nyuuzyou/wikis huggingface dataset:
1681

== Number ==
$1681$ (one thousand six hundred and eighty-one) is:
:$41^2$
:The $6$th square after $49$, $169$, $361$, $1225$, $1444$ whose decimal representation can be split into two parts which are each themselves square:
::$1681 = 41^2$; $16 = 4^2$, $81 = 9^2$
:and the only such number of $4$ digits whose two parts are both of $2$ digits
:The $36$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $706$, $730$, $771$, $784$, $841$, $1024$, $1089$, $1225$, $1255$, $1351$, $1414$, $1444$, $1521$, $1681$, $\ldots$
:The $41$st square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1024$, $1089$, $1156$, $1225$, $1296$, $1369$, $1444$, $1521$, $1600$:
::$1681 = 41 \times 41$
== Also see ==
* 
* 
* 
== Sources ==
* : $1681$
* : $1681$
Category:Specific Numbers
Category:1681
169

== Number ==
$169$ (one hundred and sixty-nine) is:
:$13^2$
:Can be expressed as the sum of $n$ non-zero squares for all $n$ from $1$ to $155$.
:$1! + 6! + 9! = 363 \, 601$, $3! + 6! + 3! + 6! + 0! + 1! = 1454$, $1! + 4! + 5! + 4! = 169$
:The $1$st square number which is the difference between two cubes:
::$169 = 8^3 - 7^3$
:The $2$nd power of $13$ after $(1)$, $13$:
::$169 = 13^2$
:The $2$nd centered hexagonal number after $1$ which is also square:
::$169 = 8^3 - 7^3 = 13^2$
:The $2$nd square after $49$ whose decimal representation can be split into two parts which are each themselves square:
::$169 = 13^2$; $16 = 4^2$, $9 = 3^2$
:The $5$th square lucky number:
::$1$, $9$, $25$, $49$, $169$, $\ldots$
:The $8$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$:
::$169 = 1 + 6 + 12 + 18 + 24 + 30 + 36 + 42 = 8^3 - 7^3$
:The $11$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $\ldots$
:The $13$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$, $121$, $144$:
::$169 = 13 \times 13$
:The $34$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $115$, $127$, $129$, $133$, $135$, $141$, $151$, $159$, $163$, $169$, $\ldots$
:The $21$st powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$, $100$, $108$, $121$, $125$, $128$, $144$
== Also see ==
* 169 as Sum of up to 155 Squares
* Sum of Factorials of Digits of 169
* Squares which are Difference between Two Cubes
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $169$
* : $169$
Category:Specific Numbers
Category:169
1695

== Number ==
$1695$ (one thousand, six hundred and ninety-five) is:
:$3 \times 5 \times 113$
:The magic constant of a magic square of order $15$, after $1$, $(5)$, $15$, $34$, $65$, $111$, $175$, $260$, $369$, $505$, $671$, $870$, $1105$, $1379$:
::$\ds 1695 = \dfrac 1 {15} \sum_{k \mathop = 1}^{15^2} k = \dfrac {15 \paren {15^2 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:1695
169 as Sum of up to 155 Squares

