nyuuzyou/wikis huggingface dataset:
492 is Sum of 3 Cubes in 3 Ways/Historical Note

== Historical Note on 492 is Sum of 3 Cubes in 3 Ways ==
 and reported in $2009$ on a systematic investigation they performed on all the solutions to the equation:
:$x^3 + y^3 + z^3 = n$
for all $0 < n < 1000$ and such that $\size x, \size y, \size z \le 10^{14}$.
Within that range, they discovered that there are exactly $3$ solutions for $n = 492$.
Of all the numbers from $0$ to $1000$, most have many more than $3$ such ways, although a few still have no known solutions.
The full results can be found at https://www.uni-math.gwdg.de/jahnel/Arbeiten/Liste/threecubes_20070419.txt.
== Sources ==
* 
Category:Historical Notes
492 is Sum of 3 Cubes in 3 Ways/Mistake

== Source Work ==
:
:The Dictionary
::$492$
== Mistake ==
:$492$ is the sum of $3$ cubes, one or two of which may be negative, in no fewer than $10$ different ways. [Madachy]
== Correction ==
 appears to have conflated $2$ separate results.
The number of known ways $492$ can be expressed as the sum of $3$ cubes, either positive or negative, is $3$:
The result given in 's $1966$ work is that it is possible to express $492^3$ in no fewer than $10$ different ways:
(except that the $5$th expression is incorrect: $144$ should read $114$).
Three further such expressions for $492^3$ have since been found:
$492$ is not the only integer whose cube can be expressed as the sum of $3$ positive cubes in $13$ ways, but it is the smallest.
This entry has been removed from .
== Sources ==
* : $492$
Category:Mistakes/Curious and Interesting Numbers
495

== Number ==
$495$ (four hundred and ninety-five) is:
:$3^2 \times 5 \times 11$
:The $9$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$, $330$:
::$495 = \ds \sum_{k \mathop = 1}^9 \dfrac {k \paren {k + 1} \paren {k + 2} } 6 = \dfrac {9 \paren {9 + 1} \paren {9 + 2} \paren {9 + 3} } {24}$
:The $18$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$, $100$, $126$, $155$, $187$, $222$, $260$, $301$, $345$, $392$, $442$:
::$495 = \dfrac {18 \paren {3 \times 18 + 1} } 2$
:The $36$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $222$, $247$, $260$, $287$, $301$, $330$, $345$, $376$, $392$, $425$, $442$, $477$:
::$495 = \dfrac {18 \paren {3 \times 18 + 1} } 2$
:Kaprekar's process, when applied to a $3$-digit integer whose digits are not all the same, results in $495$ after no more than $6$ iterations.
== Also see ==
* Kaprekar's Process on 3 Digit Number ends in 495
* 
* 
* 
== Sources ==
* : $495$
* : $495$
Category:Specific Numbers
Category:495
4950

== Number ==
$4950$ (four thousand, nine hundred and fifty) is:
:$2 \times 3^2 \times 5^2 \times 11$
:The $6$th hexamorphic number after $1$, $45$, $66$, $1225$, $1326$:
::$4950 = H_{50}$
:The $12$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$, $703$, $999$, $2223$, $2728$, $4879$:
::$4950^2 = 24 \, 502 \, 500 \to 2450 + 2500 = 4950$
:The $50$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $3160$, $3321$, $3486$, $3655$, $3828$, $4005$, $4186$, $4371$, $4560$:
::$4950 = \ds \sum_{k \mathop = 1}^{50} \paren {4 k - 3} = 50 \paren {2 \times 50 - 1}$
:The $99$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4095$, $4186$, $4278$, $4371$, $4465$, $4560$, $4656$, $4753$, $4851$:
::$4950 = \ds \sum_{k \mathop = 1}^{99} k = \dfrac {99 \times \paren {99 + 1} } 2$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:4950
496

