nyuuzyou/wikis huggingface dataset:
1729/Historical Note

== Historical Note on $1729$ (One Thousand, Seven Hundred and Twenty-Nine) ==
In the opinion of some writers, "among the most famous of all numbers".
This is all down to the influence of the writings of , who documents an anecdote about a time when he visited in hospital.
He reports the incident thus:
:[ ] could remember the idiosyncrasies of numbers in an almost uncanny way. It was who said that every positive integer was one of 's personal friends. I remember going to see him once when he was lying ill in Putney. I had ridden in a taxi-cab No. $1729$, and remarked that the number seemed to me a rather dull one, and that I hoped it was not an unfavourable omen. 'No,' he reflected, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'
::$1729 = 12^3 + 1^3 = 10^3 + 9^3$
Because of this, $1729$ is often seen referred to as a taxicab number (sometimes hyphenated: taxi-cab).
 then asked whether he knew the answer to the same problem for $4$th powers. thought for a moment, then said he did not, but he believed the number would be very large.
In fact it is $635 \, 318 \, 657$.
This property of $1729$ appears occasionally in mainstream entertainment either as a subject of mathematics as discussed by supposed mathematicians, or (in more subtle fare) as a mathematical in-joke.
== Sources ==
* 
* : $1729$
* : $1729$
* 
Category:Historical Notes
173

== Number ==
$173$ (one hundred and seventy-three) is:
:The $40$th prime number
:The $4$th balanced prime after $5$, $53$, $157$:
::$173 = \dfrac {167 + 179} 2$
:The $8$th inconsummate number after $62$, $63$, $65$, $75$, $84$, $95$, $161$:
::$\nexists n \in \Z_{>0}: n = 173 \times \map {s_{10} } n$
:The $10$th of $11$ primes of the form $2 x^2 + 11$:
::$2 \times 9^2 + 11 = 173$ ()
:The $11$th prime number after $53$, $71$, $103$, $107$, $109$, $149$, $151$, $157$, $163$, $167$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$
:The $13$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$, $131$:
::$2 \times 173 + 1 = 347$, which is prime.
:The $19$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$, $113$, $137$, $167$
:The $34$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Sources ==
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:173
174

== Number ==
$174$ (one hundred and seventy-four) is:
:$2 \times 3 \times 29$
:The $15$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$, $114$, $130$, $138$, $154$, $165$, $170$:
::$174 = 2 \times 3 \times 29$
:The $24$th nontotient:
::$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 174$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:174
175

== Number ==
$175$ (one hundred and seventy-five) is:
:$5^2 \times 7$
:The magic constant of a magic square of order $7$, after $1$, $(5)$, $15$, $34$, $65$, $111$:
::$175 = \ds \dfrac 1 7 \sum_{k \mathop = 1}^{7^2} k = \dfrac {7 \paren {7^2 + 1} } 2$
:The $13$th number after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $89$, $135$ which is the sum of the increasing powers of its digits taken in order:
::$1^1 + 7^2 + 5^3 = 175$
:The $22$nd Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$, $112$, $115$, $128$, $132$, $135$, $144$:
::$175 = 5 \times 35 = 5 \times \paren {1 \times 7 \times 5}$
== Also see ==
* 
* 
* 
== Sources ==
* : $135$
* : $175$
* : $135$
* : $175$
Category:Specific Numbers
Category:175
175,560

== Number ==
$175 \, 560$ (one hundred and seventy-five thousand, five hundred and sixty) is:
:$2^3 \times 3 \times 5 \times 7 \times 11 \times 19$
:The $3$rd and last after $24$, $120$ of the $3$ integers which can be expressed as the product of both 3 and 4 consecutive integers:
::$175 \, 560 = 55 \times 56 \times 57 = 19 \times 20 \times 21 \times 22$
== Also see ==
* 
== Sources ==
* : $175,560$
Category:Specific Numbers
Category:175,560
1751

