nyuuzyou/wikis huggingface dataset:
8256

== Number ==
$8256$ (eight thousand, two hundred and fifty-six) is:
:$2^6 \times 3 \times 43$
:The $128$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$, $8128$:
::$8256 = \ds \sum_{k \mathop = 1}^{128} k = \dfrac {128 \times \paren {128 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:8256
8275

== Number ==
$8275$ (eight thousand, two hundred and seventy-five) is:
:$5^2 \times 331$
:The $21$st integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$, $105$, $160$, $661$, $2653$, $3069$, $3943$, $4053$, $4998$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$ is prime
== Also see ==
* 
Category:Specific Numbers
Category:8275
828,828

== Number ==
$828 \, 828$ (eight hundred and twenty-eight thousand, eight hundred and twenty-eight) is:
:$2^2 \times 3^2 \times 7 \times 11 \times 13 \times 23$
:The $16$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$, $666$, $3003$, $5995$, $8778$, $15 \, 051$, $66 \, 066$, $617 \, 716$:
::$828 \, 828 = \dfrac {1287 \times \paren {1287 + 1} } 2$
:The $3$rd of those after $55$, $66$, and largest known, which can be split into two palindromic halves
:The $1287$th triangular number:
::$828 \, 828 = \dfrac {1287 \times \paren {1287 + 1} } 2$
== Also see ==
* 
* 
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== Sources ==
* : $828,828$
* : $828,828$
Category:Specific Numbers
Category:828,828
8281

== Number ==
$8281$ (eight thousand, two hundred and eighty-one) is:
:$7^2 \times 13^2$
:The $91$st square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $7396$, $7569$, $7744$, $7921$, $8100$:
::$8281 = 91 \times 91$
:The sum of the first $13$ cubes:
::$8281 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 + 10^3 + 11^3 + 12^3 + 13^3$
:The $1$st square number whose digits form $2$ consecutive integers, and the only one with $4$ digits:
::$\mathbf {82} \, \mathbf {81} = 91^2$
:The $1$st square number whose digits form $2$ consecutive decreasing integers:
::$\mathbf {82} \, \mathbf {81} = 91^2$
== Also see ==
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== Sources ==
* : $8281$
* : $8281$
Category:Specific Numbers
Category:8281
829

== Number ==
$829$ (eight hundred and twenty-nine) is:
:The $145$th prime number
:The $4$th after $199$, $409$, $619$ of the smallest $10$ primes in arithmetic sequence:
::$829 = 3 \times 210 + 199$
:The smaller of the $15$th pair of primes whose prime gap is $10$:
::$839 - 829 = 10$
:The $21$st of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 20^2 + 29 = 829$ ()
:The larger of the $33$rd pair of twin primes:
::$829 - 827 = 2$
== Also see ==
* 
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Category:Specific Numbers
Category:829
8294

== Number ==
$8294$ (eight thousand, two hundred and ninety-four) is:
:$2 \times 11 \times 13 \times 29$
:The smaller of the $2$nd pair of consecutive positive integers which are each the product of exactly $4$ distinct prime numbers:
::$8294 = 2 \times 11 \times 13 \times 29$, $8295 = 3 \times 5 \times 7 \times 79$
== Also see ==
* 
Category:Specific Numbers
Category:8294
8295

== Number ==
$8295$ (eight thousand, two hundred and ninety-five) is:
:$3 \times 5 \times 7 \times 79$
:The larger of the $2$nd pair of consecutive positive integers which are each the product of exactly $4$ distinct prime numbers:
::$8294 = 2 \times 11 \times 13 \times 29$, $8295 = 3 \times 5 \times 7 \times 79$
== Also see ==
* 
Category:Specific Numbers
Category:8295
83