== Number ==
$496$ (four hundred and ninety-six) is:
:$2^4 \times 31$
:The $1$st triangular number which serves as a counterexample to Greenwood's Conjecture:
::$496 = T_{31}$ but $496 + 1 = 497 = 7 \times 71$ is not prime
:The $3$rd perfect number after $6$, $28$:
::$496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 31 \times 16 = \paren {2^5 - 1} \paren {2^{5 - 1} }$
:The $6$th Ore number after $1$, $6$, $28$, $140$, $270$:
::$\dfrac {496 \times \map {\sigma_0} {496} } {\map {\sigma_1} {496} } = 5$
:The $12$th primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$, $304$, $350$, $368$, $464$, $490$:
::$496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$
:The $16$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $120$, $153$, $190$, $231$, $276$, $325$, $378$, $435$:
::$496 = \ds \sum_{k \mathop = 1}^{16} \paren {4 k - 3} = 16 \paren {2 \times 16 - 1}$
:The $31$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$:
::$496 = \ds \sum_{k \mathop = 1}^{31} k = \dfrac {31 \times \paren {31 + 1} } 2$
== Also see ==
* Greenwood's Conjecture
* 
* 
* 
* 
* 
== Sources ==
* : $5$
* : $28$
* : $31$
* : $496$
* : $5$
* : $28$
* : $31$
* : $496$
Category:Specific Numbers
Category:496
4960

== Number ==
$4960$ (four thousand, nine hundred and sixty) is:
:$2^5 \times 5 \times 31$
:The $30$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $816$, $969$, $1140$, $1330$, $1540$, $1771$, $2024$, $2300$, $2600$, $2925$, $3276$, $3276$, $4060$, $4495$:
::$4960 = \ds \sum_{k \mathop = 1}^{30} \frac {k \paren {k + 1} } 2 = \dfrac {30 \paren {30 + 1} \paren {30 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:4960
4961

== Number ==
$4961$ (four thousand, nine hundred and sixty-one) is:
:$11^2 \times 41$
:The $18$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$, $1105$, $1541$, $1729$, $1891$, $2465$, $2665$, $2701$, $2821$, $3281$, $3367$, $3751$:
::$3^{4961} \equiv 3 \pmod {4961}$
:The $41$st octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $3008$, $3201$, $3400$, $3605$, $3816$, $4033$, $4256$, $4485$, $4720$:
::$4961 = \ds \sum_{k \mathop = 1}^{41} \paren {6 k - 5} = 41 \paren {3 \times 41 - 2}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4961
497

== Number ==
$497$ (four hundred and ninety-seven) is:
:$7 \times 71$
:The $1$st counterexample to Greenwood's Conjecture:
::$497 = T_{31} + 1 = \dfrac {31 \times \paren {31 + 1} } 2 + 1$ but is not prime
:The $2$nd number such that the result when adding $2$ is the reversal of when multiplying by $2$:
::$497 + 2 = 499$; $497 \times 2 = 994$
:The $26$th number whose divisor sum is square:
:
== Also see ==
* Greenwood's Conjecture
* 
* 
== Sources ==
* : $496$
* : $496$
Category:Specific Numbers
Category:497
498

== Number ==
$498$ (four hundred and ninety-eight) is:
:$2 \times 3 \times 83$
:The $38$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $324$, $326$, $336$, $342$, $372$, $406$, $408$, $426$, $430$, $448$, $472$, $474$
== Also see ==
* 
Category:Specific Numbers
Category:498
499

== Number ==
$499$ (four hundred and ninety-nine) is:
:The $95$th prime number
:The larger of the $7$th pair of primes whose prime gap is $8$:
::$499 - 491 = 8$
:The $12$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$
:The $18$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $24$, $25$, $49$, $51$, $75$, $76$, $99$, $125$, $249$, $251$, $375$, $376$:
::$499^3 = 124 \, 251 \, \mathbf {499}$
:The $24$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $353$, $373$, $383$, $433$, $443$, $449$
:The $35$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $383$, $389$, $419$, $433$, $461$, $487$, $491$
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:499
4995

== Number ==
$4995$ (four thousand, nine hundred and ninety-five) is:
:$3^3 \times 5 \times 37$
:The $45$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $3010$, $3186$, $3367$, $3744$, $3940$, $4141$, $4347$, $4558$, $4774$:
::$4995 = \ds \sum_{k \mathop = 1}^{45} \paren {5 k - 4} = \dfrac {45 \paren {5 \times 45 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:4995
4998

== Number ==
$4998$ (four thousand, nine hundred and ninety-eight) is:
:$2 \times 3 \times 7^2 \times 17$
:The $20$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$, $105$, $160$, $661$, $2653$, $3069$, $3943$, $4053$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$ is prime
== Also see ==
* 
Category:Specific Numbers
Category:4998
4 Consecutive Integers cannot be Square-Free