== Number ==
$1751$ (one thousand, seven hundred and fifty-one) is:
:$17 \times 103$
:The $34$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $1107$, $1190$, $1276$, $1365$, $1457$, $1552$, $1650$:
::$1751 = \dfrac {34 \paren {3 \times 34 + 1} } 2$
:The $68$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $1335$, $1365$, $1426$, $1457$, $1520$, $1552$, $1617$, $1650$, $1717$:
::$1751 = \dfrac {34 \paren {3 \times 34 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1751
176

== Number ==
$176$ (one hundred and seventy-six) is:
:$2^4 \times 11$
:The $8$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$:
::$176 = 1 + 7 + 13 + 19 + 25 + 31 + 37 + 43 = 8 \paren {3 \times 8 - 2}$
:The $11$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$:
::$176 = 1 + 4 + 7 + 10 + 13 + 16 + 19 + 22 + 25 + 28 + 31 = \dfrac {11 \paren {3 \times 11 - 1} } 2$
:The number of integer partitions for $15$:
::$\map p {15} = 176$
:The $21$st generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$, $40$, $51$, $57$, $70$, $77$, $92$, $100$, $117$, $126$, $145$, $155$:
::$176 = \dfrac {11 \paren {3 \times 11 - 1} } 2$
:The $28$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $91$, $94$, $97$, $100$, $103$, $109$, $129$, $130$, $133$, $139$, $167$:
::$176 \to 1^2 + 7^2 + 6^2 = 1 + 49 + 36 = 86 \to 8^2 + 6^2 = 64 + 36 = 100 \to 1^2 + 0^2 + 0^2 = 1$
== Also see ==
* 
* 
* 
* 
* 
== Sources ==
Category:Specific Numbers
Category:176
1764

== Number ==
$1764$ (one thousand seven hundred and sixty-four) is:
:$2^2 \times 3^2 \times 7^2$
:The $16$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $324$, $400$, $576$, $784$, $900$, $961$, $1024$, $1296$, $1600$ to be the divisor sum value of some (strictly) positive integer:
::$1764 = \map {\sigma_1} {820}$
:The $42$nd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1024$, $1089$, $1156$, $1225$, $1296$, $1369$, $1444$, $1521$, $1600$, $1681$:
::$1764 = 42 \times 42$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1764
177,147

== Number ==
$177 \, 147$ (one hundred and seventy-seven thousand, one hundred and forty-seven) is:
:$3^{11}$
:The $11$th power of $3$ after $(1)$, $3$, $9$, $27$, $81$, $243$, $729$, $2187$, $6561$, $19 \, 683$, $59 \, 049$:
::$177 \, 147 = 3^{11}$
== Also see ==
* 
Category:Specific Numbers
Category:177,147
177,792

== Number ==
$177 \, 792$ (one hundred and seventy-seven thousand, seven hundred and ninety-two) is:
:$2^7 \times 3 \times 7 \times 463$
:The $6$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:177,792
1770

== Number ==
$1770$ (one thousand, seven hundred and seventy) is:
:$2 \times 3 \times 5 \times 59$
:The $30$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $703$, $780$, $861$, $946$, $1035$, $1225$, $1326$, $1431$, $1540$, $1653$:
::$1770 = \ds \sum_{k \mathop = 1}^{30} \paren {4 k - 3} = 30 \paren {2 \times 30 - 1}$
:The $59$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $1326$, $1378$, $1431$, $1485$, $1540$, $1596$, $1653$, $1711$:
::$1770 = \ds \sum_{k \mathop = 1}^{59} k = \dfrac {59 \times \paren {59 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1770
1771

== Number ==
$1771$ (one thousand, seven hundred and seventy-one) is:
:$7 \times 11 \times 23$
:The $21$st tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $816$, $969$, $1140$, $1330$, $1540$:
::$1771 = \ds \sum_{k \mathop = 1}^{21} \frac {k \paren {k + 1} } 2 = \dfrac {21 \paren {21 + 1} \paren {21 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:1771
1777