== Theorem ==
$169$ can be expressed as the sum of $n$ non-zero squares for all $n$ from $1$ to $155$.
== Proof ==
We note the following:
Let $41 \le n \le 155$.
Let $n \equiv 0 \pmod 3$.
Then $169$ can be represented as:
:$169 = 2 \times 3^2 + \left({37 - \dfrac {n - 42} 3}\right) \times 2^2 + \left({\dfrac {4 \left({n - 42}\right)} 3 + 3}\right) \times 1^2$
Let $n \equiv 1 \pmod 3$.
Then $169$ can be represented as:
:$169 = \left({42 - \dfrac {n - 43} 3}\right) \times 2^2 + \left({\dfrac {4 \left({n - 43}\right)} 3 + 1}\right) \times 1^2$
Let $n \equiv 2 \pmod 3$.
Then $169$ can be represented as:
:$169 = 3^2 + \left({40 - \dfrac {n - 41} 3}\right) \times 2^2 + \left({\dfrac {4 \left({n - 41}\right)} 3}\right) \times 1^2$
Thus in each case $169$ has been expressed in terms of $n$ squares: a combination of varying numbers of $1^2$, $2^2$ and $3^2$.
For $n > 155$ this construction does not work, because it results in a negative number of instances of $2^2$.
Similarly, for $n < 41$ this construction does not work, because it results in a negative number of instances of $1^2$.
For $n < 41$, we have:
:$169 = 25 + k$
where $k$ equals one of:
:$1 \times 12^2$
:$9 \times 4^2$
:$16 \times 3^2$
:$14 \times 3^2 + 3 \times 2^2 + 6 \times 1^2$
:$10 \times 3^2 + 11 \times 2^2 + 10 \times 1^2$
That is, $169$ equals $25$ plus $1$, $9$, $16$, $23$ or $31$ other squares.
But from 25 as Sum of 4 to 11 Squares, $25$ can also be written as a sum of $n$ squares for $4 \le n \le 11$.
Thus, by using judicious combinations of:
: sums of squares totalling $25$
: sums of squares totalling $169 - 25$
we can obtain a representation of $169$ as the sum of $n$ squares where $5 \le n \le 40$.
In summary, therefore, there exists a representation of $169$ as the sum of $n$ squares for $1 \le n \le 155$.
== Sources ==
* 
* : $169$
Category:Sums of Squares
Category:169
169 as Sum of up to 155 Squares/Mistake

== Source Work ==
:
:The Dictionary
::$169$
== Mistake ==
:In fact, $169$ can be written as the sum of $n$ non-zero squares, for all values of $n$ from $1$ to $155$, but for no larger values. [ Jackson, Masat and Mitchell, MM v61 41 ]
This is apparently untrue: $169$ can be expressed as the sum of $n$ non-zero squares for several values of $n$ greater than $155$, for example:
The citation given is also wrong. It should refer to volume $66$, not $61$:
== Sources ==
* : $169$
Category:Mistakes/Curious and Interesting Numbers
17

== Number ==
$17$ (seventeen) is:
:The $7$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$
:The $1$st integer to be the sum of $2$ distinct powers of $4$:
::$17 = 1 + 16 = 1^4 + 2^4$
:The $1$st odd positive integer not expressible in the form $2 n^2 + p$ where $p$ is prime
:The $2$nd emirp after $13$
:The $2$nd long period prime after $7$:
::$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$
:The $3$rd Stern number after $1$, $3$
:The $3$rd Stern prime after $1$, $3$
:The $3$rd Fermat number and Fermat prime after $3$, $5$:
::$17 = 2^{2^2} + 1$
:The $3$rd positive integer $n$ after $5$, $11$ such that no factorial of an integer can end with $n$ zeroes
:The $3$rd prime number of the form $n^2 + 1$ after $2$, $5$:
::$17 = 4^2 + 1$
:The $3$rd integer after $2$, $5$ at which the prime number race between primes of the form $4 n + 1$ and $4 n - 1$ are tied
:The $4$th Proth prime after $3$, $5$, $13$:
::$17 = 1 \times 2^4 + 1$
:The $4$th Dudeney number after $0$, $1$, $8$, and only prime Dudeney number:
::$17 = 4 + 9 + 1 + 3$, while $17^3 = 4913$
:The smaller of the $4$th pair of twin primes, with $19$
:The $5$th of the lucky numbers of Euler after $2$, $3$, $5$, $11$:
::$n^2 + n + 17$ is prime for $0 \le n < 15$
:The index of the $6$th Mersenne prime after $2$, $3$, $5$, $7$, $13$:
::$M_{17} = 2^{17} - 1 = 131 \, 071$
:The $6$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$
:The $7$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$
:The $7$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $\ldots$
:The index of the $7$th Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$ which asserted to be prime
:The $9$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $\ldots$
:The $12$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes
:The $12$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $10$, $11$ such that $5^n$ contains no zero in its decimal representation:
::$5^{17} = 762 \, 939 \, 453 \, 125$
:There are $17$ wallpaper groups
== Also see ==
* Construction of Regular Heptadecagon
* Prime Dudeney Number
* Smallest Odd Number not of form 2 a squared plus p
* Reciprocal of 17
* 17 Wallpaper Groups
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $\S 13$: The fundamental theorem of arithmetic
* : $17$
* : $17$
Category:Specific Numbers
Category:17
17,163

