nyuuzyou/wikis huggingface dataset:
1027

== Number ==
$1027$ (one thousand and twenty-seven) is:
:$13 \times 79$
:The $8$th after $4$, $13$, $38$, $87$, $208$, $377$, $666$ in the sequence formed by adding the squares of the first $n$ primes:
::$1027 = \ds \sum_{i \mathop = 1}^8 {p_i}^2 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 + 19^2$
:The $19$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$, $331$, $397$, $469$, $547$, $631$, $721$, $817$, $919$:
::$1027 = \ds 1 + \sum_{k \mathop = 1}^{19 - 1} 6 k = 19^3 - 18^3$
:The $26$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $610$, $672$, $737$, $805$, $876$, $950$:
::$1027 = \dfrac {26 \paren {3 \times 26 + 1} } 2$
:The $52$nd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $610$, $651$, $672$, $715$, $737$, $782$, $805$, $852$, $876$, $925$, $950$, $1001$:
::$1027 = \dfrac {26 \paren {3 \times 26 + 1} } 2$
== Also see ==
* 
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Category:Specific Numbers
Category:1027
103

== Number ==
$103$ (one hundred and three) is:
:The $27$th prime number
:The smallest prime number the period of whose decimal expansion is $\dfrac 1 3$ of its maximum length, that is: $34$:
::$\dfrac 1 {103} = 0 \cdotp \dot 00970 \, 87378 \, 64077 \, 66990 \, 29126 \, 21359 \, 223 \dot 3$
:The $3$rd prime number after $53, 71$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$.
:The larger of the $9$th pair of twin primes, with $101$
:The $21$st happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$, $70$, $79$, $82$, $86$, $91$, $94$, $97$, $100$:
::$103 \to 1^2 + 0^2 + 3^2 = 10 \to 1^2 + 0^2 = 1$
:The $37$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $77$, $83$, $89$, $97$, $101$, $103$, $\ldots$
== Also see ==
* 103 is Smallest Prime whose Period of Reciprocal is One Third of Maximal
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== Sources ==
* : $103$
* : $103$
Category:Specific Numbers
Category:103
103,040

== Number ==
$103 \, 040$ (one hundred and three thousand and forty) is:
:$2^7 \times 5 \times 7 \times 23$
:The $25$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
::$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$, $38$, $94$, $166$, $324$, $379$, $469$, $546$, $974$, $1963$, $3507$, $3610$, $6917$, $21 \, 480$, $34 \, 790$, $94 \, 550$, $103 \, 040$
== Also see ==
* 
Category:Specific Numbers
Category:103,040
1031

== Number ==
$1031$ (one thousand and thirty-one) is:
:The $173$rd prime number
:The index of the $5$th repunit prime after $R_2$, $R_{19}$, $R_{23}$, $R_{317}$
:The larger of the $17$th pair of primes whose prime gap is $10$:
::$1031 - 1021 = 10$
:The $36$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $733$, $739$, $743$, $751$, $761$, $769$, $907$, $937$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$
:The smaller of the $37$th pair of twin primes:
::$1033 - 1031 = 2$
:The $40$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $653$, $659$, $683$, $719$, $743$, $761$, $809$, $911$, $953$, $1013$, $1019$:
::$2 \times 1031 + 1 = 2063$, which is prime.
== Historical Note ==
== Also see ==
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== Sources ==
* : $1111 \ldots 111111$
Category:Specific Numbers
Category:1031
1031/Historical Note

== Historical Note on $1031$ ==
The prime nature of the repunit $R_{1031}$ was proved by and in $1986$.
== Sources ==
* 
* : $1,111,111,111,111,111,111$
* : $1,111,111,111,111,111,111$
* : $1111 \ldots 111111$
Category:Historical Notes
1033

== Number ==
$1033$ (one thousand and thirty-three) is:
:The $174$th prime number
:The larger of the $37$th pair of twin primes:
::$1033 - 1031 = 2$
:The $37$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $733$, $739$, $743$, $751$, $761$, $769$, $907$, $937$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$, $1031$
== Also see ==
* 
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Category:Specific Numbers
Category:1033
1034