== Number ==
$1777$ (one thousand, seven hundred and seventy-seven) is:
:The $275$th prime number
:The $2$nd after $587$ of a sequence of $11$ primes formed by tripling and adding $16$:
::$1777 = 3 \times 587 + 16$
:The $52$nd near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $877$, $881$, $883$, $887$, $911$, $919$, $929$, $977$, $991$, $997$, $1117$, $1171$, $1181$, $1511$, $1511$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1777
1782

== Number ==
$1782$ (one thousand, seven hundred and eighty-two) is:
:$2 \times 3^4 \times 11$
:$3$ times the sum of all the $2$-digit integers that can be formed from its digits:
::$1782 = 3 \times 594 = 3 \times \paren {17 + 18 + 12 + 71 + 78 + 72 + 81 + 87 + 82 + 21 + 27 + 28}$
:The $27$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $874$, $970$, $1071$, $1177$, $1288$, $1404$, $1525$, $1651$:
::$1782 = \ds \sum_{k \mathop = 1}^{27} \paren {5 k - 4} = \dfrac {27 \paren {5 \times 27 - 3} } 2$
== Also see ==
* 1782 is 3 Times Sum of all 2-Digit Numbers from its Digits
* 
== Sources ==
* : $1782$
* : $1782$
Category:Specific Numbers
Category:1782
1782 is 3 Times Sum of all 2-Digit Numbers from its Digits

== Theorem ==
$1782$ equals $3$ multiplied by the sum of all the $2$-digit integers that can be formed from its digits.
== Proof ==
The number of $2$-digit integers that can be formed from the digits of $1782$ equals the number of $2$-permutations of $\set {1, 7, 8, 2}$.
That is:
:$\set {17, 18, 12, 71, 78, 72, 81, 87, 82, 21, 27, 28}$
Hence:
:$17 + 18 + 12 + 71 + 78 + 72 + 81 + 87 + 82 + 21 + 27 + 28 = 594 = \dfrac {1782} 3$
== Sources ==
* : $1782$
* : $1782$
Category:Recreational Mathematics
Category:1782
1785

== Number ==
$1785$ (one thousand, seven hundred and eighty-five) is:
:$3 \times 5 \times 7 \times 17$
:The $17$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $91$, $140$, $204$, $285$, $385$, $506$, $650$, $819$, $1015$, $1240$, $1496$:
::$1785 = \ds \sum_{k \mathop = 1}^{17} k^2 = \dfrac {17 \paren {17 + 1} \paren {2 \times 17 + 1} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:1785
179

== Number ==
$179$ (one hundred and seventy-nine) is:
:The $41$st prime number
:The $2$nd term after $89$ of the smallest Cunningham chain of the first kind of length $6$:
::$\tuple {89, 179, 359, 719, 1439, 2879}$
:The $10$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$:
::$179 = 2 \times 89 + 1$
:The smaller of the $13$th pair of twin primes, with $181$
:The $14$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$, $131$, $173$:
::$2 \times 179 + 1 = 359$, which is prime.
:The $14$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$, $157$, $167$
:The $15$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$, $109$, $113$, $131$, $149$, $167$
:The $35$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
Category:Specific Numbers
Category:179
179,931,895,322

== Number ==
$179 \, 931 \, 895 \, 322$ is:
:$2 \times 61 \times 929 \times 1 \, 587 \, 569$
:The largest term of the aliquot sequence that starts with $138$, reached after $117$ steps.
== Sources ==
* Aliquot Sequence starting from $138$
== Sources ==
* : $276$
* : $276$
Category:Specific Numbers
Category:179,931,895,322
17 Consecutive Integers each with Common Factor with Product of other 16

== Theorem ==
The $17$ consecutive integers from $2184$ to $2200$ have the property that each one is not coprime with the product of the other $16$.
== Proof ==
We obtain the prime decomposition of all $17$ of these integers:
It can be seen by inspection that each of the integers in this sequence shares at least one prime factor with at least one other.
It is then worth noting that:
and it can be seen that the sequence can be extended neither upwards nor downwards.
== Sources ==
* 
* : $2185$
Category:Recreational Mathematics
17 Consecutive Integers each with Common Factor with Product of other 16/Mistake