== Number ==
$83$ (eighty-three) is:
:The $23$rd prime number
:The $7$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$:
::$83 = 2 \times 41 + 1$
:The $7$th of $11$ primes of the form $2 x^2 + 11$:
::$2 \times 6^2 + 11 = 83$ ()
:The $9$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$:
::$2 \times 83 + 1 = 167$, which is prime.
:The $14$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$, $53$, $59$, $67$, $71$, $73$, $79$ such that the Mersenne number $2^p - 1$ is composite
:The $14$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$
:The $33$rd odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $77$, $83$, $\ldots$
== Also see ==
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Category:Specific Numbers
Category:83
83,328

== Number ==
$83 \, 328$ (eighty-eight thousand, three hundred and twenty-eight) is:
:$2^7 \times 3 \times 7 \times 31$
:The $5$th 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:83,328
83,521

== Number ==
$83 \, 521$ (eighty-three thousand, five hundred and twenty-one) is:
:$17^4$
:The $289$th square number:
::$83 \, 521 = 289 \times 289$
:The $17$th fourth power after $1$, $16$, $81$, $256$, $625$, $\ldots$, $28 \, 561$, $38 \, 416$, $50 \, 625$, $65 \, 536$:
::$83 \, 521 = 17 \times 17 \times 17 \times 17$
== Also see ==
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* 
Category:Specific Numbers
Category:83,521
83,623,935

== Number ==
$83 \, 623 \, 935$ is:
:$3 \times 5 \times 17 \times 353 \times 929$
:Has a Euler $\phi$ value which is a divisor of itself plus $1$:
::$2 \times \map \phi {83 \, 623 \, 935} = 83 \, 623 \, 935 + 1$
== Also see ==
* Numbers n whose Euler Phi value Divides n + 1
== Sources ==
* : $83,623,935$
Category:Specific Numbers
Category:83,623,935
83,952

== Number ==
$83 \, 952$ (eighty-three thousand, nine hundred and fifty-two) is:
:$2^4 \times 3^2 \times 11 \times 53$
:One of the elements of one of the $4$-cycles to which Kaprekar's process on some $5$-digit numbers converges:
::$83 \, 952 \to 74 \, 943 \to 62 \, 964 \to 71 \, 973 \to 83 \, 952$
== Also see ==
* 
Category:Specific Numbers
Category:83,952
8321

== Number ==
$8321$ (eight thousand, three hundred and twenty-one) is:
:$53 \times 157$
:The $20$th Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$, $2465$, $2701$, $2821$, $3277$, $4033$, $4369$, $4371$, $4681$, $5461$, $6601$, $7957$:
::$2^{8321} \equiv 2 \pmod {8321}$: $8321 = 53 \times 157$
:The $53$rd octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $5985$, $6256$, $6533$, $6816$, $7105$, $7400$, $7701$, $8008$:
::$8321 = \ds \sum_{k \mathop = 1}^{53} \paren {6 k - 5} = 53 \paren {3 \times 53 - 2}$
== Also see ==
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* 
Category:Specific Numbers
Category:8321
833

== Number ==
$833$ (eight hundred and thirty-three) is:
:$7^2 \times 17$
:The $17$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$, $176$, $225$, $280$, $341$, $408$, $481$, $560$, $645$, $736$:
::$833 = \ds \sum_{k \mathop = 1}^{17} \paren {6 k - 5} = 17 \paren {3 \times 17 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:833
836

== Number ==
$836$ (eight hundred and thirty-six) is:
:$2^2 \times 11 \times 19$
:The $1$st positive integer whose square is a palindromic number with an even number of digits:
::$836^2 = 698 \, 896$ ($6$ digits)
:The $2$nd weird number after $70$:
::$\map {\sigma_1} {836} - 836 = 844$: its aliquot parts are $1$, $2$, $4$, $11$, $19$, $22$, $38$, $44$, $76$, $209$, $418$, from which $836$ cannot be made.
:The $4$th non-palindromic square root after $26$, $264$, $307$ of a palindromic square:
::$836^2 = 698 \, 896$
:The $13$th primitive abundant number after $20$, $70$, $88$, $104$, $272$, $304$, $368$, $464$, $550$, $572$, $650$, $748$:
::$1 + 2 + 4 + 11 + 19 + 22 + 38 + 44 + 76 + 209 + 418 = 844 > 836$
== Also see ==
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== Sources ==
* : $836$
* : $836$
Category:Specific Numbers
Category:836
8373