== Number ==
$1034$ (one thousand and thirty-four) is:
:$2 \times 11 \times 47$
:The $13$th positive integer after $1$, $7$, $102$, $110$, $142$, $159$, $187$, $381$, $690$, $714$, $770$, $994$ the sum of whose divisors is a cube:
::$\map {\sigma_1} {1034} = 1728 = 12^3$
== Also see ==
* 
Category:Specific Numbers
Category:1034
1035

== Number ==
$1035$ (one thousand and thirty-five) is:
:$3^2 \times 5 \times 23$
:The $23$rd hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $703$, $780$, $861$, $946$:
::$1035 = \ds \sum_{k \mathop = 1}^{23} \paren {4 k - 3} = 23 \paren {2 \times 23 - 1}$
:The $45$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$, $780$, $820$, $861$, $903$, $946$, $990$:
::$1035 = \ds \sum_{k \mathop = 1}^{45} k = \dfrac {45 \times \paren {45 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1035
1039

== Number ==
$1039$ (one thousand and thirty-nine) is:
:The $175$th prime number
:The $5$th after $199$, $409$, $619$, $829$ of the smallest $10$ primes in arithmetic sequence:
::$1039 = 4 \times 210 + 199$
:The smaller of the $18$th pair of primes whose prime gap is $10$:
::$1049 - 1039 = 10$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1039
103 is Smallest Prime whose Period of Reciprocal is One Third of Maximal

== Theorem ==
$103$ is the smallest prime number the period of whose decimal expansion of its reciprocal is $\dfrac 1 3$ of its maximum, that is: $34$:
:$\dfrac 1 {103} = 0 \cdotp \dot 00970 \, 87378 \, 64077 \, 66990 \, 29126 \, 21359 \, 223 \dot 3$
== Proof ==
From Reciprocal of $103$:
and so by counting it can be seen that its period of recurrence is $34$.
By Maximum Period of Reciprocal of Prime, the maximum period of recurrence of the reciprocal of $p$ when expressed in decimal notation is $p - 1$.
Therefore in order for a prime number to have its period of its reciprocal to equal $\dfrac 1 3$ of its maximum length, we must have:
:$3 \divides p - 1$
The prime numbers less than $103$ with this property are $7$, $13$, $19$, $31$, $37$, $43$, $61$, $67$, $73$, $79$ and $97$.
We have:
* $7$ is Long Period Prime
* $19$ is Long Period Prime
* $61$ is Long Period Prime
* $97$ is Long Period Prime
which shows that each of these primes has maximum period $p - 1$.
We have:
:Reciprocal of $13$: $\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3$: recurring with period $6 = \dfrac {p - 1} 2$.
:Reciprocal of $31$: $\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$: recurring with period $15 = \dfrac {p - 1} 2$.
:Reciprocal of $37$: $\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$: recurring with period $3 = \dfrac {p - 1} {12}$.
:Reciprocal of $43$: $\dfrac 1 {43} = 0 \cdotp \dot 02325 \, 58139 \, 53488 \, 37209 \dot 3$: recurring with period $21 = \dfrac {p - 1} 2$.
:Reciprocal of $67$: $\dfrac 1 {67} = 0 \cdotp \dot 01492 \, 53731 \, 34328 \, 35820 \, 89552 \, 23880 \, 59 \dot 7$: recurring with period $33 = \dfrac {p - 1} 2$.
:Reciprocal of $73$: $\dfrac 1 {73} = 0 \cdotp \dot 01369 \, 86 \dot 3$: recurring with period $8 = \dfrac {p - 1} 9$.
:Reciprocal of $79$: $\dfrac 1 {79} = 0 \cdotp \dot 01265 \, 82278 \, 48 \dot 1$: recurring with period $13 = \dfrac {p - 1} 6$.
Thus $103$ is the smallest prime number with this property.
== Sources ==
* : $103$
* : $103$
Category:103
Category:Examples of Reciprocals
104