== Theorem ==
Let $n, n + 1, n + 2, n + 3$ be four consecutive positive integers.
At least one of these is not square-free.
== Proof ==
Exactly one of $n, n + 1, n + 2, n + 3$ is divisible by $4 = 2^2$.
Thus, by definition, one of these is not square-free.
== Sources ==
* : $29$
Category:Square-Free Integers
4 Integers whose Euler Phi Value is 10,368

== Theorem ==
:$\map \phi {25 \, 930} = \map \phi {25 \, 935} = \map \phi {25 \, 940} = \map \phi {25 \, 942} = 10 \, 368 = 2^7 \times 3^4$
where $\phi$ denotes the Euler $\phi$ function.
== Proof ==
The significance of this result escapes the author of this page.
== Sources ==
* : $25,930$
Category:Euler Phi Function
Category:10,368
4 Pints from 5 Pints and 3 Pints

== Classic Problem ==
You have $2$ jars: one holds $5$ pints and one holds $3$ pints.
Neither jar is marked in any way.
You also have a cask with (to all intents and purposes) an unlimited supply of liquid, from which you may fill either jar.
You may also pour the liquid back from either jar into the cask.
The question is: how do you measure $4$ pints?
== Solution ==
Fill the $3$ pint jug from the cask, and empty its contents into the $5$ pint jug.
Fill the $3$ pint jug from the cask again, and fill the $5$ pint jug from it.
There will then be $1$ pint in the $3$ pint jug.
Empty the $5$ pint jug into the cask.
Pour the $1$ pint from the $3$ pint jug into the $5$ pint jug, which now contains that $1$ pint of liquid.
Fill the $3$ pint jug from the cask, and pour its contents into the $5$ pint jug.
The $5$ pint jug now contains the required $4$ pints.
== Historical Note ==
== Sources ==
* 
* : Liquid Pouring: $97$
Category:Classic Problems
Category:Liquid Pouring Problems
4 Pints from 5 Pints and 3 Pints/Historical Note

== Historical Note on $4$ Pints from $5$ Pints and $3$ Pints ==
This problem appears for the first time in of $1484$, by .
It has appeared in anthologies of puzzles and problems of recreational mathematics countless times since.
== Sources ==
* : Liquid Pouring: $97$
Category:Historical Notes
4 Positive Integers in Arithmetic Sequence which have Same Euler Phi Value

== Theorem ==
The following sets of $4$ positive integers which form an arithmetic sequence are the smallest which all have the same Euler $\phi$ value:
:$72, 78, 84, 90$
:$216, 222, 228, 234$
:$76 \, 326, 76 \, 332, 76 \, 338, 76 \, 344$
== Proof ==
demonstrating that this is indeed an arithmetic sequence, with a common difference of $6$.
Now we show:
demonstrating that this is indeed an arithmetic sequence, with a common difference of $6$.
Now we show:
demonstrating that this is indeed an arithmetic sequence, with a common difference of $6$.
Now we show:
== Sources ==
* 
* : $72$
* : $72$
Category:Euler Phi Function
4 Positive Integers in Arithmetic Sequence which have Same Euler Phi Value/Mistake

== Source Work ==
:
:The Dictionary
:: $72$
:
:The Dictionary
:: $72$
== Mistake ==
:$\map \phi {72} = \map \phi {78} = \map \phi {84} = \map \phi {90} = 24$. This is the smallest set of four numbers in arithmetical progression whose $\phi$ values are equal. The next two $4$-term arithmetical progressions with equal $\phi$ values start at $216$ and $76,236$ and each has a common difference, $6$.
== Correction ==
The last one of those should say $76,326$.
== Sources ==
* : $72$
* : $72$
Category:Mistakes/Curious and Interesting Numbers
4 Sine Pi over 10 by Cosine Pi over 5

== Theorem ==
:$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
Category:4 Sine Pi over 10 by Cosine Pi over 5
Category:Sine Function
Category:Cosine Function
4 Sine Pi over 10 by Cosine Pi over 5/Proof 2