== Number ==
$8373$ (eight thousand, three hundred and seventy-three) is:
:$3 \times 2791$
:The $1$st term, with $8433$ and $8493$, of the $1$st arithmetic sequence of $3$ integers with common difference of $60$ all of which can be expressed as the sum of $4$ fourth powers:
::$8373^4 = 4450^4 + 5500^4 + 5670^4 + 7123^4$
== Also see ==
* 
== Sources ==
* : $8373$
Category:Specific Numbers
Category:8373
8385

== Number ==
$8385$ (eight thousand, three hundred and eighty-five) is:
:$3 \times 5 \times 13 \times 43$
:The $65$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $6903$, $7140$, $7381$, $7626$, $7875$, $8128$:
::$8385 = \ds \sum_{k \mathop = 1}^{65} \paren {4 k - 3} = 65 \paren {2 \times 65 - 1}$
:The $129$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$, $8128$, $8256$:
::$8385 = \ds \sum_{k \mathop = 1}^{129} k = \dfrac {129 \times \paren {129 + 1} } 2$
== Also see ==
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* 
Category:Specific Numbers
Category:8385
839

== Number ==
$839$ (eight hundred and thirty-nine) is:
:The $146$th prime number
:The smaller of the $5$th pair of primes whose prime gap is $14$:
::$853 - 839 = 14$
:The larger of the $15$th pair of primes whose prime gap is $10$:
::$839 - 829 = 10$
:The $22$nd safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$, $467$, $479$, $503$, $563$, $587$, $719$:
::$839 = 2 \times 419 + 1$
== Also see ==
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Category:Specific Numbers
Category:839
84

== Number ==
$84$ (eighty-four) is:
:$2^2 \times 3 \times 7$
:The $3$rd 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 $5$th and last after $21$, $29$, $61$, $69$ of the $5$ $2$-digit positive integers which can occur as a $5$-fold repetition at the end of a square number
:The $5$th inconsummate number after $62$, $63$, $65$, $75$:
::$\nexists n \in \Z_{>0}: n = 84 \times \map {s_{10} } n$
:The $7$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$:
::$84 = 1 + 3 + 6 + 10 + 15 + 21 + 28 = \dfrac {7 \paren {7 + 1} \paren {7 + 2} } 6$
:The smallest positive integer which can be expressed as the sum of $2$ odd primes in $8$ ways.
:The $19$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$:
::$\map {\sigma_1} {84} = 224$
:The $19$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$, $66$, $72$, $78$, $80$:
::$84 = 14 + 28 + 42$
:The $44$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $61$, $65$, $66$, $67$, $72$, $77$, $80$, $81$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $47$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $77$, $78$, $79$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
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== Historical Note ==
Category:Specific Numbers
Category:84
840

== Number ==
$840$ (eight hundred and forty) is:
:$2^3 \times 3 \times 5 \times 7$
:The $15$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$:
::$\map {\sigma_0} {840} = 32$
:The $15$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$:
::$\dfrac {\map {\sigma_1} {840} } {840} = \dfrac {2880} {840} \approx 3 \cdotp 429$
:The $42$nd highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $210$, $216$, $240$, $288$, $300$, $336$, $360$, $420$, $480$, $504$, $600$, $630$, $660$, $720$:
::$\map {\sigma_1} {840} = 2880$
:Has the largest divisor count of all the positive integers up to $1000$:
::$\map {\sigma_0} {840} = 32$
== Also see ==
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== Sources ==
* : $840$
* : $840$
Category:Specific Numbers
Category:840
841

== Number ==
$841$ (eight hundred and forty-one) is:
:$29^2$
:The $29$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $625$, $676$, $729$, $784$:
::$841 = 29 \times 29$
:The $27$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $400$, $526$, $529$, $625$, $676$, $706$, $730$, $771$, $784$, $841$, $\ldots$
== Also see ==
* 
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Category:Specific Numbers
Category:841
8415

