, $4010$, $4641$, $5335$, $6095$, $6924$, $7825$, $8801$:
::$9855 = \ds \dfrac 1 {27} \sum_{k \mathop = 1}^{27^2} k = \dfrac {27 \paren {27^2 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:9855
987

== Number ==
$987$ (nine hundred and eighty-seven) is:
:$3 \times 7 \times 47$
:The $16$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$, $34$, $55$, $89$, $144$, $233$, $377$, $610$:
::$987 = 377 + 610$
== Also see ==
* 
Category:Specific Numbers
Category:987
987,654,321

== Number ==
$987 \, 654 \, 321$ is:
:$3^2 \times 17^2 \times 379 \, 721$
:Remains penholodigital when multiplied by $2$, $4$, $5$, $7$ and $8$
== Also see ==
* Penholodigital Properties of 987,654,321
== Sources ==
* : $987,654,321$
* : $987,654,321$
Category:Specific Numbers
Category:987,654,321
9870

== Number ==
$9870$ (nine thousand, eight hundred and seventy) is:
:$2 \times 3 \times 5 \times 7 \times 47$
:The $140$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $8911$, $9045$, $9180$, $9316$, $9453$, $9591$, $9730$:
::$9870 = \ds \sum_{k \mathop = 1}^{140} k = \dfrac {140 \times \paren {140 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:9870
9880

== Number ==
$9880$ (nine thousand, eight hundred and eighty) is:
:$2^3 \times 5 \times 13 \times 19$
:The $38$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $3276$, $3276$, $4060$, $4495$, $4960$, $5456$, $5984$, $6545$, $7140$, $7770$, $8436$, $9139$:
::$9880 = \ds \sum_{k \mathop = 1}^{38} \frac {k \paren {k + 1} } 2 = \dfrac {38 \paren {38 + 1} \paren {38 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:9880
99

== Number ==
$99$ (ninety-nine) is:
::$3^2 \times 11$
:The $2$nd of the $3$rd pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {98} = \map {\sigma_0} {99} = 6$
:The $5$th Kaprekar number after $1$, $9$, $45$, $55$:
::$99^2 = 9801 \to 98 + 01 = 99$
:The $6$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $99$, $\ldots$
:The $8$th integer after $0$, $1$, $3$, $5$, $7$, $9$, $33$ which is palindromic in both decimal and binary:
::$99_{10} = 1 \, 100 \, 011_2$
:The $12$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $24$, $25$, $49$, $51$, $75$, $76$:
::$99^3 = 970 \, 2 \mathbf {99}$
:The $22$nd lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $75$, $79$, $87$, $93$, $99$, $\ldots$
== Also see ==
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $99$
* : $99$
Category:Specific Numbers
Category:99
99,131

== Number ==
$99 \, 131$ (ninety-nine thousand, one hundred and thirty-one) is:
:The $9517$th prime number
:The $1$st term of the $38$th prime quadruple:
::$99 \, 131, 99 \, 133, 99 \, 137, 99 \, 139$
== Also see ==
* 
* 
== Sources ==
* : $99,131$
Category:Specific Numbers
Category:99,131
99,359

== Number ==
$99 \, 359$ (ninety-nine thousand, three hundred and fifty-nine) is:
:$13 \times 7643$
:The $4$th integer $m$ after $1$, $47$, $2161$ whose cube can be expressed as the sum of $m$ consecutive squares:
::$99 \, 359^3 = \ds \sum_{k \mathop = 1}^{99 \, 359} \paren {45 \, 449 + k}^2$
== Also see ==
* 
Category:Specific Numbers
Category:99,359
99,990,001

== Number ==
$99 \, 990 \, 001$ (ninety-nine million, nine hundred and ninety thousand and one) is:
:The $5 \, 760 \, 923$rd prime number
:The $9$th unique period prime after $3$, $11$, $37$, $101$, $9091$, $9901$, $333 \, 667$, $909 \, 091$: its period is $24$:
::$\dfrac 1 {99 \, 990 \, 001} = 0 \cdotp \dot 00000 \, 00100 \, 00999 \, 99998 \, 999 \dot 9$
== Also see ==
* Reciprocal of 99,990,001
* 
* 
Category:Specific Numbers
Category:99,990,001
99/Historical Note