== Number ==
$104$ (one hundred and four) is:
:$2^3 \times 13$
:The $4$th primitive abundant number after $20$, $70$, $88$:
::$1 + 2 + 4 + 8 + 13 + 26 + 52 = 106 > 104$
:The $5$th primitive semiperfect number after $6$, $20$, $28$, $88$:
::$104 = 1 + 4 + 8 + 13 + 26 + 52$
:The $1$st of the $2$nd ordered quadruple of consecutive integers that have divisor sums which are strictly decreasing:
::$\map {\sigma_1} {104} = 210, \ \map {\sigma_1} {105} = 192, \ \map {\sigma_1} {106} = 162, \ \map {\sigma_1} {107} = 108$
:The $4$th positive integer solution after $1$, $3$, $15$ to $\map \phi n = \map \phi {n + 1}$:
::$\map \phi {104} = 48 = \map \phi {105}$
:The $20$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$, $61$, $67$, $73$, $79$, $85$, $91$, $92$, $98$ such that no factorial of an integer can end with $n$ zeroes
== Also see ==
* 
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== Sources ==
* : $104$
* : $104$
Category:Specific Numbers
Category:104
1045

== Number ==
$1045$ (one thousand, one hundred and forty-five) is:
:$5 \times 11 \times 19$
:The $19$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$, $176$, $225$, $280$, $341$, $408$, $481$, $560$, $645$, $736$, $833$, $936$:
::$1045 = \ds \sum_{k \mathop = 1}^{19} \paren {6 k - 5} = 19 \paren {3 \times 19 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:1045
1049

== Number ==
$1049$ (one thousand and forty-nine) is:
:The $176$th prime number
:The larger of the $18$th pair of primes whose prime gap is $10$:
::$1049 - 1039 = 10$
:The smaller of the $38$th pair of twin primes:
::$1051 - 1049 = 2$
:The $41$st Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $653$, $659$, $683$, $719$, $743$, $761$, $809$, $911$, $953$, $1013$, $1019$, $1031$:
::$2 \times 1049 + 1 = 2099$, which is prime.
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:1049
105

== Number ==
$105$ (one hundred and five) is:
:$3 \times 5 \times 7$
:The $1$st positive integer $n$ such that $1$ can be expressed as the sum of reciprocals of distinct odd integers such that none are less than $\dfrac 1 n$:
::$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {33} + \dfrac 1 {35} + \dfrac 1 {45} + \dfrac 1 {55} + \dfrac 1 {77} + \dfrac 1 {105}$
:The $1$st of the $1$st ordered triple of consecutive integers that have Euler $\phi$ values which are strictly increasing:
::$\map \phi {105} = 48$, $\map \phi {106} = 52$, $\map \phi {107} = 106$
:The $2$nd positive integer $n$ such that $\map {\sigma_1} n = \dfrac {\map \phi n \times \map {\sigma_0} n} 2$:
::$\map {\sigma_1} {105} = 192 = \dfrac {\map \phi {105} \times \map {\sigma_0} {105} } 2$
:The $1$st of the $2$nd pair of triangular numbers whose sum and difference are also both triangular:
::$105 = T_{14}$, $171 = T_{18}$, $105 + 171 = T_{23}$, $171 - 105 = T_{11}$
:The $2$nd of the $2$nd ordered quadruple of consecutive integers that have divisor sums which are strictly decreasing:
::$\map {\sigma_1} {104} = 210, \ \map {\sigma_1} {105} = 192, \ \map {\sigma_1} {106} = 162, \ \map {\sigma_1} {107} = 108$
:The $4$th integer after $1$, $14$, $30$ whose divisor sum divided by its Euler $\phi$ value is a square:
::$\dfrac {\map {\sigma_1} {105} } {\map \phi {105} } = \dfrac {192} {48} = 4 = 2^2$
:The $7$th positive integer $n$ after $4$, $7$, $15$, $21$, $45$, $75$, and largest known, such that $n - 2^k$ is prime for all $k$
:The $7$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$:
::$105 = 3 \times 5 \times 7$
:The $9$th integer $n$ after $1$, $3$, $15$, $30$, $35$, $56$, $70$, $78$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} {105} = 8$, $\map \phi {105} = 48$, $\map {\sigma_1} {105} = 192$
:The $9$th and largest odd positive integer after $1$, $3$, $5$, $7$, $9$, $15$, $21$, $45$ such that all smaller odd integers greater than $1$ which are coprime to it are prime.
:The $13$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime
:The $14$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$, $78$, $91$:
::$105 = \ds \sum_{k \mathop = 1}^{14} k = \dfrac {14 \times \paren {14 + 1} } 2$
:The $23$rd lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $75$, $79$, $87$, $93$, $99$, $105$, $\ldots$
:The largest integer such that all smaller odd integers greater than $1$ which are coprime to it are prime
== Also see ==
* Reciprocals of Odd Numbers adding to 1
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== Sources ==
* : $105$
* : $105$
Category:Specific Numbers
Category:105
1050