== Theorem ==
== Proof ==
Rewrite the :
We then solve for $\theta = \dfrac \pi {10}$:
Having realized that the above process is the same as the derivation of the expansion formula for $\cos {5 \theta}$, we can generalize this expression and then plug in $\dfrac \pi {10}$.
We were able to divide off the $u$ from both sides of that last line because we know that $\cos \dfrac \pi {10}$ cannot possibly equal $0$. From here, we use the quadratic formula to solve for $u^2$.
Taking the square root of both sides, we can solve for $u$.
We know that $\dfrac \pi {10}$ is in the first quadrant, so we can change the exterior $±$ sign to a $+$ sign:
We know that $\dfrac \pi {10}$ is a lot closer to $0$ than to $\dfrac \pi 2$:
$\dfrac \pi {10} - 0 < \dfrac \pi 2 - \dfrac \pi {10}$
Therefore, the larger of the possible interior quantities of the radical should be correct.
Now that we have evaluated $\cos \dfrac \pi {10}$, we can substitute it for the most simplified expression from above with only $\cos \dfrac \pi {10}$:
Category:4 Sine Pi over 10 by Cosine Pi over 5
5,134,240

== Number ==
$5 \, 134 \, 240$ (five million, one hundred and thirty-four thousand, two hundred and forty) is:
:$2^5 \times 5 \times 32 \, 089$
:The largest positive integer that cannot be expressed as the sum of distinct $4$th powers
== Also see ==
* Largest Integer Not Expressible as Sum of Distinct 4th Powers
== Sources ==
* : $5,134,240$
Category:Specific Numbers
Category:5,134,240
5,391,411,025

== Number ==
$5 \, 391 \, 411 \, 025$ is:
:$5^2 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29$
:The smallest odd abundant number not divisible by $3$:
::$\map {\sigma_1} {5 \, 391 \, 411 \, 025} = 10 \, 799 \, 308 \, 800 = 2 \times 5 \, 391 \, 411 \, 025 + 16 \, 486 \, 750$
== Also see ==
* Smallest Odd Abundant Number not Divisible by 3
== Sources ==
* : $5,391,411,025$
Category:Specific Numbers
Category:5,391,411,025
5,442,889

== Number ==
$5 \, 442 \, 889$ (five million, four hundred and forty-two thousand, eight hundred and eighty-nine) is:
:$2333^2$
:The $2333$rd square number:
::$5 \, 442 \, 889 = 2333 \times 2333$
== Also see ==
* 
== Sources ==
* : $2333$
* : $2333$
Category:Specific Numbers
Category:5,442,889
5,500,002,244

== Number ==
$5 \, 500 \, 002 \, 244$:
:$2^2 \times 11^2 \times 3371^2$
:The $74 \, 162$nd square number:
::$5 \, 500 \, 002 \, 244 = 74 \, 162 \times 74 \, 162$
:Has the property that it is a square whose digits occur in $5$ adjacent pairs:
::$74 \, 162^2 = 5 \, 500 \, 002 \, 244$
== Also see ==
* 
== Sources ==
* : $74,162$
Category:Specific Numbers
Category:5,500,002,244
5,559,060,566,555,523

== Number ==
$5 \, 559 \, 060 \, 566 \, 555 \, 523$ is:
:$3^{33}$
:The $33$rd power of $3$:
::$5 \, 559 \, 060 \, 566 \, 555 \, 523 = 3^{33}$
:The $177 \, 147$th cube number:
::$5 \, 559 \, 060 \, 566 \, 555 \, 523 = 177 \, 147 \times 177 \, 147 \times 177 \, 147$
== Also see ==
* Properties of $5\,559\,060\,566\,555\,523$
* 
* 
== Sources ==
* : $5,559,060,566,555,523$
Category:Specific Numbers
Category:5,559,060,566,555,523
5,704,689,200,685,129,054,721

== Number ==
$5 \, 704 \, 689 \, 200 \, 685 \, 129 \, 054 \, 721$ is:
:The $n$th prime, where $n$ remains to be determined
:One of the two prime factors of the $7$th Fermat number $2^{\left({2^7}\right)} + 1$, the other being $59 \, 649 \, 589 \, 127 \, 497 \, 217$:
::$5 \, 704 \, 689 \, 200 \, 685 \, 129 \, 054 \, 721 = 11 \, 141 \, 971 \, 095 \, 088 \, 142 \, 685 \times 2^9 + 1$
== Also see ==
* 
Category:Specific Numbers
Category:5,704,689,200,685,129,054,721
5,740,816

== Number ==
$5 \, 740 \, 816$ (five million, seven hundred and forty thousand, eight hundred and sixteen) is:
:$2^4 \times 599^2$
:The $4$th after $1 \, 048 \, 576$, $1 \, 056 \, 784$, $1 \, 085 \, 764$ of a sequence of $7$ numbers which are square and anagrams of each other
:The $2396$th square number:
::$5 \, 740 \, 816 = 2396 \times 2396$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5,740,816
5,761,455

