nyuuzyou/wikis huggingface dataset:
109,418,989,131,512,359,209

== Number ==
$109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209$ is:
:$3^{42}$
:The $10 \, 460 \, 353 \, 203$th square number:
::$109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209 = 10 \, 460 \, 353 \, 203 \times 10 \, 460 \, 353 \, 203$
:The $4 \, 782 \, 969$th cube number:
::$109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209 = 4 \, 782 \, 969 \times 4 \, 782 \, 969 \times 4 \, 782 \, 969$
:The $2187$th sixth power:
::$109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209 = 2187 \times 2187 \times 2187 \times 2187 \times 2187 \times 2187$
:The $729$th seventh power:
::$109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209 = 729 \times 729 \times 729 \times 729 \times 729 \times 729 \times 729$
:The $42$nd power of $3$:
::$109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209 = 3^{42}$
:The $21$st power of $9$:
::$109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209 = 9^{21}$
:The largest $n$th power which has $n$ digits:
::$9^{21} = 109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209$ has $21$ digits
== Also see ==
* Largest nth Power which has n Digits
* 
* 
* 
* 
* 
* 
== Sources ==
* : $109,418,989,131,512,359,209$
* : $109,418,989,131,512,359,209$
Category:Specific Numbers
Category:109,418,989,131,512,359,209
1091

== Number ==
$1091$ (one thousand and ninety-one) is:
:The $182$nd prime number
:The smaller of the $40$th pair of twin primes:
::$1093 - 1091 = 2$
:The $40$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$, $1031$, $1033$, $1061$, $1069$
:The equation $x^6 + 1091$ yields a composite number for all integer values of $x$ from $1$ to $3095$
== Also see ==
* Numbers for which Sixth Power plus 1091 is Composite
* 
* 
* 
== Sources ==
* : $1091$
Category:Specific Numbers
Category:1091
1093

== Number ==
$1093$ (one thousand and ninety-three) is:
:The $183$rd prime number
:The $1$st of the only $2$ known Wieferich primes:
::$1093^2 \divides 2^{1092} - 1$
:The larger of the $40$th pair of twin primes:
::$1093 - 1091 = 2$
== Also see ==
* 
* 
* 
== Sources ==
* : $1093$
* : $3511$
* : $1093$
* : $3511$
* 
* 
Category:Specific Numbers
Category:1093
1097

== Number ==
$1097$ (one thousand and ninety-seven) is:
:The $184$th prime number
:The $41$st emirp after $13$, $17$, $31$, $37$, $\ldots$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$, $1031$, $1033$, $1061$, $1069$, $1091$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1097
10 Consecutive Integers contain Coprime Integer

== Theorem ==
Let $n \in \Z$ be an integer.
Let $S := \set {n, n + 1, n + 2, \ldots, n + 9}$ be the set of $10$ consecutive integers starting from $n$.
Then at least one element of $S$ is coprime to every other element of $S$.
== Proof ==
Consider $2$ elements $a, b$ of $S$ which share a common divisor $d$.
Then $d \divides \size {a - b}$ and so $d < 10$.
Now from the Fundamental Theorem of Arithmetic, $d$ must have a prime factor which is strictly less than $10$.
So for $a$ and $b$ to have a common divisor, at least one such common divisor is in $\set {2, 3, 5, 7}$.
There are exactly $5$ elements of $S$ which have a common divisor of $2$.
There are either $3$ or $4$ elements of $S$ common divisor of $3$.
The case where there are $4$ happens when $n = 3 k$ and $n + 9 = 3 \paren {k + 3}$.
Of these $3$ or $4$, no more than $2$ are odd and so have not been accounted for.
There are exactly $2$ elements of $S$ which have $5$ as a common divisor.
One of those is even and so has been counted already.
There are at most $2$ elements of $S$ which have $7$ as a common divisor.
One of those is even and so has been counted already.
Thus we have a count of how many elements of $S$ which can possibly share a common divisor with another element of $S$:
:$5$ are divisible by $2$
:$2$ or fewer are divisible by $3$ and not $2$
:$1$ at the most is divisible by $5$ and not by $2$ or $3$
:$1$ at the most is divisible by $7$ and not by $2$, $3$ or $5$.
That makes a total of $9$ elements of $S$ which are divisible by $2$, $3$, $5$ or $7$.
Thus there exists (at least) $1$ element of $S$ which is not divisible by $2$, $3$, $5$ or $7$ and so cannot share a common divisor with another element of $S$.
Hence the result.
== Historical Note ==
== Sources ==
* 
* : $10$
* : $10$
Category:Coprime Integers
Category:10
10 Consecutive Integers contain Coprime Integer/Historical Note