== Number ==
$1050$ (one thousand and fifty) is:
:$2 \times 3 \times 5^2 \times 7$
:With $1925$, an element of the $3$rd quasiamicable pair:
::$\map {\sigma_1} {1050} = \map {\sigma_1} {1925} = 2976 = 1050 + 1925 + 1$
== Also see ==
* 
Category:Specific Numbers
Category:1050
1051

== Number ==
$1051$ (one thousand and fifty-one) is:
:The $177$th prime number
:The smaller of the $19$th pair of primes whose prime gap is $10$:
::$1061 - 1051 = 10$
:The larger of the $38$th pair of twin primes:
::$1051 - 1049 = 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1051
1056

== Number ==
$1056$ (one thousand and fifty-six) is:
:$2^5 \times 3 \times 11$
:The $1$st (with $43 \, 056$) of the only $2$ solutions $n$ to the equation $n = x \paren {x + 1} = y \paren {y + 5} \paren {y + 10} \paren {y + 15}$:
::$1056 = 32 \times 33 = 1 \times 6 \times 11 \times 16$
== Also see ==
* 
== Sources ==
* : $1056$
Category:Specific Numbers
Category:1056
106

== Number ==
$106$ (one hundred and six) is:
:$2 \times 53$
:The $34$th semiprime:
::$106 = 2 \times 53$
:The $2$nd of the $1$st ordered triple of consecutive integers that have Euler $\phi$ values which are strictly increasing:
::$\map \phi {105} = 48$, $\map \phi {106} = 52$, $\map \phi {107} = 106$
:The $3$rd of the $2$nd ordered quadruple of consecutive integers that have divisor sums which are strictly decreasing:
::$\map {\sigma_1} {104} = 210, \ \map {\sigma_1} {105} = 192, \ \map {\sigma_1} {106} = 162, \ \map {\sigma_1} {107} = 108$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:106
1061

== Number ==
$1061$ (one thousand and sixty-one) is:
:The $178$th prime number
:The larger of the $19$th pair of primes whose prime gap is $10$:
::$1061 - 1051 = 10$
:The $38$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $761$, $769$, $907$, $937$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$, $1031$, $1033$
:The smaller of the $39$th pair of twin primes:
::$1063 - 1061 = 2$
== Also see ==
* 
* 
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Category:Specific Numbers
Category:1061
1069

== Number ==
$1087$ (one thousand and eighty-seven) is:
:The $180$th prime number
:The $39$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$, $1031$, $1033$, $1061$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1069
107