== Number ==
$8415$ (eight thousand, four hundred and fifteen) is:
:$3^2 \times 5 \times 11 \times 17$
:The $18$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$, $4095$, $4725$, $5355$, $5775$, $5985$, $6435$, $6615$, $6825$, $7245$, $7425$, $7875$, $8085$:
::$\map {\sigma_1} {8415} - 8415 = 8433 > 8415$
== Also see ==
* 
Category:Specific Numbers
Category:8415
843

== Number ==
$843$ (eight hundred and forty-three) is:
:$3 \times 281$
:The $14$th Lucas number after $(2)$, $1$, $3$, $4$, $7$, $11$, $18$, $29$, $47$, $76$, $123$, $199$, $322$, $521$:
::$843 = 322 + 521$
== Also see ==
* 
Category:Specific Numbers
Category:843
8433

== Number ==
$8433$ (eight thousand, four hundred and thirty-three) is:
:$3^2 \times 937$
:The $2$nd term, with $8373$ and $8493$, of the $1$st arithmetic sequence of $3$ integers with common difference of $60$ all of which can be expressed as the sum of $4$ fourth powers:
::$8433^4 = 4730^4 + 4806^4 + 5230^4 + 7565^4$
== Also see ==
* 
== Sources ==
* : $8373$
Category:Specific Numbers
Category:8433
8436

== Number ==
$8436$ (eight thousand, four hundred and thirty-six) is:
:$2^2 \times 3 \times 19 \times 37$
:The $36$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $3276$, $3276$, $4060$, $4495$, $4960$, $5456$, $5984$, $6545$, $7140$, $7770$:
::$8436 = \ds \sum_{k \mathop = 1}^{36} \frac {k \paren {k + 1} } 2 = \dfrac {36 \paren {36 + 1} \paren {36 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:8436
845

== Number ==
$845$ (eight hundred and forty-five) is:
:$5 \times 13^2$
:The $5$th positive integer after $325$, $425$, $650$, $725$ which can be expressed as the sum of two square numbers in $3$ distinct ways:
::$845 = 29^2 + 2^2 = 26^2 + 13^2 = 22^2 + 19^2$
== Also see ==
* 
Category:Specific Numbers
Category:845
846

== Number ==
$846$ (eight hundred and forty-six) is:
:$2 \times 3^2 \times 47$
:The $4$th integer after $428$, $573$, $727$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive increasing integers:
::$846^2 = 715 \, 716$
:The $5$th integer after $91$, $428$, $573$, $727$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive integers:
::$846^2 = 715 \, 716$
== Also see ==
* 
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Category:Specific Numbers
Category:846
8464

== Number ==
$8464$ (eight thousand, four hundred and sixty-four) is:
:$2^4 \times 23^2$
:The $92$nd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $7396$, $7569$, $7744$, $7921$, $8100$, $8281$
::$8464 = 92 \times 92$
== Also see ==
* 
Category:Specific Numbers
Category:8464
8481

== Number ==
$8481$ (eight thousand, four hundred and eighty-one) is:
:$3 \times 11 \times 257$
:The $21$st Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$, $2465$, $2701$, $2821$, $3277$, $4033$, $4369$, $4371$, $4681$, $5461$, $6601$, $7957$, $8321$:
::$2^{8481} \equiv 2 \pmod {8481}$: $8481 = 3 \times 11 \times 257$
== Also see ==
* 
Category:Specific Numbers
Category:8481
8493

== Number ==
$8493$ (eight thousand, four hundred and ninety-three) is:
:$3 \times 19 \times 149$
:The $3$rd term, with $8373$ and $8433$, of the $1$st arithmetic sequence of $3$ integers with common difference of $60$ all of which can be expressed as the sum of $4$ fourth powers:
::$8493^4 = 524^4 + 4910^4 + 5925^4 + 7630^4$
== Also see ==
* 
== Sources ==
* : $8373$
Category:Specific Numbers
Category:8493
85