== Source Work ==
:
:The Dictionary
::$2185$
== Mistake ==
:The start of a sequence of $17$ consecutive integers, each of which has a common factor, greater than $1$, with the product of the remaining $16$.
It is easiest to see this is wrong if one is first to obtain the prime decomposition of all $17$ of these integers:
We have that:
: $2197 = 13^3$, and that none of the other $16$ have $13$ as a divisor.
: $2201 = 31 \times 71$: none of the other $16$ has either $31$ or $71$ as a divisor.
Thus neither $2197$ nor $2201$ have a common factor, greater than $1$, with the product of the remaining $16$.
The key number here is that the sequence of $17$ starts at $2184$ and not $2185$:
:$2184 = 2^3 \times 3 \times 7 \times 13$
which then provides that necessary common factor with $2197$, and allows $2201$ to be removed from the other end of the sequence.
== Sources ==
* : $2185$
Category:Mistakes/Curious and Interesting Numbers
18,144

== Number ==
$18 \, 144$ (eighteen thousand, one hundred and forty-four) is:
:$2^5 \times 3^4 \times 7$
:The volume of the only rational tetrahedron whose edge lengths are less than $157$
== Also see ==
* Volume of Smallest Rational Tetrahedron
== Sources ==
* : $18,144$
Category:Specific Numbers
Category:18,144
18,416

== Number ==
$18 \, 416$ (eighteen thousand, four hundred and sixteen) is:
:$2^4 \times 1151$
:The larger 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:18,416
18,446,744,073,709,551,615

== Number ==
$18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 615$ is:
:$3 \times 5 \times 17 \times 257 \times 641 \times 65 \, 537 \times 6 \, 700 \, 417$
:$2^{64} - 1$
:The number of moves it would take the monks to move all the disks onto a different peg in the Tower of Hanoi.
== Also see ==
* Prime Factors of $2^{64} - 1$
== Historical Note ==
== Sources ==
* : $31$
* : $18,446,744,073,709,551,615$
* : $31$
* : $18,446,744,073,709,551,615$
Category:Specific Numbers
Category:18,446,744,073,709,551,615
18,446,744,073,709,551,615/Historical Note

== Historical Note on $18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 615$ ==
The story goes that the inventor of chess, Sissa ben Dahir, was offered a reward by King Shirham of India.
King Shirham asked Sissa to name his reward.
Sissa replied:
"Please give me one grain (of rice, or wheat, etc.) placed on the first square of the chessboard. Then two grains placed on the second square. Then four placed on the third square and eight placed on the fourth square, and so doubling the number of grains on each subsequent square until all $64$ squares are so covered."
King Shirham acceded to what he viewed as a foolishly paltry request, until such time came as to attempt to actually carry out such a task.
By Sum of Geometric Sequence, The total number of grains evaluates to $2^{64} - 1$, which is more grains of wheat than existed.
It is a popular classroom exercise to require students to calculate the volume or weight of grain that Sissa asked for.
Note that the number of grains is the same number as that of the number of moves needed to complete the task of the Tower of Hanoi.
 and proceed to point out, in their of $1940$, that if it is assumed there have been $64$ generations since $0 \text {CE}$, this is also the number of ancestors of each person alive since that time.
As puts it in his of $1992$:
:The ratio of $2^{64}$ to the actual population of the earth at that time is therefore a measure of the amount of unintentional interbreeding that has taken place.
== Sources ==
* : Arithmetical Puzzles: The Progression of Numbers
* 
* : $18,446,744,073,709,551,615$
* : Sissa and the Chessboard: $46$
* : $18,446,744,073,709,551,615$
Category:Historical Notes
18,446,744,073,709,551,617

