nyuuzyou/wikis huggingface dataset:
117,649

== Number ==
$117 \, 649$ (one hundred and seventeen thousand, six hundred and forty-nine) is:
:$7^6$
:The $6$th power of $7$ after $(1)$, $7$, $49$, $343$, $2401$, $16 \, 807$:
::$117 \, 649 = 7^6$
:The $7$th $6$th power after $1$, $64$, $729$, $4096$, $15 \, 625$, $46 \, 656$:
::$117 \, 649 = 7 \times 7 \times 7 \times 7 \times 7 \times 7$
:The $49$th cube number:
::$117 \, 649 = 49 \times 49 \times 49$
:The $343$rd square number:
::$117 \, 649 = 343 \times 343$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:117,649
1171

== Number ==
$1171$ (one thousand one hundred and seventy-one) is:
:The $193$nd prime number
:The larger of the $17$th pair of primes whose prime gap is $8$:
::$1171 - 1163 = 8$
:The smaller of the $21$st pair of primes whose prime gap is $10$:
::$1181 - 1171 = 10$
:The $49$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $877$, $881$, $883$, $887$, $911$, $919$, $929$, $977$, $991$, $997$, $1117$, $1151$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:1171
1176

== Number ==
$1176$ (one thousand, one hundred and seventy-six) is:
:$2^3 \times 3 \times 7^2$
:The $48$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$, $780$, $820$, $861$, $903$, $946$, $990$, $1035$, $1081$, $1128$:
::$1176 = \ds \sum_{k \mathop = 1}^{48} k = \dfrac {48 \times \paren {48 + 1} } 2$
:The $41$st Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $1111$, $1112$, $1113$, $1115$, $1116$, $1131$:
::$1176 = 28 \times 42 = 28 \times \paren {1 \times 1 \times 7 \times 6}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1176
1177

== Number ==
$1177$ (one thousand, one hundred and seventy-seven) is:
:$11 \times 107$
:The $22$nd heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$, $235$, $286$, $342$, $403$, $469$, $540$, $616$, $697$, $783$, $874$, $970$, $1071$:
::$\ds 1177 = \sum_{k \mathop = 1}^{22} \paren {5 k - 4} = \dfrac {22 \paren {5 \times 22 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:1177
118

== Number ==
$118$ (one hundred and eighteen) is:
:$2 \times 59$
:The $37$th semiprime:
::$118 = 2 \times 59$
:The $15$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$, $68$, $74$, $76$, $86$, $90$, $94$, $98$, $114$:
::$\nexists m \in \Z_{>0}: \map \phi m = 118$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $4$th term of the $2$nd $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
::$\tuple {115, 116, 117, 118, 119}$
:The smallest positive integer which is the sum of $4$ distinct ordered triples, each of which has the same product:
::$118 = 14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 + 72$
:each of which has a product of $37 \, 800$
== Also see ==
* Smallest Number which is Sum of 4 Triples with Equal Products
* 
* 
* 
== Sources ==
* : $118$
* : $118$
Category:Specific Numbers
Category:118
118,132,654,413,675,138,222

== Number ==
$118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222$ is:
:$2 \times 3 \times 41 \times 480 \, 214 \, 042 \, 332 \, 012 \, 757$
:The $3$rd of $5$ integers which have the property that the sum of any $3$ of them is square:
== Also see ==
* 
Category:Specific Numbers
Category:118,132,654,413,675,138,222
1181

== Number ==
$1181$ (one thousand one hundred and eighty-one) is:
:The $194$th prime number
:The larger of the $22$nd pair of primes whose prime gap is $10$:
::$1181 - 1171 = 10$
:The $25$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 24^2 + 29 = 1181$ ()
:The $46$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$, $1031$, $1033$, $1061$, $1069$, $1091$, $1097$, $1103$, $1109$, $1151$, $1153$
:The $50$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $877$, $881$, $883$, $887$, $911$, $919$, $929$, $977$, $991$, $997$, $1117$, $1171$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:1181
1183

== Number ==
$1183$ (one thousand, one hundred and eighty-three) is:
:$7 \times 13^2$
:The $13$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $126$, $196$, $288$, $405$, $550$, $726$, $936$:
::$1183 = \ds \sum_{k \mathop = 1}^{13} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {13^2 \paren {13 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:1183
1184