== Number ==
$107$ (one hundred and seven) is:
:The $28$th prime number
:The $3$rd of the $1$st ordered triple of consecutive integers that have Euler $\phi$ values which are strictly increasing:
::$\map \phi {105} = 48$, $\map \phi {106} = 52$, $\map \phi {107} = 106$
:The $4$th prime number after $53, 71, 103$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$.
:The $4$th of the $2$nd ordered quadruple of consecutive integers that have divisor sums which are strictly decreasing:
::$\map {\sigma_1} {104} = 210, \ \map {\sigma_1} {105} = 192, \ \map {\sigma_1} {106} = 162, \ \map {\sigma_1} {107} = 108$
:The $8$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$:
::$107 = 2 \times 53 + 1$
:The $9$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$
:The smaller of the $10$th pair of twin primes, with $109$
:The index of the $11$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$:
::$M_{107} = 2^{107} - 1 = 162 \, 259 \, 276 \, 829 \, 213 \, 363 \, 391 \, 578 \, 010 \, 288 \, 127$
:The $52$nd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $77$, $80$, $81$, $84$, $89$, $94$, $95$, $96$, $100$, $101$, $102$ which cannot be expressed as the sum of distinct pentagonal numbers.
== Also see ==
* 
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Category:Specific Numbers
Category:107
107,928,278,317

== Number ==
$107\,928\,278\,317$ is:
:A prime number
:The $1$st of the smallest $18$ primes in arithmetic sequence:
::$107\,928\,278\,317 = 107\,928\,278\,317 + 0 \times 9\,922\,782\,870$
== Also see ==
* 
* Smallest 18 Primes in Arithmetic Sequence
== Sources ==
* : $107,928,278,317$
Category:Specific Numbers
Category:107,928,278,317
1071

== Number ==
$1071$ (one thousand and seventy-one) is:
:$3^2 \times 7 \times 17$
:The $21$st heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$, $235$, $286$, $342$, $403$, $469$, $540$, $616$, $697$, $783$, $874$, $970$:
::$1071 = \ds \sum_{k \mathop = 1}^{21} \paren {5 k - 4} = \dfrac {21 \paren {5 \times 21 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:1071
108

== Number ==
$108$ (one hundred and eight) is:
:$2^2 \times 3^3$
:The number of distinct free heptominoes
:The $15$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$, $100$
:The $22$nd highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$:
::$\map {\sigma_1} {108} = 280$
:The $29$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $67$, $72$, $76$, $92$, $96$, $108$, $\ldots$
:The $53$rd (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $77$, $78$, $79$, $84$, $90$, $91$, $95$, $96$, $102$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* Number of Heptominoes
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== Sources ==
* : $35$
* : $108$
* : $35$
* : $108$
Category:Specific Numbers
Category:108
1080

== Number ==
$1080$ (one thousand and eighty) is:
:$2^3 \times 3^3 \times 5$
:The $27$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $477$, $532$, $590$, $651$, $715$, $782$, $852$, $925$, $1001$:
::$1080 = \ds \sum_{k \mathop = 1}^{27} \paren {3 k - 2} = \dfrac {27 \paren {3 \times 27 - 1} } 2$
:The $45$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $336$, $360$, $420$, $480$, $504$, $600$, $630$, $660$, $720$, $840$, $840$, $960$, $1008$:
::$\map {\sigma_1} {1080} = 3600$
:The $53$rd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $610$, $651$, $672$, $715$, $737$, $782$, $805$, $852$, $876$, $925$, $950$, $1001$, $1027$:
::$1080 = \ds \sum_{k \mathop = 1}^{27} \paren {3 k - 2} = \dfrac {27 \paren {3 \times 27 - 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1080
1081

== Number ==
$1081$ (one thousand and eighty-one) is:
:$23 \times 47$
:The $46$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$, $780$, $820$, $861$, $903$, $946$, $990$, $1035$:
::$1081 = \ds \sum_{k \mathop = 1}^{46} k = \dfrac {46 \times \paren {46 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:1035
1086

== Number ==
$1086$ (one thousand and eighty-six) is:
:$2 \times 3 \times 181$
:The $50$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $778$, $825$, $852$, $895$, $913$, $915$, $922$, $958$, $985$:
::$1 + 0 + 8 + 6 = 2 + 3 + 1 + 8 + 1 = 15$
== Also see ==
* 
Category:Specific Numbers
Category:1086
1087

== Number ==
$1087$ (one thousand and eighty-seven) is:
:The $181$st prime number
:The $21$st obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$, $509$, $599$, $701$, $757$, $757$, $809$, $877$, $905$, $907$, $959$, $977$, $997$, $1019$
:The $24$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 23^2 + 29 = 1087$ ()
== Also see ==
* 
* 
Category:Specific Numbers
Category:1087
1089