== Number ==
$85$ (eighty-five) is:
:$5 \times 17$
:The $2$nd Fermat pseudoprime to base $4$ after $15$:
::$4^{85} \equiv 4 \pmod {85}$
:The $3$rd positive integer after $50$, $65$ which can be expressed as the sum of two square numbers in two or more different ways:
::$85 = 9^2 + 2^2 = 7^2 + 6^2$
:The $4$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $85$, $\ldots$
:The $5$th Smith number after $4$, $22$, $27$, $58$:
::$8 + 5 = 5 + 1 + 7 = 13$
:The $7$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$
:The $10$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$, $65$ such that the Fibonacci number $F_n$ ends in $n$
:The $16$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$, $61$, $67$, $73$, $79$ such that no factorial of an integer can end with $n$ zeroes
:The $27$th semiprime:
::$85 = 5 \times 17$
== Also see ==
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== Sources ==
* : $85$
* : $85$
Category:Specific Numbers
Category:85
8505

== Number ==
$8505$ (eight thousand, five hundred and five) is:
:$3^5 \times 5 \times 7$
:The $19$th odd abundant number after $945$, $1575$, $2205$, $\ldots$, $6615$, $6825$, $7245$, $7425$, $7875$, $8085$, $8415$:
::$\map {\sigma_1} {8505} - 8505 = 8967 > 8505$
== Also see ==
* 
Category:Specific Numbers
Category:8505
8515

== Number ==
$8515$ (eight thousand, five hundred and fifteen) is:
:$5 \times 13 \times 131$
:The $130$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$, $8128$, $8256$, $8385$:
::$8515 = \ds \sum_{k \mathop = 1}^{130} k = \dfrac {130 \times \paren {130 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:8515
8517

== Number ==
$8517$ (eight thousand, five hundred and seventeen) is:
:$3 \times 17 \times 169$
:The $1$st term, with $8577$ and $8637$, 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:
::$8517^4 = 1642^4 + 3440^4 + 6100^4 + 7815^4$
== Also see ==
* 
== Sources ==
* : $8373$
Category:Specific Numbers
Category:8517
852

== Number ==
$852$ (eight hundred and fifty-two) is:
:$2^2 \times 3 \times 71$
:The $24$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $477$, $532$, $590$, $651$, $715$, $782$:
::$852 = \ds \sum_{k \mathop = 1}^{24} \left({3 k - 2}\right) = \dfrac {24 \paren {3 \times 24 - 1} } 2$
:The $47$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$, $532$, $551$, $590$, $610$, $651$, $672$, $715$, $737$, $782$, $805$:
::$852 = \ds \sum_{k \mathop = 1}^{24} \left({3 k - 2}\right) = \dfrac {24 \paren {3 \times 24 - 1} } 2$
:The $42$nd Smith number after $4$, $22$, $27$, $58$, $\ldots$, $648$, $654$, $663$, $666$, $690$, $706$, $728$, $729$, $762$, $778$, $825$:
::$8 + 5 + 2 = 2 + 2 + 3 + 7 + 1 = 15$
== Also see ==
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Category:Specific Numbers
Category:852
853

== Number ==
$853$ (eight hundred and fifty-three) is:
:The $147$th prime number
:The larger of the $5$th pair of primes whose prime gap is $14$:
::$853 - 839 = 14$
:The $47$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $443$, $467$, $523$, $547$, $613$, $617$, $643$, $647$, $653$, $673$, $683$, $743$, $773$, $797$, $823$
== Also see ==
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Category:Specific Numbers
Category:853
854

== Number ==
$854$ (eight hundred and fifty-four) is:
:$2 \times 7 \times 61$
:The $2$nd, with $567$, of $2$ positive integers which, together with their square, use all the digits from $1$ to $9$:
::$854^2 = 729 \, 316$
== Also see ==
* 
== Sources ==
* : $567$
* : $854$
* : $567$
Category:Specific Numbers
Category:854
8555