== Number ==
$1184$ (one thousand, one hundred and eighty-four) is:
:$2^5 \times 37$
:The smaller of the $2$nd amicable pair, with $1210$:
::$\map {\sigma_1} {1184} = \map {\sigma_1} {1210} = 2394 = 1184 + 1210$
:The $15$th primitive abundant number after $20$, $70$, $88$, $104$, $272$, $304$, $368$, $464$, $550$, $572$, $650$, $748$, $836$, $945$:
::$1 + 2 + 4 + 8 + 16 + 32 + 37 + 74 + 148 + 296 + 592 = 1210 > 1184$
:The $20$th primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$, $304$, $350$, $368$, $464$, $490$, $496$, $550$, $572$, $650$, $748$, $770$, $910$, $945$:
::$1184 = 1 + 4 + 32 + 37 + 74 + 148 + 296 + 592$
:The $42$nd Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $1111$, $1112$, $1113$, $1115$, $1116$, $1131$, $1176$:
::$1184 = 37 \times 32 = 37 \times \paren {1 \times 1 \times 8 \times 4}$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $1184$
* : $1184$
Category:Specific Numbers
Category:1184
1187

== Number ==
$1187$ (one thousand, one hundred and eighty-seven) is:
:The $195$th prime number
:The $7$th Stern number after $1$, $3$, $17$, $137$, $227$, $977$
:The $7$th Stern prime after $1$, $3$, $17$, $137$, $227$, $977$
:The $18$th balanced prime after $5$, $53$, $157$, $173$, $211$, $257$, $263$, $373$, $563$, $593$, $607$, $653$, $733$, $947$, $977$, $1103$, $1123$:
::$1187 = \dfrac {1181 + 1193} 2$
:The $27$th safe prime after $5$, $7$, $11$, $23$, $47$, $\ldots$, $479$, $503$, $563$, $587$, $719$, $839$, $863$, $887$, $983$, $1019$:
::$1187 = 2 \times 553 + 1$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:1187
1189

== Number ==
$1189$ (one thousand, one hundred and eighty-nine) is:
:$29 \times 41$
:The $3$rd positive integer after $1$, $35$ whose square can be expressed as the sum of a sequence of odd cubes from $1$:
::$1189^2 = 1 \, 413 \, 721 = 1^3 + 3^3 + 5^3 + \dotsb + 55^3 + 57^3$
== Also see ==
* 
Category:Specific Numbers
Category:1189
119

== Number ==
$119$ (one hundred and nineteen) is:
:$7 \times 17$
:The $5$th term of the $2$nd $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
::$\tuple {115, 116, 117, 118, 119}$
:The $9$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$, $115$ whose divisor sum is square:
:
:The $38$th semiprime:
::$119 = 7 \times 17$
:The $55$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $77$, $78$, $79$, $84$, $90$, $91$, $95$, $96$, $102$, $108$, $114$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:119
1190

== Number ==
$1190$ (one thousand, one hundred and ninety) is:
:$2 \times 5 \times 7 \times 17$
:The $21$st primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$, $304$, $350$, $368$, $464$, $490$, $496$, $550$, $572$, $650$, $748$, $770$, $910$, $945$, $1184$:
::$1190 = 1 + 2 + 5 + 10 + 14 + 17 + 34 + 70 + 85 + 119 + 238 + 595$
:The $28$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $610$, $672$, $737$, $805$, $876$, $950$, $1027$, $1107$:
::$1190 = \dfrac {28 \paren {3 \times 28 + 1} } 2$
:The $56$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $610$, $651$, $672$, $715$, $737$, $782$, $805$, $852$, $876$, $925$, $950$, $1001$, $1027$, $1080$, $1107$, $1162$:
::$1190 = \dfrac {28 \paren {3 \times 28 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1190
1193

== Number ==
$1193$ (one thousand, one hundred and ninety-three) is:
:The $196$th prime number
:The smaller of the $18$th pair of primes whose prime gap is $8$:
::$1201 - 1193 = 8$
:The $47$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $1061$, $1069$, $1091$, $1097$, $1103$, $1109$, $1151$, $1153$, $1181$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1193
1197

== Number ==
$1197$ (one thousand, one hundred and ninety-seven) is:
:$3^2 \times 7 \times 19$
:The $43$rd Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $1111$, $1112$, $1113$, $1115$, $1116$, $1131$, $1176$, $1184$:
::$1197 = 19 \times 63 = 19 \times \left({1 \times 1 \times 9 \times 7}\right)$
== Also see ==
* 
Category:Specific Numbers
Category:1197
11 is Only Palindromic Prime with Even Number of Digits