== Number ==
$1089$ (one thousand and eighty-nine) is:
:$3^2 \times 11^2$
:The $1$st number which is reversed when multiplied by $9$:
::$1089 \times 9 = 9801$
:The $1$st of only $2$ numbers with $4$ digits or fewer which has a multiple which is its reversal:
::$1089 \times 9 = 9801$
:The $29$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $400$, $526$, $529$, $625$, $676$, $706$, $730$, $771$, $784$, $841$, $1024$, $1089$, $\ldots$
:The $33$rd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $625$, $676$, $729$, $784$, $841$, $900$, $961$, $1024$:
::$1089 = 33 \times 33$
:Take a $3$-digit number, reverse it and subtract the smaller from the larger, reverse that and add it to itself, and you end up with $1089$.
:The only square of a $2$-digit number which has this property:
::$33^2 = 65^2 - 56^2$
== Also see ==
* 1089 Trick
* Square which is Difference between Square and Square of Reversal
* 
* 
* 
* 
== Sources ==
* : $1089$
* : $1089$
Category:Specific Numbers
Category:1089
1089 Trick

== Classic Puzzle ==
Take a three-digit number (one where the first and last digits are different by at least $2$).
Reverse it, and get the difference between that and the first number.
Reverse that, and add that to the difference you calculated just now.
You get $1089$.
== Proof ==
Let the number you started with be expressed in decimal notation as $[abc]_{10}$.
Then it is $10^2 a + 10 b + c$.
Its reversal is:
:$10^2 c + 10 b + a$
The difference between the two is $99a - 99c$ (or $99c - 99a$, it matters not).
This is a multiple of $99$.
The three-digit multiples of $99$ are:
:$198$
:$297$
:$396$
:$495$
:$594$
:$693$
:$792$
:$891$
By adding any one of these to its reversal, you get:
:$9 \times 100 + 2 \times 9 \times 10 + 9 = 1089$
Note: You need to make sure the difference between the first and last digits of the number you started with is at least $2$ so as to make sure that the first difference you calculate is definitely a $3$-digit number.
== Sources ==
* : $1089$
* : No. $2$ Vol. $1$: The Late Seventies Masonettes: The Second Chapter: 'My Brain Hurts'
:::Jocasta Dribble has been passed the list of number codes for the secret society she has just joined. The code for "weird" is $1089$: "a funny number". Examples of its use ($234$, $742$ and $876$) are illustrated on the piece of paper.
* : $1089$
* 
Category:Classic Problems
Category:Reversals
Category:1089
Category:Think of a Number Puzzles
109

== Number ==
$109$ (one hundred and nine) is:
:The $29$th prime number
:The $5$th prime number after $53$, $71$, $103$, $107$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$.
:The $8$th of $11$ primes of the form $2 x^2 + 11$:
::$2 \times 7^2 + 11 = 109$ ()
:The larger of the $10$th pair of twin primes, with $107$
:The $10$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$
:The $22$nd happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $91$, $94$, $97$, $100$, $103$:
::$109 \to 1^2 + 0^2 + 9^2 = 1 + 0 + 81 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:109
109,297

== Number ==
$109 \, 297$ (one hundred and nine thousand, two hundred and ninety-seven) is:
:The $10 \, 390$th prime number
:The index of the $8$th (probable) repunit prime after $R_2$, $R_{19}$, $R_{23}$, $R_{317}$, $R_{1031}$, $R_{49 \, 081}$, $R_{86 \, 453}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:109,297
109,376

== Number ==
$109 \, 376$ (one hundred and nine thousand, three hundred and seventy-six) is:
:$2^6 \times 1709$
:The $10$th automorphic number after $1$, $5$, $6$, $25$, $76$, $376$, $625$, $9376$, $90 \, 625$:
::$109 \, 376^2 = 11 \, 963 \, \mathbf{109 \, 376}$
== Also see ==
* 
Category:Specific Numbers
Category:109,376