== Number ==
$17 \, 163$ (seventeen thousand, one hundred and sixty-three) is:
:$3^2 \times 1907$
:The largest positive integer which cannot be expressed as the sum of the squares of distinct prime numbers
== Also see ==
* Largest Number not Sum of Squares of Distinct Primes
== Sources ==
* : $17,163$
* : $17,163$
Category:Specific Numbers
Category:17,163
17,296

== Number ==
$17 \, 296$ (seventeen thousand, two hundred and ninety-six) is:
:$2^4 \times 23 \times 47$
:The smaller of the $8$th amicable pair, with $18 \, 416$:
::$\map {\sigma_1} {17 \, 296} = \map {\sigma_1} {18 \, 416} = 35 \, 712 = 17 \, 296 + 18 \, 416$
:which is also the $2$nd Thabit pair:
::$17 \, 296 = 2^4 \times 23 \times 47 = 2^4 \paren {3 \times 2^4 - 1} \paren {3 \times 2^{3 - 1} - 1}, 18 \, 416 = 2^4 \paren {9 \times 2^{2 \times 4 - 1} - 1}$
== Also see ==
* 
* 
== Sources ==
* : $17,296$
* : $17,296$
Category:Specific Numbers
Category:17,296
17,576

== Number ==
$17 \, 576$ (seventeen thousand, five hundred and seventy-six) is:
:$2^3 \times 13^3$
:The $26$th cube number:
::$17 \, 576 = 26 \times 26 \times 26$
:The $6$th cube after $0$, $1$, $512$, $4913$, $5832$ whose digits add up to its root:
::$17 \, 576 = 26^3$, while $1 + 7 + 5 + 7 + 6 = 26$
== Also see ==
* 
* 
Category:Specific Numbers
Category:17,576
17,716

== Number ==
$17 \, 716$ (seventeen thousand, seven hundred and sixteen) is:
:$2^2 \times 43 \times 103$
:The $28$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:17,716
17,796,126,877,482,329,126,044

== Number ==
$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044$ is:
:$2^2 \times 7 \times 4327 \times 456 \, 293 \times 321 \, 911 \, 699 \, 243$
:The $1$st in a sequence of $9$ consecutive integers all of which have the same number of divisors:
::$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044} = 48$
== Also see ==
* 
== Sources ==
* : $17,796,126,877,482,329,126,044$
Category:Specific Numbers
Category:17,796,126,877,482,329,126,044
17,796,126,877,482,329,126,045

== Number ==
$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 045$ is:
:$5 \times 17^2 \times 47 \times 53 \times 4 \, 944 \, 062 \, 119 \, 125 \, 691$
:The $2$nd in a sequence of $9$ consecutive integers all of which have the same number of divisors:
::$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 045} = 48$
== Also see ==
* 
Category:Specific Numbers
Category:17,796,126,877,482,329,126,045
17,796,126,877,482,329,126,046

== Number ==
$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 046$ is:
:$2 \times 3^2 \times 179 \times 5171 \times 1 \, 068 \, 133 \, 213 \, 285 \, 183$
:The $3$rd in a sequence of $9$ consecutive integers all of which have the same number of divisors:
::$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 046} = 48$
== Also see ==
* 
Category:Specific Numbers
Category:17,796,126,877,482,329,126,046
17,796,126,877,482,329,126,047

== Number ==
$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 047$ is:
:$11^5 \times 23 \times 107 \times 44 \, 900 \, 425 \, 217 \, 777$
:The $4$th in a sequence of $9$ consecutive integers all of which have the same number of divisors:
::$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 047} = 48$
== Also see ==
* 
Category:Specific Numbers
Category:17,796,126,877,482,329,126,047
17,796,126,877,482,329,126,048