== Historical Note on 10 Consecutive Integers contain Coprime Integer ==
Since this result appeared in $1975$, it has reappeared in a number of places, including examinations and quizzes.
In Volume $149$, pages $313$ to $317$, refines this result, by going on to demonstrate that any block of $n$ consecutive integers for $2 \le n \le 16$ contains at least one integer which is coprime to the others.
Category:Historical Notes
10 is Only Triangular Number that is Sum of Consecutive Odd Squares

== Theorem ==
$10$ is the only triangular number which is the sum of two consecutive odd squares:
:$10 = 1^2 + 3^2$
== Proof ==
for $n \in \Z_{\ge 0}$.
The expression for the $n$th odd square number is:
:$4 n^2 + 4 n + 1$
again, for $n \in \Z_{\ge 0}$.
Therefore the closed-form expression for the $n$th sum of two consecutive odd squares is:
:$4 n^2 + 4 n + 1 + 4 \paren {n + 1}^2 + 4 \paren {n + 1} + 1$
This simplifies to:
:$8 n^2 + 16 n + 10$
Equate the two with a variable replacing $n$:
:$8 x^2 + 16 x + 10 = \dfrac {y \paren {y + 1} } 2$
This simplifies to:
:$16 x^2 + 32 x + 20 = y^2 + y$
We then apply Solutions to Diophantine Equation $16 x^2 + 32 x + 20 = y^2 + y$:
Due to the restrictions on the variables, solutions with negative inputs are invalid.
This leaves one solution:
:$\tuple {0, 4}$
as follows:
== Sources ==
* : $10$
* : $10$
Category:10
Category:Odd Squares
Category:Triangular Numbers
11,000,001,446,613,353

== Number ==
$11\,000\,001\,446\,613\,353$ is:
:A prime number
:The lower of a pair of consecutive prime numbers separated by a prime gap of $654$:
::$11\,000\,001\,446\,614\,007 - 11\,000\,001\,446\,613\,353 = 654$
== Also see ==
* Prime Gap of 654
* 
== Sources ==
* : $11,000,001,446,613,353$
Category:Specific Numbers
Category:11,000,001,446,613,353
11,111

== Number ==
$11 \, 111$ (eleven thousand, one hundred and eleven) is:
:$41 \times 271$
:The $5$th repuint after $1$, $11$, $111$, $1111$
:The $19$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$, $111$, $121$, $202$, $212$, $1001$, $1111$, $2002$, $10 \, 001$, $10 \, 101$, $10 \, 201$, $11 \, 011$ whose square is also palindromic integer
::$11 \, 111^2 = 123 \, 454 \, 321$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,111
11,111,111