== Historical Note on $99$ (Ninety-Nine) ==
The number $99$ is associated with the word , used at the end of prayers in the Christian, Judaic and Muslim traditions to denote a strong sense of affirmation: verily or truly with perhaps a sense of let it be made to be so.
The attribution arises from the Greek transliteration $\alpha \mu \eta \nu$, which bear the numerical values $1$, $40$, $8$ and $50$ respectively, totalling $99$.
Hence, in many old editions of the , instead of amen at the end of a prayer, the number $99$ can often be found.
== Sources ==
* : Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory
Category:Historical Notes
990

== Number ==
$990$ (nine hundred and ninety) is:
:$2 \times 3^2 \times 5 \times 11$
:The second of the $4$th pair of triangular numbers whose sum and difference are also both triangular:
::$780 = T_{39}$, $990 = T_{44}$, $780 + 990 = T_{59}$, $990 - 780 = T_{20}$
:The $4$th triangular number after $6$, $120$, $210$ which can be expressed as the product of $3$ consecutive integers:
::$990 = T_{44} = 9 \times 10 \times 11$
:The $44$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$, $780$, $820$, $861$, $903$, $946$:
::$990 = \ds \sum_{k \mathop = 1}^{44} k = \dfrac {44 \times \paren {44 + 1} } 2$
== Also see ==
* 
* 
* 
== Sources ==
* : $780$
* : $780$
Category:Specific Numbers
Category:990
9901

== Number ==
$9901$ (nine thousand, nine hundred and one) is:
:The $1221$st prime number
:The $3$rd integer after $91$, $9079$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive decreasing integers:
::$9901^2 = 98 \, 029 \, 801$
:The $6$th unique period prime after $3$, $11$, $37$, $101$, $9091$: its period is $12$:
::$\dfrac 1 {9901} = 0 \cdotp \dot 00010 \, 09998 \, 9 \dot 9$
:The $8$th integer after $91$, $428$, $573$, $727$, $846$, $7810$, $9079$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive integers:
::$9901^2 = 98 \, 029 \, 801$
== Also see ==
* Reciprocal of 9901
* 
* 
* 
* 
Category:Specific Numbers
Category:9901
991

== Number ==
$991$ (nine hundred and ninety-one) is:
:The $167$th prime number
:The $9$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$3$, $5$, $11$, $13$, $41$, $89$, $317$, $337$, $991$
:The $14$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$
:The larger of the $15$th pair of primes whose prime gap is $8$:
::$991 - 983 = 8$
:The $22$nd permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $113$, $131$, $199$, $311$, $337$, $373$, $733$, $919$
:The $33$rd emirp after $13$, $17$, $31$, $37$, $\ldots$, $733$, $739$, $743$, $751$, $761$, $769$, $907$, $937$, $941$, $953$, $967$, $971$, $983$
:The $45$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $727$, $733$, $757$, $773$, $787$, $797$, $811$, $877$, $881$, $883$, $887$, $911$, $919$, $929$, $977$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:991
994

== Number ==
$994$ (nine hundred and ninety-four) is:
:$2 \times 7 \times 71$
:The $12$th positive integer after $1$, $7$, $102$, $110$, $142$, $159$, $187$, $381$, $690$, $714$, $770$ the sum of whose divisors is a cube:
::$\map {\sigma_1} {994} = 1728 = 12^3$
:The $209$th nontotient after $992$:
::$\nexists m \in \Z_{>0}: \map \phi m = 994$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:994
9941

== Number ==
$9941$ (nine thousand, nine hundred and forty-one) is:
:The $1226$th prime number
:The index of the $22$nd Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$, $107$, $127$, $521$, $607$, $1279$, $2203$, $2281$, $3217$, $4253$, $4423$, $9689$:
::$M_{9941} = 2^{9941} - 1 \approx 3 \cdotp 461 \times 10^{2992}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9941
9949

== Number ==
$9949$ (nine thousand, nine hundred and forty-nine) is:
:The $1227$th prime number
:The $20$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$, $991$, $6469$, $6949$, $9001$, $9049$, $9649$
== Also see ==
* 
* 
Category:Specific Numbers
Category:9949
997

== Number ==
$997$ (nine hundred and ninety-seven) is:
:The $168$th prime number
:The $19$th obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$, $509$, $599$, $701$, $757$, $757$, $809$, $877$, $905$, $907$, $959$, $977$
:The $23$rd of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 22^2 + 29 = 997$ ()
:The $46$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $727$, $733$, $757$, $773$, $787$, $797$, $811$, $877$, $881$, $883$, $887$, $911$, $919$, $929$, $977$, $991$
:The $54$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $743$, $773$, $797$, $823$, $853$, $883$, $937$, $947$, $953$, $967$, $983$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:997
9976