== Number ==
$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 048$ is:
:$2^5 \times 19 \times 4 \, 590 \, 338 \, 339 \times 6 \, 376 \, 424 \, 429$
:The $5$th in a sequence of $9$ consecutive integers all of which have the same number of divisors:
::$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 048} = 48$
== Also see ==
* 
Category:Specific Numbers
Category:17,796,126,877,482,329,126,048
17,796,126,877,482,329,126,049

== Number ==
$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 049$ is:
:$3 \times 13^2 \times 241 \times 557 \times 261 \, 484 \, 106 \, 225 \, 711$
:The $6$th in a sequence of $9$ consecutive integers all of which have the same number of divisors:
::$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 049} = 48$
== Also see ==
* 
Category:Specific Numbers
Category:17,796,126,877,482,329,126,049
17,796,126,877,482,329,126,050

== Number ==
$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 050$ is:
:$2 \times 5^2 \times 11 \, 831 \times 189 \, 043 \times 159 \, 137 \, 830 \, 837$
:The $7$th in a sequence of $9$ consecutive integers all of which have the same number of divisors:
::$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 050} = 48$
== Also see ==
* 
Category:Specific Numbers
Category:17,796,126,877,482,329,126,050
17,796,126,877,482,329,126,051

== Number ==
$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 051$ is:
:$7^5 \times 29 \times 351 \, 121 \times 103 \, 987 \, 345 \, 177$
:The $8$th in a sequence of $9$ consecutive integers all of which have the same number of divisors:
::$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 051} = 48$
== Also see ==
* 
Category:Specific Numbers
Category:17,796,126,877,482,329,126,051
17,796,126,877,482,329,126,052

== Number ==
$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 052$ is:
:$2^2 \times 3 \times 149 \times 991 \, 723 \times 10 \, 036 \, 160 \, 394 \, 373$
:The $9$th and last in a sequence of $9$ consecutive integers all of which have the same number of divisors:
::$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 052} = 48$
== Also see ==
* 
Category:Specific Numbers
Category:17,796,126,877,482,329,126,052
17,962,559

== Number ==
$17 \, 962 \, 559$ (seventeen million, nine hundred and sixty-two thousand, five hundred and fifty-nine) is:
:The $1 \, 149 \, 096$th prime number
:The $5$th term of the smallest Cunningham chain of the first kind of length $7$:
::$1 \, 122 \, 659$, $2 \, 245 \, 319$, $4 \, 490 \, 639$, $8 \, 981 \, 279$, $17 \, 962 \, 559$, $35 \, 925 \, 119$, $71 \, 850 \, 239$
== Also see ==
* 
* 
Category:Specific Numbers
Category:17,962,559
17/Historical Note

== Historical Note on $17$ ==
$17$ was the highest number whose square root was proved irrational by .
 reports that the Pythagoreans had a horror of the number $17$.
That was because it lay half way between $16$ and $18$, which are the areas of rectangles whose areas equal their perimeters.
$17$ seems to be a popular age to write a pop song about:
: by 
:7-Teen by The Regents
: by 
to name but three.
== Sources ==
* 
* : $17$
* : $17$
Category:Historical Notes
170

== Number ==
$170$ (one hundred and seventy) is:
:$2 \times 5 \times 17$
:The $7$th positive integer after $50$, $65$, $85$, $125$, $130$, $145$ which can be expressed as the sum of two square numbers in two or more different ways:
::$145 = 12^2 + 1^2 = 9^2 + 8^2$
:The $10$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$, $115$, $119$ whose divisor sum is square:
:
:The $14$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$, $114$, $130$, $138$, $154$, $165$:
::$170 = 2 \times 5 \times 17$
:The $15$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$, $100$, $116$, $122$, $130$, $134$, $146$, $154$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 170$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $23$rd nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 170$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:170
170, 141,183, 460,469, 231,731, 687,303, 715,884, 105,727