== Number ==
$18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 617$ is:
:$274 \, 177 \times 67 \, 280 \, 421 \, 310 \, 721$
:The $6$th Fermat number after $3$, $5$, $17$, $257$, $65 \, 537$, $4 \, 294 \, 967 \, 297$:
::$18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 617 = 2^{\paren {2^6} } + 1 = 2^{64} + 1$
:and the $2$nd not to be prime.
== Also see ==
* Prime Decomposition of 6th Fermat Number
* 
Category:Specific Numbers
Category:18,446,744,073,709,551,617
18,465,126,293

== Number ==
$18 \, 465 \, 126 \, 293$ is:
:A prime number
:The lower end of the highest known stretch of integers where $4 n + 1$ is not less than $4 n - 1$ in their prime number race
== Also see ==
* Largest Known Lead by 4n+1 in Prime Number Race
* 
== Sources ==
* : $18,465,126,293$
* : $18,465,126,293$
Category:Specific Numbers
Category:18,465,126,293
18,496

== Number ==
$18 \, 496$ (eighteen thousand, four hundred and ninety-six) is:
:$2^6 \times 17^2$
:The index of the $6$th Cullen prime after $1$, $141$, $4713$, $5795$, $6611$:
::$18 \, 496 \times 2^{18 \, 496} + 1$, also expressible as $\paren {17 \times 2^{9251} }^2 + 1$
:The $136$th square number:
::$18 \, 496 = 136 \times 136$
== Also see ==
* 
* 
== Sources ==
* 
* : $18,496$
Category:Specific Numbers
Category:18,496
18,523

== Number ==
$18 \, 523$ (eighteen thousand, five hundred and twenty-three) is:
:The $2122$nd prime number
:The $16$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$, $1021$, $2657$, $3229$, $4547$, $4787$, $11 \, 549$, $13 \, 649$, $18 \, 523$
== Also see ==
* 
* 
Category:Specific Numbers
Category:18,523
180

== Number ==
$180$ (one hundred and eighty) is:
:$2^2 \times 3^2 \times 5$
:The $4$th number after $1$, $9$, $20$ whose square has a divisor sum which is itself square:
:The $11$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$:
::$\map \tau {180} = 18$
:The $11$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$:
::$\dfrac {\map {\sigma_1} {180} } {180} = \dfrac {546} {180} = 3 \cdotp 0 \dot 3$
:The $26$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$, $120$, $144$, $168$:
::$\map {\sigma_1} {180} = 546$
:$180^3 = 6^3 + 7^3 + \cdots + 67^3 + 68^3 + 69^3$
:The number of degrees in the sum of the internal angles of a triangle
== Also see ==
* Measurement of Straight Angle
* Sum of Angles of Triangle equals Two Right Angles
* Cube of 180 is Sum of Sequence of Consecutive Cubes
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $180$
* : $180$
Category:Specific Numbers
Category:180
180/Historical Note

== Historical Note on $180$ ==
There are $180 \degrees$ in a straight angle.
Thus the colloquial expression doing a $180$, meaning turning completely around (either literally or metaphorically) to face in the opposite direction, can be found in a variety of contexts.
$180 \fahr$ is the number of degrees Fahrenheit between the melting point ($32 \fahr$) and boiling point ($212 \fahr$) of water.
== Sources ==
* : $180$
* : $180$
Category:Historical Notes
1800

== Number ==
$1800$ (one thousand eight hundred or eighteen hundred) is:
:$2^3 \times 3^2 \times 5^2$
:The $15$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $126$, $196$, $288$, $405$, $550$, $726$, $936$, $1183$, $1470$:
::$1800 = \ds \sum_{k \mathop = 1}^{15} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {15^2 \paren {15 + 1} } 2$
:The product of the number of edges, edges per face and faces of a regular dodecahedron.
::$30 \times 12 \times 5$
:The product of the number of edges, edges per face and faces of a regular icosahedron.
::$30 \times 20 \times 3$
== Also see ==
* Product of Number of Edges, Edges per Face and Faces of Regular Dodecahedron
* Product of Number of Edges, Edges per Face and Faces of Regular Icosahedron
* 
== Sources ==
* : $72$
Category:Specific Numbers
Category:1800
1801