== Theorem ==
$11$ is the only palindromic prime with an even number of digits when expressed in decimal notation.
== Proof ==
Let $P$ be a palindromic number with $2 n$ digits:
:$P = \sqbrk {a_{2 n - 1} a_{2 n - 2} \ldots a_2 a_1 a_0}_{10}$
Thus:
:$P = \ds \sum_{j \mathop = 0}^{n - 1} a_j + 10^{2 n - 1 - j}$
Consider the summation:
:$S = \ds \sum_{k \mathop = 0}^{2 n - 1} \paren {-1}^k a_k$
As $a_k = a_{2 n - 1 - k}$ we have:
As $2 n - 1$ is odd, it follows that $k$ and $2 n - 1 - k$ are of opposite parity.
Thus:
:$\paren {-1}^k = -\paren {-1}^{2 n - 1 - k}$
and it follows that:
:$S = 0$
It follows by Divisibility by 11 that $P$ is divisible by $11$.
Thus, except for $11$ itself, a palindromic number with an even number of digits cannot be prime.
The result follows.
== Sources ==
* : $11$
* : $11$
Category:Palindromic Primes
Category:11
12,157,692,622,039,623,539

== Number ==
$12 \, 157 \, 692 \, 622 \, 039 \, 623 \, 539$ is:
:$179 \times 4547 \times 14 \, 937 \, 336 \, 818 \, 603$
:The $20$th and last number after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $89$, $135$, $175$, $518$, $598$, $1306$, $1676$, $2427$, $2 \, 646 \, 798$ which is the sum of the increasing powers of its digits taken in order:
== Also see ==
* 
Category:Specific Numbers
Category:12,157,692,622,039,623,539
12,285

== Number ==
$12 \, 285$ (twelve thousand, two hundred and eighty-five) is:
:$3^3 \times 5 \times 7 \times 13$
:The smaller of the $7$th amicable pair and the $1$st odd amicable pair, with $14 \, 595$:
::$\map {\sigma_1} {12 \, 285} = \map {\sigma_1} {14 \, 595} = 26 \, 880 = 12 \, 285 + 14 \, 595$
== Also see ==
* 
* 
== Sources ==
* : $12,285$
* : $12,285$
Category:Specific Numbers
Category:12,285
12,321

== Number ==
$12 \, 321$ (twelve thousand, three hundred and twenty-one) is:
:$3^2 \times 37^2$
:The $3$rd Wonderful Demlo number after $1$, $121$
:The $111$th square number:
::$12 \, 321 = 111 \times 111$
:A palindromic square of a palindromic number
== Also see ==
* Square of Small-Digit Palindromic Number is Palindromic/Examples/111
* 
* 
== Sources ==
* : $12,321$
Category:Specific Numbers
Category:12,321
12,345,678,901,234,567,891

== Number ==
$12 \, 345 \, 678 \, 901 \, 234 \, 567 \, 891$ is:
:The $2$nd of the only $3$ known prime numbers consisting of strings of consecutive ascending digits starting from $1$ (allowing either $0$ or $1$ to follow $9$)
== Also see ==
* 
* 
== Sources ==
* : $1,234,567,891$
* : $1,234,567,891$
Category:Specific Numbers
Category:12,345,678,901,234,567,891
12,345,679

== Number ==
$12 \, 345 \, 679$ (twelve million, three hundred and forty-five thousand, six hundred and seventy-nine) is:
:$37 \times 333 \, 667$
== Also see ==
* Properties of 12,345,679
* Properties of Family of 333,667 and Related Numbers
== Sources ==
* : $12,345,679$
* : $12,345,679$
Category:Specific Numbers
Category:12,345,679
12,367

== Number ==
$12 \, 367$ (twelve thousand, three hundred and sixty-seven) is:
:$83 \times 149$
:The index of the $1$st harmonic number to be greater than $10$:
::$H_{12 \, 367} = \ds \sum_{k \mathop = 1}^{12 \, 367} \frac 1 k \approx 10 \cdotp 00004 \, 30083$
== Also see ==
* First Harmonic Number to exceed 10
== Sources ==
* : $272,400,600$
Category:Specific Numbers
Category:12,367
12,496