== Number ==
$170 \, 141 \, 183 \, 460 \, 469 \, 231 \, 731 \, 687 \, 303 \, 715 \, 884 \, 105 \, 727$ is:
:A prime number
:The $31$st Mersenne number:
::$170 \, 141 \, 183 \, 460 \, 469 \, 231 \, 731 \, 687 \, 303 \, 715 \, 884 \, 105 \, 727 = 2^{127} - 1$
:The $12$th Mersenne prime:
::$170 \, 141 \, 183 \, 460 \, 469 \, 231 \, 731 \, 687 \, 303 \, 715 \, 884 \, 105 \, 727 = 2^{127} - 1$
== Historical Note ==
== Also see ==
* 
* 
* 
== Sources ==
* : $2^{127} - 1$
* : $2^{127} - 1$
Category:Specific Numbers
Category:170, 141,183, 460,469, 231,731, 687,303, 715,884, 105,727
170, 141,183, 460,469, 231,731, 687,303, 715,884, 105,727/Historical Note

== Historical Note on $170 \, 141 \, 183 \, 460 \, 469 \, 231 \, 731 \, 687 \, 303 \, 715 \, 884 \, 105 \, 727$ ==
It was who demonstrated in $1876$ that $2^{127} - 1$ is prime.
He was able to do this by using a new technique he designed, which was a precursor to the Lucas-Lehmer Test.
He himself expressed some doubt over the fact of this result, but it was confirmed in $1914$ by .
This number held the record for the highest known prime number for longer than any other: $1876$ to $1951$, when discovered that $\dfrac {2^{148} + 1} {17}$ is prime.
It also remains the largest prime number to be discovered without the help of modern calculation machines.
The following year ($1952$) discovered the prime nature of $2^{521} - 1$, and followed it up with $4$ others, of whch $M_{2281}$ was the largest.
To put the question into perspective, the author of this page used an online integer factorization calculator to find out the prime numbers immediately before and after $2^{127} - 1$ in a matter of seconds.
== Sources ==
* : Preface
* : $2^{127} - 1$
* : $2^{2281} - 1$
* : $2^{127} - 1$
Category:Historical Notes
171

== Number ==
$171$ (one hundred and seventy-one) is:
:$3^2 \times 19$
:The second 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 $1$st of the $6$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {171} = \map {\sigma_0} {172} = 6$
:The $7$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$
:The $10$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $99$, $111$, $141$, $151$, $171$, $\ldots$
:The $18$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$, $78$, $91$, $105$, $120$, $136$, $153$:
::$171 = \ds \sum_{k \mathop = 1}^{18} k = \dfrac {18 \times \paren {18 + 1} } 2$
:The $35$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $133$, $135$, $141$, $151$, $159$, $163$, $169$, $171$, $\ldots$
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:171
1711

== Number ==
$1711$ (one thousand, seven hundred and eleven) is:
:$29 \times 59$
:The $58$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $1326$, $1378$, $1431$, $1485$, $1540$, $1596$, $1653$:
::$1711 = \ds \sum_{k \mathop = 1}^{58} k = \dfrac {58 \times \left({58 + 1}\right)} 2$
== Also see ==
* 
Category:Specific Numbers
Category:1711
1715

== Number ==
$1715$ (one thousand seven hundred and fifteen) is:
:$5 \times 7^3$
:The $51$st Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $1176$, $1184$, $1197$, $1212$, $1296$, $1311$, $1332$, $1344$, $1416$, $1575$:
::$1715 = 47 \times 35 = 47 \times \left({1 \times 7 \times 1 \times 5}\right)$
== Also see ==
* 
Category:Specific Numbers
Category:1715
1717