== Number ==
$1801$ (one thousand, eight hundred and one) is:
:The $279$th prime number
:The $4$th prime number after $1009$, $1129$, $1201$ which can be expressed in the form $x^2 + n y^2$ for all values of $n$ from $1$ to $10$:
:The $25$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $817$, $919$, $1027$, $1141$, $1261$, $1387$, $1519$, $1657$:
::$1801 = \ds 1 + \sum_{k \mathop = 1}^{25 - 1} 6 k = 25^3 - 24^3$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1801
180 x (2^127 - 1) + 1/Mistake

== Source Work ==
:
:The Dictionary
::$180 \times \paren {2^{127} - 1} + 1$
== Mistake ==
:The largest known prime in July $1951$, discovered by and of Cambridge University on the EDSAC. They had a test for numbers of the form $k \times \text M_{127} + 1$ where $\text M_{127}$ is the $127$th Mersenne number. This was the largest prime found.
:In the same month , using a desk calculator only, showed the primality of $\paren {2^{143} + 1} / 17$.
== Correction ==
There are a number of problems here.
=== Definition of Mersenne Number ===
For a start, a number of the form $\text M_n = 2^n - 1$ is usually only defined to be a Mersenne number $n$ is prime.
Hence, as $127$ is the $31$st prime number, $M_{127}$ is generally identified as the $31$st Mersenne number.
=== Non-Primality of $180 \times \paren {2^{127} - 1} + 1$ ===
Secondly, $180 \times \paren {2^{127} - 1} + 1$ is not actually a prime.
We have:
The number in question is actually:
:$180 \times \paren {2^{127} - 1}^2 + 1$
which evaluates to:
:$5 \, 210 \, 644 \, 015 \, 679 \, 228 \, 794 \, 060 \, 694 \, 325 \, 390 \, 955 \, 853 \, 335 \, 898 \, 483 \, 908 \, 056 \, 458 \, 352 \, 183 \, 851 \, 018 \, 372 \, 555 \, 735 \, 221$
which has $79$ digits.
=== Incorrect reporting of 's Number ===
Thirdly, $\paren {2^{143} + 1} / 17$ has been reported incorrectly.
We have that:
and so is not actually a multiple of $17$.
But interestingly:
:$\paren {2^{143} + 1} / 17 = 655\,904\,270\,545\,018\,327\,692\,227\,008\,019\,069\,456\,058\,247$ remainder $10$
and $655\,904\,270\,545\,018\,327\,692\,227\,008\,019\,069\,456\,058\,247$ is indeed prime, and has $42$ digits.
However, a search of the online literature identifies for us what number it was that found.
It was in fact:
which has $44$ digits.
This paragraph has been removed from of $1997$.
== Sources ==
* : $180 \times \paren {2^{127} - 1} + 1$
Category:Mistakes/Curious and Interesting Numbers
181

== Number ==
$181$ (one hundred and eighty-one) is:
:The $42$nd prime number
:The smaller of the $2$nd pair of primes whose prime gap is $10$:
::$191 - 181 = 10$
:The $5$th near-repdigit prime after $101$, $113$, $131$, $151$
:The $5$th and final positive integer solution after $1$, $3$, $5$, $11$ to the Ramanujan-Nagell equation $x^2 + 7 = 2^n$ for integral $n$:
::$181^2 + 7 = 32 \, 768 = 2^{15}$
:The $9$th palindromic prime:
::$2$, $3$, $5$, $7$, $11$, $101$, $131$, $151$, $181$, $\ldots$
:The $12$th prime number after $53$, $71$, $103$, $107$, $109$, $149$, $151$, $157$, $163$, $167$, $173$ 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 $13$th pair of twin primes, with $179$
:The $16$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$, $109$, $113$, $131$, $149$, $167$, $179$
== Also see ==
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Category:Specific Numbers
Category:181