== Number ==
$11 \, 111 \, 111$ (eleven million, one hundred and eleven thousand, one hundred and eleven) is:
:$11 \times 73 \times 101 \times 137$
:The $8$th repuint after $1$, $11$, $111$, $1111$, $11 \, 111$, $111 \, 111$, $1 \, 111 \, 111$
:The $12$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $77$, $363$, $1111$, $2662$, $111 \, 111$, $246 \, 642$ which is the index of a palindromic triangular number
::$T_{11 \, 111 \, 111} = 61 \, 728 \, 399 \, 382 \, 716$
:A palindromic number whose square is also palindromic:
::$11 \, 111 \, 111^2 = 123 \, 456 \, 787 \, 654 \, 321$
== Also see ==
* Square of Small-Digit Palindromic Number is Palindromic
* 
* 
Category:Specific Numbers
Category:11,111,111
11,111,111,111,111

== Number ==
$11 \, 111 \, 111 \, 111 \, 111$ is:
:$11 \times 239 \times 4649 \times 909 \, 091$
:The $14$th repunit
 
== Also see ==
* 
Category:Specific Numbers
Category:11,111,111,111,111
11,111,111,111,111,111

== Number ==
$11 \, 111 \, 111 \, 111 \, 111 \, 111$ is:
:$2 \, 071 \, 723 \times 5 \, 363 \, 222 \, 357$
:The $17$th repunit
 
== Also see ==
* 
Category:Specific Numbers
Category:11,111,111,111,111,111
11,111,111,111,111,111,111

== Number ==
$11 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$ is:
:$11 \times 41 \times 101 \times 271 \times 3541 \times 9091 \times 27 \, 961$
:The $20$th repunit
== Also see ==
* 
Category:Specific Numbers
Category:11,111,111,111,111,111,111
11,111,111,111,111,111,111,111

== Number ==
$11 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$ is:
:A prime number
:The $3$rd repunit prime
:The $23$rd repunit
:The $24$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $113$, $131$, $199$, $311$, $337$, $373$, $733$, $919$, $991$, $1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$
== Also see ==
* 
* 
* 
== Sources ==
* : $11,111,111,111,111,111,111,111$
* : $11,111,111,111,111,111,111,111$
Category:Specific Numbers
Category:11,111,111,111,111,111,111,111
11,164

== Number ==
$11 \, 164$ (eleven thousand, one hundred and sixty-four) is:
:$2^2 \times 2791$
:The $23$rd integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$, $105$, $160$, $661$, $2653$, $3069$, $3943$, $4053$, $4998$, $8275$, $9158$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$ is prime
== Also see ==
* 
Category:Specific Numbers
Category:11,164
11,213

== Number ==
$11 \, 213$ (eleven thousand, two hundred and thirteen) is:
:The $1357$th prime number
:The index of the $23$rd Mersenne prime after $2$, $3$, $5$, $7$, $13$, $\ldots$, $2203$, $2281$, $3217$, $4253$, $4423$, $9689$, $9941$:
::$M_{11 \, 213} = 2^{11 \, 213} - 1 \approx 2 \cdotp 814 \times 10^{3375}$
== Also see ==
* 
* 
== Historical Note ==
== Sources ==
* : $2^{11213} - 1$
* 
* : $2^{11,213} - 1$
Category:Specific Numbers
Category:11,213
11,213/Historical Note

== Historical Note on $11 \, 213$ ==
The Mersenne number $M_{11 \, 213} = 2^{11 \, 213} - 1$ was demonstrated to be a Mersenne prime by on $2$ June $1963$ at the University of Illinois.
The University celebrated this event by using a special postmark to frank its letters.
== Sources ==
* : $\text {3-5}$ The Use of Computers in Number Theory
* : $2^{11213} - 1$
* : $2^{11,213} - 1$
Category:Historical Notes
11,390,625