== Number ==
$1717$ (one thousand, seven hundred and seventeen) is:
:$17 \times 101$
:The $34$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $925$, $1001$, $1080$, $1162$, $1247$, $1335$, $1426$, $1520$, $1617$:
::$1717 = \ds \sum_{k \mathop = 1}^{34} \paren {3 k - 2} = \dfrac {34 \paren {3 \times 34 - 1} } 2$
:The $67$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $1335$, $1365$, $1426$, $1457$, $1520$, $1552$, $1617$, $1650$:
::$1717 = \ds \sum_{k \mathop = 1}^{34} \paren {3 k - 2} = \dfrac {34 \paren {3 \times 34 - 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1717
172

== Number ==
$172$ (one hundred and seventy-two) is:
:$2^2 \times 43$
:The $2$nd of the $6$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {171} = \map {\sigma_0} {172} = 6$
:The $16$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$, $100$, $116$, $122$, $130$, $134$, $146$, $154$, $170$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 172$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
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Category:Specific Numbers
Category:172
1722

== Number ==
$1722$ (one thousand, seven hundred and twenty-two) is:
:$2 \times 3 \times 7 \times 41$
:The $3$rd Giuga number:
::$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {41} - \dfrac 1 {1722} = 1$
== Also see ==
* 
Category:Specific Numbers
Category:1722
1728

== Number ==
$1728$ (one thousand, seven hundred and twenty-eight) is:
:$2^6 \times 3^3$
:The $3$rd power of $12$ after $(1)$, $12$, $144$:
::$1728 = 12^3$
:The $3$rd cube after $6^3$, $9^3$ which can be expressed as the sum of $3$ positive cubes
:The $12$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$, $1331$:
::$1728 = 12 \times 12 \times 12 = \paren {2^2 \times 3}^3$
:Hence in duodecimal notation:
::$1000$
:The $363$rd harshad number:
::$1728 = 96 \times 18 = 96 \times \paren {1 + 7 + 2 + 8}$
== Also see ==
* 
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== Historical Note ==
== Sources ==
* : $1728$
* : $1728$
Category:Specific Numbers
Category:1728
1729

== Number ==
$1729$ (one thousand, seven hundred and twenty-nine) is:
:$7 \times 13 \times 19$
:The $1$st taxicab number: a positive integer which can be expressed as the sum of $2$ cubes in $2$ different ways:
::$1729 = 12^3 + 1^3 = 10^3 + 9^3$
:The $1$st Fermat pseudoprime to each of the bases $2$, $3$ and $5$:
::$2^{1729} \equiv 2 \pmod {1729}$, $3^{1729} \equiv 3 \pmod {1729}$, $5^{1729} \equiv 5 \pmod {1729}$
:The $2$nd Hardy-Ramanujan number after $2$: the smallest positive integer which can be expressed as the sum of $2$ cubes in $2$ different ways:
::$1729 = \map {\operatorname {Ta} } 2 = 12^3 + 1^3 = 10^3 + 9^3$
:The $3$rd Carmichael number after $561$, $1105$:
::$\forall a \in \Z: a \perp 1729: a^{1728} \equiv 1 \pmod {1729}$
:The $4$th and largest, after $1$, $81$, $1458$ of the $4$ harshad numbers which can each be expressed as the product of the sum of its digits and the reversal of the sum of its digits:
::$1729 = 91 \times 19 = 91 \times \paren {1 + 7 + 2 + 9}$
:The $6$th Poulet number after $341$, $561$, $645$, $1105$, $1387$:
::$2^{1729} \equiv 2 \pmod {1729}$: $1729 = 7 \times 13 \times 19$
:The $7$th Fermat pseudoprime to base $5$ after $4$, $124$, $217$, $561$, $781$, $1541$:
::$5^{1729} \equiv 5 \pmod {1729}$
:The $9$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$, $1105$, $1541$:
::$3^{1729} \equiv 3 \pmod {1729}$
:The $364$th harshad number:
::$1729 = 91 \times 19 = 91 \times \paren {1 + 7 + 2 + 9}$
== Also see ==
* Smallest Fermat Pseudoprime to Bases 2, 3 and 5
* 
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== Historical Note ==
== Sources ==
* : $1729$
* : $1729$
Category:Specific Numbers
Category:1729