== Number ==
$9976$ (nine thousand, nine hundred and seventy-six) is:
:$2^3 \times 29 \times 43$
:The $58$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $7701$, $8008$, $8321$, $8640$, $8965$, $9296$, $9633$:
::$9976 = \ds \sum_{k \mathop = 1}^{58} \paren {6 k - 5} = 58 \paren {3 \times 58 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:9976
998

== Number ==
$998$ (nine hundred and ninety-eight) is:
:$2 \times 449$
:The $210$th nontotient after $992$, $994$:
::$\nexists m \in \Z_{>0}: \map \phi m = 998$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
Category:Specific Numbers
Category:998
999

== Number ==
$999$ (nine hundred and ninety-nine) is:
:$3^3 \times 37$
:The smallest integer which is the sum of a set of $3$ three-digit primes using all $9$ digits from $1$ to $9$ once each:
::$149 + 263 + 587 = 999$
:The smallest multiple of $27$ whose digits add up to $27$:
::$999 = 37 \times 27$; $27 = 9 + 9 + 9$
:The $8$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$, $703$:
::$999^2 = 998 \, 001 \to 998 + 001 = 999$
:The $25$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $\ldots$, $501$, $624$, $625$, $749$, $751$, $875$:
::$999^3 = 997 \, 002 \, \mathbf {999}$
== Also see ==
* 
* 
== Sources ==
* : $999$
* : $999$
Category:Specific Numbers
Category:999
9996

== Number ==
$9996$ (nine thousand, nine hundred and ninety-six) is:
:$2^2 \times 3 \times 7^7 \times 173$
:The $4$th after $6999$, $7998$, $8997$ of the $4$-digit integers $m$ which need the largest number of reverse-and-add process iterations ($20$) before reaching a palindromic number:
::$9996$, $16995$, $76956$, $\ldots$, $16668488486661$
== Also see ==
* 
Category:Specific Numbers
Category:9996
9999

== Number ==
$9999$ (nine thousand, nine hundred and ninety-nine) is:
:$3^2 \times 11 \times 101$
:The $17$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$, $703$, $999$, $2223$, $2728$, $4879$, $4950$, $5050$, $5292$, $7272$, $7777$:
::$9999^2 = 99 \, 980 \, 001 \to 9998 + 0001 = 9999$
== Also see ==
* 
== Sources ==
* : $9999$
* : $9999$
Category:Specific Numbers
Category:9999
9 is Only Square which is Sum of 2 Consecutive Positive Cubes

== Theorem ==
Discounting the trivial solution:
:$1^2 = 1 = 0^3 + 1^3$
$9$ is the only square number which is the sum of $2$ consecutive positive cube numbers:
:$3^2 = 9 = 1^3 + 2^3$
== Proof ==
The expression for the $n$th square number is:
:$n^2$
for $n \in \Z$.
The expression for the $n$th cube number is:
:$n^3$
again, for $n \in \Z$.
Therefore the closed-form expression for the $n$th sum of two consecutive cubes is:
:$n^3 + \paren {n + 1}^3$
This simplifies to:
:$2 n^3 + 3 n^2 + 3 n + 1$
Equate the two expressions with a variable replacing $n$:
Substituting:
:$u = 2 x + 1$
:$v = 2 y$
the equation becomes:
:$v^2 = u^3 + 3 u$
We then apply Integral Points of Elliptic Curve $y^2 = x^3 + 3 x$:
We then write them in terms of $x$ and $y$:
:$\tuple {-\frac 1 2, 0}, \tuple {0, \pm 1}, \tuple {1, \pm 3}, \tuple {\frac {11} 2, \pm 21}$
Due to the restrictions on the variables, solutions with non-integer inputs are invalid.
This leaves $4$ solutions:
:$\tuple {0, \pm 1}, \tuple {1, \pm 3}$
as follows:
:$\paren {\pm1}^2 = 1 = 0^3 + 1^3$
:$\paren {\pm3}^2 = 9 = 1^3 + 2^3$
== Sources ==
* : $9$
* : $9$
Category:9
Category:Square Numbers
Category:Cube Numbers
A.E. Equal Positive Measurable Functions have Equal Integrals

== Theorem ==
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f, g: X \to \overline \R_{\ge 0}$ be positive $\Sigma$-measurable functions.
Suppose that $f = g$ almost everywhere.
Then:
:$\ds \int f \rd \mu = \int g \rd \mu$
=== Corollary 1 ===
=== Corollary 2 ===
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $10.10$
Category:Measure Theory
Category:A.E. Equal Positive Measurable Functions have Equal Integrals