== Number ==
$12 \, 496$ (twelve thousand, four hundred and ninety-six) is:
:$2^4 \times 11 \times 71$
:The $1$st term of a sociable chain of order $5$ whose smallest term is $12 \, 496$:
::$12 \, 496 \to 14 \, 288 \to 15 \, 472 \to 14 \, 536 \to 14 \, 264 \to 12 \, 496$
== Also see ==
* 
== Sources ==
* : $12,496$
* : $12,496$
Category:Specific Numbers
Category:12,496
12,758

== Number ==
$12 \, 758$ (twelve thousand, seven hundred and fifty-eight) is:
:$2 \times 6379$
:The largest positive integer that cannot be expressed as the sum of distinct cubes
== Also see ==
* Largest Positive Integer not Sum of Distinct Cubes
== Sources ==
* : $12,758$
* : $12,758$
Category:Specific Numbers
Category:12,758
12,890,625

== Number ==
$12 \, 890 \, 625$ (twelve million, eight hundred and ninety thousand, six hundred and twenty-four) is:
:$3 \times 5^8 \times 11$
:The $14$th automorphic number after $1$, $5$, $6$, $25$, $76$, $376$, $625$, $9376$, $90 \, 625$, $109 \, 376$, $890 \, 625$, $2 \, 890 \, 625$, $7 \, 109 \, 376$:
::$12 \, 890 \, 625^2 = 166 \, 168 \, 2\mathbf {12 \, 890 \, 625}$
== Also see ==
* 
Category:Specific Numbers
Category:12,890,625
12,960,000

== Number ==
$12 \, 960 \, 000$ is:
:$2^8 \times 3^4 \times 5^4$
:The $60$th fourth power:
::$12 \, 960 \, 000 = 60 \times 60 \times 60 \times 60$
:The $3600$th square number:
::$12 \, 960 \, 000 = 3600 \times 3600$
== Also see ==
* 
* 
== Historical Note ==
Category:Specific Numbers
Category:12,960,000
12,960,000/Historical Note

== Historical Note on $12 \, 960 \, 000$ ==
$12 \, 960 \, 000$ is one of the two numbers (the other being $216$) generally believed to be 's geometrical number:
:... which has control over the good and evil of births.
::::-- 's : $546$
It can be arrived at either by noting that it is $60^4$ or that it is $4800 \times 2700$
A great year of Plato is a time period of $36 \, 000$ years.
If one year is specified as being $360$ days, as it was by , then a great year is $12 \, 960 \, 000$ days.
Note, however, that there is no indication that ever mentioned it.
== Sources ==
* : $12,960,000$
* : Chapter $\text {A}.2$: Pythagoras (ca. B.C.): Footnote $5$
* : $12,960,000$
Category:Historical Notes
12,988,816

== Number ==
$12 \, 988 \, 816$ is:
:$2^4 \times 17^2 \times 53^2$
:The $3604$th square number:
::$12 \, 988 \, 816 = 3604 \times 3604$
:The number of ways to tile a standard chessboard with dominoes
== Also see ==
* Number of Ways to Tile Standard Chessboard with Dominoes
* 
== Sources ==
* : $12,988,816$
Category:12,988,816
Category:Specific Numbers
Category:Chessboard Puzzles
Category:Dominoes
12/Historical Note

== Historical Note on $12$ (Twelve) ==
There are many occurrences of the number $12$ in various cultures in history:
:There are $12$ months in a year, divided into $4$ seasons of $3$ months each.
:There are $12$ signs of the Zodiac, divided into $3$ sets of $4$ each, and categorising them another way, into $4$ sets of $3$ each.
:There are $12$ hours in each day and in each night.
:There are $12$ semitones in an octave of a well-tempered scale.
:There are $12$ inches in a foot.
:There were $12$ pence in $1$ shilling in pre-decimal British coinage.
The word dozen, now falling into disuse, means a set of $12$.
== Also see ==
* Definition:Duodecimal System
== Sources ==
* : $12$
* : $12$
Category:Historical Notes
120