== Number ==
$5 \, 761 \, 455$ (five million, seven hundred and sixty-one thousand, four hundred and fifty-five) is:
:$3 \times 5 \times 7 \times 37 \times 1483$
:The number of primes with no more than $8$ digits:
::$2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $\ldots$, $99 \, 999 \, 839$, $99 \, 999 \, 847$, $99 \, 999 \, 931$, $99 \, 999 \, 941$, $99 \, 999 \, 959$, $99 \, 999 \, 971$, $99 \, 999 \, 989$
== Also see ==
* 
== Sources ==
* : $5,761,455$
Category:Specific Numbers
Category:5,761,455
5,764,801

== Number ==
$5 \, 764 \, 801$ (five million, seven hundred and sixty-four thousand, eight hundred and one) is:
:$7^8$
:The $5$th after $1 \, 048 \, 576$, $1 \, 056 \, 784$, $1 \, 085 \, 764$, $5 \, 740 \, 816$ of a sequence of $7$ numbers which are square and anagrams of each other
:The $7$th eighth power after $1$, $256$, $6561$, $65 \, 536$, $390 \, 625$, $1 \, 679 \, 616$:
::$5 \, 764 \, 801 = 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7$
:The $8$th power of $7$ after $(1)$, $7$, $49$, $343$, $2401$, $16 \, 807$, $117 \, 649$, $823 \, 543$:
::$5 \, 764 \, 801 = 7^8$
:The $49$th fourth power:
::$5 \, 764 \, 801 = 49 \times 49 \times 49 \times 49$
:The $2401$st square number:
::$5 \, 764 \, 801 = 2401 \times 2401$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:5,764,801
5/Historical Note

== Historical Note on $5$ ==
The number $5$ was the number associated by the Pythagoreans with marriage, the sum of $2$, the first female number, and $3$, the first male number.
According to , the Pythagoreans also associated $5$ with nature, on account of its automorphic property.
The number $5$ is expressed in Roman numerals as $\mathrm V$.
It has been suggested that this originates from the shape of one hand held up showing the full $5$ fingers.
== Sources ==
* : Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory
* : $5$
* : $50$
* : Chapter $\text {A}.2$: Pythagoras (ca. B.C.)
* : $5$
* : $50$
Category:Historical Notes
50

== Number ==
$50$ (fifty) is:
:$2 \times 5^2$
:The $1$st positive integer which can be expressed as the sum of two square numbers in two or more different ways:
::$50 = 7^2 + 1^2 = 5^2 + 5^2$
:The $2$nd term of the $1$st $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
::$\tuple {49, 50, 51, 52, 53}$
:The $4$th hexagonal pyramidal number after $1$, $7$, $22$:
::$50 = 1 + 6 + 15 + 28$
:The $4$th noncototient after $10$, $26$, $34$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 50$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $5$th nontotient after $14$, $26$, $34$, $38$:
::$\nexists m \in \Z_{>0}: \map \phi m = 50$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $31$st positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $33$, $37$, $38$, $42$, $43$, $44$, $45$, $46$, $49$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $34$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $37$, $38$, $42$, $43$, $44$, $48$, $49$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Historical Note ==
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $50$
* : $50$
Category:Specific Numbers
Category:50
50,625

== Number ==
$50 \, 625$ (fifth thousand, six hundred and twenty-five) is:
:$3^4 \times 5^4$
:The $1$st fourth power which can be expressed as the sum of $5$ fourth powers:
::$50 \, 625 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$
:The $4$th power of $15$ after $(1)$, $15$, $225$, $3375$:
::$50 \, 625 = 15^4$
:The $15$th fourth power after $1$, $16$, $81$, $256$, $625$, $1296$, $2401$, $4096$, $6561$, $10 \, 000$, $14 \, 641$, $20 \, 736$, $28 \, 561$, $38 \, 416$:
::$50 \, 625 = 15 \times 15 \times 15 \times 15$
:The $225$th square number:
::$50 \, 625 = 225 \times 225$
== Also see ==
* Smallest Fourth Power as Sum of 5 Distinct Fourth Powers
* 
* 
* 
== Sources ==
* : $50,625$
* : $50,625$
Category:Specific Numbers
Category:50,625
50,847,534