== Number ==
$11 \, 390 \, 625$ (eleven million, three hundred and ninety thousand, six hundred and twenty-five) is:
:$3^6 \times 5^6$
:The $6$th power of $15$ after $(1)$, $15$, $225$, $3375$, $50 \, 625$, $759 \, 375$:
::$11 \, 390 \, 625 = 15^6$
:The $15$th sixth power after $1$, $64$, $729$, $4096$, $15 \, 625$, $46 \, 656$, $117 \, 649$, $262 \, 144$, $531 \, 441$, $1 \, 000 \, 000$, $1 \, 771 \, 561$, $2 \, 985 \, 984$, $4 \, 826 \, 809$, $7 \, 529 \, 536$:
::$11 \, 390 \, 625 = 15 \times 15 \times 15 \times 15 \times 15 \times 15$
:The $225$th cube number:
::$11 \, 390 \, 625 = 225 \times 225 \times 225$
:The $3375$th square number:
::$11 \, 390 \, 625 = 3375 \times 3375$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:11,390,625
11,410,337,850,553

== Number ==
$11 \, 410 \, 337 \, 850 \, 553$ is:
:A prime number
:The $1$st of the smallest $22$ primes in arithmetic sequence:
::$11 \, 410 \, 337 \, 850 \, 553 = 11 \, 410 \, 337 \, 850 \, 553 + 0 \times 4 \, 609 \, 098 \, 694 \, 200$
== Also see ==
* 
* 
== Sources ==
* : $11,410,337,850,553$
Category:Specific Numbers
Category:11,410,337,850,553
11,549

== Number ==
$11 \, 549$ (eleven thousand, five hundred and forty-nine) is:
:The $1391$st prime number
:The $14$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$, $1021$, $2657$, $3229$, $4547$, $4787$, $11 \, 549$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,549
11,593

== Number ==
$11 \, 593$ (eleven thousand, five hundred and ninety-three) is:
:The $1395$th prime number
:The $1$st of a sequence of consecutive prime numbers all of the form $4 n + 1$:
::$11 \, 593 = 4 \times 2898 + 1$
== Also see ==
* 
* 
== Sources ==
* : $11,593$
* : $11,593$
Category:Specific Numbers
Category:11,593
11,597

== Number ==
$11 \, 597$ (eleven thousand, five hundred and ninety-seven) is:
:The $1396$th prime number
:The $2$nd after $11 \, 593$ of a sequence of consecutive prime numbers all of the form $4 n + 1$:
::$11 \, 597 = 4 \times 2899 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,597
11,617

== Number ==
$11 \, 617$ (eleven thousand, six hundred and seventeen) is:
:The $1397$th prime number
:The $3$rd after $11 \, 593$, $11 \, 597$ of a sequence of consecutive prime numbers all of the form $4 n + 1$:
::$11 \, 617 = 4 \times 2904 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,617
11,621

== Number ==
$11 \, 621$ (eleven thousand, six hundred and twenty-one) is:
:The $1398$th prime number
:The $4$th after $11 \, 593$, $11 \, 597$, $11 \, 617$ of a sequence of consecutive prime numbers all of the form $4 n + 1$:
::$11 \, 621 = 4 \times 2905 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,621
11,633

== Number ==
$11 \, 633$ (eleven thousand, six hundred and thirty-three) is:
:The $1399$th prime number
:The $5$th after $11 \, 593$, $11 \, 597$, $11 \, 617$, $11 \, 621$ of a sequence of consecutive prime numbers all of the form $4 n + 1$:
::$11 \, 633 = 4 \times 2908 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,633
11,657

== Number ==
$11 \, 657$ (eleven thousand, six hundred and fifty-seven) is:
:The $1400$th prime number
:The $6$th after $11 \, 593$, $11 \, 597$, $11 \, 617$, $11 \, 621$, $11 \, 633$ of a sequence of consecutive prime numbers all of the form $4 n + 1$:
::$11 \, 657 = 4 \times 2914 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,657
11,677

== Number ==
$11 \, 677$ (eleven thousand, six hundred and seventy-seven) is:
:The $1401$st prime number
:The $7$th after $11 \, 593$, $11 \, 597$, $11 \, 617$, $11 \, 621$, $11 \, 633$, $11 \, 657$ of a sequence of consecutive prime numbers all of the form $4 n + 1$:
::$11 \, 677 = 4 \times 2919 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,677
11,681