== Number ==
$8555$ (eight thousand, five hundred and fifty-five) is:
:$5 \times 29 \times 59$
:The $29$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $\ldots$, $3311$, $3795$, $4324$, $4900$, $5525$, $6201$, $6930$, $7714$:
::$8555 = \ds \sum_{k \mathop = 1}^{29} k^2 = \dfrac {29 \paren {29 + 1} \paren {2 \times 29 + 1} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:8555
857

== Number ==
$857$ (eight hundred and fifty-seven) is:
:The $148$th prime number
:The smaller of the $34$th pair of twin primes:
::$859 - 857 = 2$
:The $52$nd long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $701$, $709$, $727$, $743$, $811$, $821$, $823$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:857
8577

== Number ==
$8577$ (eight thousand, five hundred and seventy-seven) is:
:$3^3 \times 953$
:The $2$nd term, with $8517$ and $8637$, 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:
::$8577^4 = 1050^4 + 2905^4 + 5236^4 + 8230^4$
== Also see ==
* 
== Sources ==
* : $8373$
Category:Specific Numbers
Category:8577
858

== Number ==
$858$ (eight hundred and fifty-eight) is:
:$2 \times 3 \times 11 \times 13$
:The $2$nd Giuga number:
::$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 {11} + \dfrac 1 {13} - \dfrac 1 {858} = 1$
== Also see ==
* 
Category:Specific Numbers
Category:858
859,433

== Number ==
$859 \, 433$ (eight hundred and fifty-nine thousand, four hundred and thirty-three) is:
:The $68 \, 301$st prime number
:The index of the $33$rd Mersenne prime after $2$, $3$, $5$, $7$, $13$, $\ldots$, $23 \, 209$, $44 \, 497$, $86 \, 243$, $110 \, 503$, $132 \, 049$, $216 \, 091$, $756 \, 839$:
::$M_{859 \, 433} = 2^{859 \, 433} - 1 \approx 1 \cdotp 295 \times 10^{258 \, 715}$
== Also see ==
* 
* 
== Historical Note ==
== Sources ==
* : $2^{859,433} - 1$
Category:Specific Numbers
Category:859,433
859,433/Historical Note

== Historical Note on $859 \, 433$ ==
The Mersenne number $M_{859 \, 433} = 2^{859 \, 433} - 1$ was demonstrated to be a Mersenne prime by and .
This discovery was announced by on the internet on $4$ January $1994$.
== Sources ==
* : $2^{859,433} - 1$
Category:Historical Notes
86

== Number ==
$86$ (eighty-six) is:
:$2 \times 43$
:The $7$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 86$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The $10$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$, $68$, $74$, $76$:
::$\nexists m \in \Z_{>0}: \map \phi m = 86$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $16$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$, $70$, $79$, $82$:
::$86 \to 8^2 + 6^2 = 64 + 36 = 100 \to 1^2 + 0^2 + 0^2 = 1$
:The $28$th semiprime:
::$86 = 2 \times 43$
:The $36$th integer $n$, and believed to be the largest, such that $2^n$ contains no zero in its decimal representation:
::$2^{86} = 77 \, 371 \, 252 \, 455 \, 336 \, 267 \, 181 \, 195 \, 264$
== Also see ==
* Powers of 2 with no Zero in Decimal Representation
* 
* 
* 
* 
* 
== Sources ==
* : $86$
Category:Specific Numbers
Category:86
86,243

== Number ==
$86 \, 243$ (eighty-six thousand, two hundred and forty-three) is:
:The $8384$th prime number
:The index of the $28$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $\ldots$, $2203$, $2281$, $3217$, $4253$, $4423$, $9689$, $9941$, $11 \, 213$, $19 \, 937$, $21 \, 701$, $23 \, 209$, $44 \, 497$:
::$M_{86 \, 243} = 2^{86 \, 243} - 1 \approx 5 \cdotp 369 \times 10^{25 \, 961}$
== Also see ==
* 
* 
== Historical Note ==
== Sources ==
* : $2^{86243} - 1$
* : $2^{86,243} - 1$
Category:Specific Numbers
Category:86,243