== Number ==
$120$ (one hundred and twenty) is:
:$2^3 \times 3 \times 5$
:The $1$st triperfect number:
::$\map {\sigma_1} {120} = 360 = 3 \times 120$
:The $2$nd triangular number after $6$ which can be expressed as the product of $3$ consecutive integers:
::$120 = T_{15} = 4 \times 5 \times 6$
:The $2$nd after $24$ of the $3$ integers which can be expressed as the product of both $3$ and $4$ consecutive integers:
::$120 = 4 \times 5 \times 6 = 2 \times 3 \times 4 \times 5$
:The $3$rd after $1$, $10$ of the $5$ tetrahedral numbers which are also triangular.
:The $4$th and final element of the Fermat set after $1$, $3$, $8$
:The $5$th factorial after $1$, $2$, $6$, $24$:
::$120 = 5! = 5 \times 4 \times 3 \times 2 \times 1$
:The $6$th untouchable number after $2$, $5$, $52$, $88$, $96$.
:The smallest positive integer greater than $1$ to appear $6$ times in Pascal's Triangle.
:The $6$th and last after $0$, $2$, $3$, $27$, $98$ of the integers which are the middle term of a sequence of $5$ consecutive integers whose cubes add up to a square
::$118^3 + 119^3 + 120^3 + 121^3 + 122^3 = 8 \, 643 \, 600 = 2940^2$
:The $8$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$:
::$120 = 1 + 5 + 9 + 13 + 17 + 21 + 25 + 29 = 8 \paren {2 \times 8 - 1}$
:The $8$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$:
::$120 = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 = \dfrac {8 \paren {8 + 1} \paren {8 + 2} } 6$
:The $10$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$:
::$\map {\sigma_0} {120} = 16$
:The $10$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$:
::$\dfrac {\map {\sigma_1} {120} } {120} = \dfrac {360} {120} = 3$
:The smallest positive integer which can be expressed as the sum of $2$ odd primes in $12$ ways.
:The $15$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$, $78$, $91$, $105$:
::$120 = \ds \sum_{k \mathop = 1}^{15} k = \dfrac {15 \times \paren {15 + 1} } 2$
:There are $17$ positive integers which have an Euler $\phi$ value $120$.
:The $23$rd highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$:
::$\map {\sigma_1} {120} = 360$
:The $56$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $95$, $96$, $102$, $108$, $114$, $119$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* Numbers with Euler Phi Value of 120
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $120$
* : $120$
Category:Specific Numbers
Category:120
1201

== Number ==
$1201$ (one thousand, two hundred and one) is:
:The $197$th prime number
:The $3$rd prime number after $1009$, $1129$ which can be expressed in the form $x^2 + n y^2$ for all values of $n$ from $1$ to $10$:
:The larger of the $18$th pair of primes whose prime gap is $8$:
::$1201 - 1193 = 8$
:The $48$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $1061$, $1069$, $1091$, $1097$, $1103$, $1109$, $1151$, $1153$, $1181$, $1201$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $1201$
Category:Specific Numbers
Category:1201
1209

== Number ==
$1209$ (one thousand, two hundred and nine) is:
:$3 \times 13 \times 31$
:The $6$th positive integer after $504$, $756$, $806$, $1008$, $1148$ which can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other:
::$1209 = 13 \times 93 = 39 \times 31$
== Also see ==
* 
Category:Specific Numbers
Category:1209
121

== Number ==
$121$ (one hundred and twenty-one) is:
:$11^2$
:The $1$st of the two square numbers which is the sum of consecutive powers of a positive integer:
::$121 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4$
:The $2$nd power of $11$ after $(1)$, $11$:
::$121 = 11^2$
:The $2$nd square number after $25$ of the form $n! + 1$:
::$121 = 5! + 1 = 11^2$
:where $!$ denotes the factorial function.
:The $2$nd Fermat pseudoprime to base $3$ after $91$:
::$3^{121} \equiv 3 \pmod {121}$
:The $2$nd (with $4$) of the $2$ square numbers which are $4$ less than a cube:
::$121 = 11^2 = 5^3 - 4$
:The $2$nd Wonderful Demlo number after $1$
:The $2$nd Friedman number base $10$ after $25$:
::$121 = 11^2$
:The $4$th square number after $1$, $4$, $36$ to be the divisor sum value of some (strictly) positive integer:
::$121 = \map {\sigma_1} {81}$
:The $6$th integer after $0$, $1$, $2$, $4$, $8$ which is palindromic in both decimal and ternary:
::$121_{10} = 11 \, 111_3$
:The $7$th Smith number after $4$, $22$, $27$, $58$, $85$, $94$:
::$1 + 2 + 1 = 1 + 1 + 1 + 1 = 4$
:The $9$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $\ldots$
:The $9$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$, $111$ whose square is also palindromic integer
::$121^2 = 14 \, 641$
:The $10$th square after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$ which has no more than $2$ distinct digits and does not end in $0$:
::$121 = 11^2$
:The $11$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$:
::$121 = 11 \times 11$
:The $16$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$, $100$, $108$
:The $39$th semiprime:
::$121 = 11 \times 11$
== Also see ==
* Brocard's Problem
* Squares which are 4 Less than Cubes
* Square Numbers which are Sum of Consecutive Powers
* Integer Greater than 121 is Sum of Distinct Primes of form 4 n + 1
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== Sources ==
* : $5$
* : $121$
* : $5$
* : $121$
Category:Specific Numbers
Category:121
121,174,811