== Number ==
$11 \, 681$ (eleven thousand, six hundred and eighty-one) is:
:The $1402$nd prime number
:The $8$th after $11 \, 593$, $11 \, 597$, $11 \, 617$, $11 \, 621$, $11 \, 633$, $11 \, 657$, $11 \, 677$ of a sequence of consecutive prime numbers all of the form $4 n + 1$:
::$11 \, 681 = 4 \times 2920 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,681
11,689

== Number ==
$11 \, 689$ (eleven thousand, six hundred and eighty-nine) is:
:The $1403$rd prime number
:The $9$th and last after $11 \, 593$, $11 \, 597$, $11 \, 617$, $11 \, 621$, $11 \, 633$, $11 \, 657$, $11 \, 677$, $11 \, 681$ of a sequence of consecutive prime numbers all of the form $4 n + 1$:
::$11 \, 689 = 4 \times 2922 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,689
11,826

== Number ==
$11 \, 826$ (eleven thousand, eight hundred and twenty-six) is:
:$2 \times 3^4 \times 73$
:The smallest positive integer whose square contains all the digits from $1$ to $9$:
::$11 \, 826^2 = 139 \, 854 \, 276$
== Also see ==
* Smallest Penholodigital Square
* 139,854,276
== Sources ==
* : $11,826$
* : $11,826$
Category:Specific Numbers
Category:11,826
11,881

== Number ==
$11 \, 881$ (eleven thousand, eight hundred and eighty-one) is:
:$109^2$
:The $18$th square after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $121$, $144$, $225$, $441$, $484$, $676$, $1444$, $7744$ which has no more than $2$ distinct digits and does not end in $0$:
::$11 \, 881 = 109^2$
:The $109$th square number:
::$11 \, 881 = 109 \times 109$
== Also see ==
* 
* 
Category:Specific Numbers
Category:11,881
11/Historical Note

== Historical Note on $11$ (Eleven) ==
Some of the archaic imperial units of length appear as divisors and multiples of other such units with a multiplicity of $11$, for example:
:$4$ rods, poles or perches equal $22$ yards equal $1$ chain
:$220 = 11 \times 20$ yards equal $1$ furlong
:$1760 = 11 \times 160$ yards equal $1$ (international) mile.
== Sources ==
* : $11$
* : $11$
Category:Historical Notes
110

== Number ==
$110$ (one hundred and ten) is:
:$2 \times 5 \times 11$
:The $4$th positive integer after $1$, $7$, $102$ the divisor sum is a cube:
::$\map {\sigma_1} {110} = 216 = 6^3$
:It is also the $3$rd such, after $1$, $102$, whose divisor count is also a cube:
::$\map {\sigma_0} {110} = 8 = 2^3$
:The $8$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$:
::$110 = 2 \times 5 \times 11$
:The $21$st positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
== Sources ==
* : $110$
Category:Specific Numbers
Category:110
110,059

== Number ==
$110 \, 059$ (one hundred and ten thousand and fifty-nine) is:
:The $10 \, 458$th prime number
:The $21$st integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$, $154$, $320$, $340$, $399$, $427$, $872$, $1477$, $6380$, $26 \, 951$, $110 \, 059$
== Also see ==
* 
* 
Category:Specific Numbers
Category:110,059
110,503

== Number ==
$110 \, 503$ (one hundred and ten thousand, five hundred and three) is:
:The $10 \, 489$th prime number
:The index of the $29$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $\ldots$, $11 \, 213$, $19 \, 937$, $21 \, 701$, $23 \, 209$, $44 \, 497$, $86 \, 243$:
::$M_{110 \, 503} = 2^{110 \, 503} - 1 \approx 5 \cdotp 219 \times 10^{33 \, 264}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:110,503
1103