== Number ==
$50 \, 847 \, 534$ is:
:$2 \times 3^3 \times 19 \times 49 \, 559$
:The number of primes with no more than $9$ digits
== Also see ==
* 
== Sources ==
* : $\text I$: The Series of Primes: $1.4$ The sequence of primes
:::although they give the number as $50 \, 847 \, 478$ -- more have been discovered since the $5$th edition was published in $1979$.
* : $50,847,534$
Category:Specific Numbers
Category:50,847,534
500

== Number ==
$500$ (five hundred) is:
: $2^2 \times 5^3$
:The $38$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $289$, $324$, $343$, $361$, $392$, $400$, $432$, $441$, $484$:
::$500 = 2^2 \times 5^3$
== Also see ==
* 
* 
== Historical Note ==
Category:Specific Numbers
Category:500
501

== Number ==
$501$ (five hundred and one) is:
:$3 \times 167$
:The $19$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $24$, $25$, $49$, $51$, $75$, $76$, $99$, $125$, $249$, $251$, $375$, $376$, $499$:
::$501^3 = 125 \, 751 \, \mathbf {501}$
== Also see ==
* 
Category:Specific Numbers
Category:501
5020

== Number ==
$5020$ (five thousand and twenty) is:
:$2^2 \times 5 \times 251$
:The smaller of the $4$th amicable pair, with $5564$:
::$\map {\sigma_1} {5020} = \map {\sigma_1} {5564} = 10 \, 584 = 5020 + 5564$
== Also see ==
* 
== Sources ==
* : $5020$
* : $5020$
Category:Specific Numbers
Category:5020
503

== Number ==
$503$ (five hundred and three) is:
:The $96$th prime number
:The $18$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$, $467$, $479$:
::$503 = 2 \times 251 + 1$
:The $36$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $383$, $389$, $419$, $433$, $461$, $487$, $491$, $499$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:503
504

== Number ==
$504$ (five hundred and four) is:
:$2^3 \times 3^2 \times 7$
:The $1$st positive integer 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:
::$504 = 12 \times 42 = 21 \times 24$
:The $24$th positive integer after $64$, $96$, $128$, $144$, $\ldots$, $384$, $400$, $416$, $432$, $448$, $480$, $504$ with $6$ or more prime factors:
::$504 = 2 \times 2 \times 2 \times 3 \times 3 \times 7$
:The $37$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $120$, $144$, $168$, $180$, $210$, $216$, $240$, $288$, $300$, $336$, $360$, $420$, $480$:
::$\map {\sigma_1} {504} = 1560$
== Also see ==
* 
* 
* 
== Sources ==
* : $\S 13$: The fundamental theorem of arithmetic
* : $504$
* : $504$
Category:Specific Numbers
Category:504
5040

== Number ==
$5040$ (five thousand and forty) is:
:$2^4 \times 3^2 \times 5 \times 7$
:The product of consecutive integers in $2$ different ways:
::$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$
:The $7$th factorial after $1$, $2$, $6$, $24$, $120$, $720$:
::$5040 = 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
:The $19$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$, $1680$, $2520$:
::$\map {\sigma_0} {5040} = 60$
:The $19$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$, $1680$, $2520$:
::$\dfrac {\map {\sigma_1} {5040} } {5040} = \dfrac {19 \, 344} {5040} \approx 3 \cdotp 838$
== Also see ==
* 5040 is Product of Consecutive Numbers in Two Ways
* 
* 
* 
== Historical Note ==
== Sources ==
* : $5040$
* : $5040$
Category:Specific Numbers
Category:5040
5040/Historical Note

== Historical Note on $5040$ ==
The philosopher decided that the exact number of citizens suitable for his ideal city was $5040$.
His reasons included:
:$5040$ has $59$ divisors excluding itself
:Can be divided by all numbers from $1$ to $10$ and so can be assembled for various wartime or peacetime collective activities into so many equal teams
:Subtracting two hearths (that is, people) from the total, you get $5038$, which is divisible by $11$ as well.
::::-- 's Laws: $738$, $741$, $747$, $771$, $878$
In the science of campanology, a complete sequence of Stedman triples contains $5040$ changes, and takes between $3$ and $4$ hours to accomplish.
== Sources ==
* : $5040$
* : Chapter $\text {A}.2$: Pythagoras (ca. B.C.): Footnote $5$
* : $5040$
Category:Historical Notes