== Number ==
$121 \, 174 \, 811$ is:
:The $6 \, 904 \, 737$th prime number
:The $1$st of the longest known sequence of consecutive prime numbers in arithmetic sequence:
::$121 \, 174 \, 811 = 0 \times 30 + 121 \, 174 \, 811$
== Also see ==
* 
* 
== Sources ==
* : $121,174,811$
Category:Specific Numbers
Category:121,174,811
121,174,841

== Number ==
$121 \, 174 \, 841$ is:
:The $6 \, 904 \, 738$th prime number
:The $2$nd of the longest known sequence of consecutive prime numbers in arithmetic sequence:
::$121 \, 174 \, 841 = 1 \times 30 + 121 \, 174 \, 811$
== Also see ==
* 
* 
Category:Specific Numbers
Category:121,174,841
121,174,871

== Number ==
$121 \, 174 \, 871$ is:
:The $6 \, 904 \, 739$th prime number
:The $3$rd of the longest known sequence of consecutive prime numbers in arithmetic sequence:
::$121 \, 174 \, 871 = 2 \times 30 + 121 \, 174 \, 811$
== Also see ==
* 
* 
Category:Specific Numbers
Category:121,174,871
121,174,901

== Number ==
$121 \, 174 \, 901$ is:
:The $6 \, 904 \, 740$th prime number
:The $4$th of the longest known sequence of consecutive prime numbers in arithmetic sequence:
::$121 \, 174 \, 901 = 3 \times 30 + 121 \, 174 \, 811$
== Also see ==
* 
* 
Category:Specific Numbers
Category:121,174,901
121,174,931

== Number ==
$121 \, 174 \, 931$ is:
:The $6 \, 904 \, 741$st prime number
:The $5$th of the longest known sequence of consecutive prime numbers in arithmetic sequence:
::$121 \, 174 \, 931 = 4 \times 30 + 121 \, 174 \, 811$
== Also see ==
* 
* 
Category:Specific Numbers
Category:121,174,931
121,174,961

== Number ==
$121 \, 174 \, 961$ is:
:The $6 \, 904 \, 742$nd prime number
:The $6$th and last of the longest known sequence of consecutive prime numbers in arithmetic sequence:
::$121 \, 174 \, 961 = 5 \times 30 + 121 \, 174 \, 811$
== Also see ==
* 
* 
Category:Specific Numbers
Category:121,174,961
1210

== Number ==
$1210$ (one thousand, two hundred and ten) is:
:$2 \times 5 \times 11^2$
:The larger of the $2$nd amicable pair, with $1184$:
::$\map {\sigma_1} {1184} = \map {\sigma_1} {1210} = 2394 = 1184 + 1210$
== Also see ==
* 
== Sources ==
* : $1210$
* : $1210$
Category:Specific Numbers
Category:1210
1212

== Number ==
$1212$ (one thousand, two hundred and twelve) is:
:$2^2 \times 3 \times 101$
:The $44$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $1111$, $1112$, $1113$, $1115$, $1116$, $1131$, $1176$, $1184$, $1197$:
::$1212 = 303 \times 4 = 303 \times \paren {1 \times 2 \times 1 \times 2}$
== Also see ==
* 
Category:Specific Numbers
Category:1212
121 is Square Number in All Bases greater than 2

== Theorem ==
Let $b \in \Z$ be an integer such that $b \ge 3$.
Let $n$ be a positive integer which can be expressed in base $b$ as $121_b$.
Then $n$ is a square number.
== Proof ==
Consider $11_b$.
By the Basis Representation Theorem:
:$11_b = b + 1$
Thus:
Thus:
:$121_b = {11_b}^2$
and so is a square number.
== Sources ==
* : $121$
* : $121$
Category:Square Numbers
Category:Number Bases