== Number ==
$1103$ (one thousand, one hundred and three) is:
:The $185$th prime number
:The $16$th balanced prime after $5$, $53$, $157$, $173$, $211$, $257$, $263$, $373$, $563$, $593$, $607$, $653$, $733$, $947$, $977$:
::$1103 = \dfrac {1097 + 1109} 2$
:The $42$nd Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $761$, $809$, $911$, $953$, $1013$, $1019$, $1031$, $1049$:
::$2 \times 1103 + 1 = 2207$, which is prime.
:The $42$nd emirp after $13$, $17$, $31$, $37$, $\ldots$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$, $1031$, $1033$, $1061$, $1069$, $1091$, $1097$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:1103
1104

== Number ==
$1104$ (one thousand, one hundred and four) is:
:$2^4 \times 3 \times 23$
:The $9$th Keith number after $14$, $19$, $28$, $47$, $61$, $75$, $197$, $742$:
::$1$, $1$, $0$, $4$, $6$, $11$, $21$, $42$, $80$, $154$, $297$, $573$, $1104$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:1104
1105

== Number ==
$1105$ (one thousand, one hundred and five) is:
:$5 \times 13 \times 17$
:The $2$nd Carmichael number after $561$:
::$\forall a \in \Z: a \perp 1105: a^{1104} \equiv 1 \pmod {1105}$
:The $4$th Poulet number after $341$, $561$, $645$:
::$2^{1105} \equiv 2 \pmod {1105}$: $1105 = 5 \times 13 \times 17$
:The $7$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$:
::$3^{1105} \equiv 3 \pmod {1105}$
:The $10$th Fermat pseudoprime to base $4$ after $15$, $85$, $91$, $341$, $435$, $451$, $561$, $645$, $703$:
::$4^{1105} \equiv 4 \pmod {1105}$
:The magic constant of a magic square of order $13$, after $1$, $(5)$, $15$, $34$, $65$, $111$, $175$, $260$, $369$, $505$, $671$, $870$:
::$1105 = \ds \dfrac 1 {13} \sum_{k \mathop = 1}^{13^2} k = \dfrac {13 \paren {13^2 + 1} } 2$
:The product of the first $3$ primes of the form $4 n + 1$:
::$1105 = \paren {4 + 1} \paren {12 + 1} \paren {16 + 1}$
:Can be expressed as the sum of two squares in more ways than any smaller integer:
::$1105 = 33^2 + 4^2 = 32^2 + 9^2 = 31^2 + 12^2 = 24^2 + 23^2$
== Also see ==
* 1105 as Sum of Two Squares
* 
* 
* 
* 
* 
== Sources ==
* : $1105$
* : $1105$
Category:Specific Numbers
Category:1105
1105 as Sum of Two Squares

== Theorem ==
$1105$ can be expressed as the sum of two squares in more ways than any smaller integer:
== Proof ==
Here is the source code of a program in Python that finds all positive integers up to $1105$ that can be written as a sum of two squares in more ways than any smaller positive integer:
 import numpy as np
 
 def two_sq_decomp_rich(n):
 
 bound = int(np.floor(np.sqrt(n)))
 count_of_two_sq_decomps = []
 for i in range(2*(bound + 1)*(bound + 1)):
 count_of_two_sq_decomps.append(0)
 for i in range(bound+1): 
 for j in range(i+1):
 count_of_two_sq_decomps[i*i+j*j] = count_of_two_sq_decomps[i*i+j*j] + 1
 
 max_sq_decomps = 0
 sq_decomp_rich_numbers = []
 for i in range(n+1):
 if count_of_two_sq_decomps[i] > max_sq_decomps:
 max_sq_decomps = count_of_two_sq_decomps[i]
 sq_decomp_rich_numbers.append(i)
 
 return sq_decomp_rich_numbers
 
 print(two_sq_decomp_rich(1105))
Output:
 [0, 25, 325, 1105]
Try it online!
== Sources ==
* : $1105$
* 
* : $1105$
Category:Sums of Squares
Category:1105
Category:1105 as Sum of Two Squares
1107

== Number ==
$1107$ (one thousand, one hundred and seven) is:
:$3^3 \times 41$
:The $27$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $610$, $672$, $737$, $805$, $876$, $950$, $1027$:
::$1107 = \dfrac {27 \paren {3 \times 27 + 1} } 2$
:The $54$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $610$, $651$, $672$, $715$, $737$, $782$, $805$, $852$, $876$, $925$, $950$, $1001$, $1027$, $1080$:
::$1107 = \dfrac {27 \paren {3 \times 27 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1107
1109

== Number ==
$1109$ (one thousand, one hundred and nine) is:
:The $186$th prime number
:The smaller of the $16$th pair of primes whose prime gap is $8$:
::$1117 - 1109 = 8$
:The $43$rd emirp after $13$, $17$, $31$, $37$, $\ldots$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$, $1031$, $1033$, $1061$, $1069$, $1091$, $1097$, $1103$
== Also see ==
* 
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Category:Specific Numbers
Category:1109
111

== Number ==
$111$ (one hundred and eleven) is:
:$3 \times 37$
:The magic constant of the smallest prime magic square.
:The $3$rd repuint after $1$, $11$
:The $4$th 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 $6$th positive integer after $1$, $2$, $7$, $11$, $101$ whose cube is palindromic:
::$111^3 = 1 \, 367 \, 631$
:The magic constant of a magic square of order $6$, after $1$, $(5)$, $15$, $34$, $65$:
::$111 = \ds \dfrac 1 6 \sum_{k \mathop = 1}^{6^2} k = \dfrac {6 \paren {6^2 + 1} } 2$
:The $7$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $99$, $111$, $\ldots$
:The $8$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$ whose square is also palindromic integer
::$111^2 = 12 \, 321$
:The $15$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$:
::$111 = 111 \times 1 = 111 \times \paren {1 \times 1 \times 1}$
:The $24$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $75$, $79$, $87$, $93$, $99$, $105$, $111$, $\ldots$
:The $35$th semiprime:
::$111 = 3 \times 37$
== Also see ==
* Magic Constant of Smallest Prime Magic Square
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== Sources ==
* : $111$
* : $111$
Category:Specific Numbers
Category:111
111,111

== Number ==
$111 \, 111$ (one hundred and eleven thousand, one hundred and eleven) is:
:$3 \times 7 \times 11 \times 13 \times 37$
:The $6$th repuint after $1$, $11$, $111$, $1111$, $11 \, 111$
:The $10$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $77$, $363$, $1111$, $2662$ which is the index of a palindromic triangular number
::$T_{111 \, 111} = 6 \, 172 \, 882 \, 716$
:The $26$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$, $\ldots$, $10 \, 201$, $11 \, 011$, $11 \, 111$, $11 \, 211$, $20 \, 002$, $20 \, 102$, $100 \, 001$, $101 \, 101$, $110 \, 011$ whose square is also palindromic integer
::$111 \, 111^2 = 12 \, 345 \, 654 \, 321$
== Also see ==
* 
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* 
Category:Specific Numbers
Category:111,111
111,111,111

== Number ==
$111 \, 111 \, 111$ (one hundred and eleven million, one hundred and eleven thousand, one hundred and eleven) is:
:$3^2 \times 37 \times 333 \, 667$
:The $9$th repuint after $1$, $11$, $111$, $1111$, $11 \, 111$, $111 \, 111$, $1 \, 111 \, 111$, $11 \, 111 \, 111$
:A palindromic number whose square is also palindromic:
::$111 \, 111 \, 111^2 = 12 \, 345 \, 678 \, 987 \, 654 \, 321$
== Also see ==
* Square of Small-Digit Palindromic Number is Palindromic
* 
Category:Specific Numbers
Category:111,111,111