Editing Factorial (section)

===Divisibility and digits===
{{main|Legendre's formula}}
The product formula for the factorial implies that <math>n!</math> is [[divisible]] by all [[prime number]]s that are at {{nowrap|most <math>n</math>,}} and by no larger prime numbers.<ref name=beiler>{{cite book|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|series=Dover Recreational Math Series|first=Albert H.|last=Beiler|publisher=Courier Corporation|year=1966|edition=2nd|isbn=978-0-486-21096-4|page=49|url=https://books.google.com/books?id=NbbbL9gMJ88C&pg=PA49}}</ref> More precise information about its divisibility is given by [[Legendre's formula]], which gives the exponent of each prime <math>p</math> in the prime factorization of <math>n!</math> as<ref>{{harvnb|Chvátal|2021}}. "1.4: Legendre's formula". pp. 6–7.</ref><ref name=padic>{{cite book | last = Robert | first = Alain M. | author-link = Alain M. Robert | contribution = 3.1: The {{nowrap|<math>p</math>-adic}} valuation of a factorial | doi = 10.1007/978-1-4757-3254-2 | isbn = 0-387-98669-3 | mr = 1760253 | pages = 241–242 | publisher = Springer-Verlag | location = New York | series = [[Graduate Texts in Mathematics]] | title = A Course in {{nowrap|<math>p</math>-adic}} Analysis | volume = 198 | year = 2000}}</ref>
<math display=block>\sum_{i=1}^\infty \left \lfloor \frac n {p^i} \right \rfloor=\frac{n - s_p(n)}{p - 1}.</math>
Here <math>s_p(n)</math> denotes the sum of the {{nowrap|[[radix|base]]-<math>p</math>}} digits {{nowrap|of <math>n</math>,}} and the exponent given by this formula can also be interpreted in advanced mathematics as the [[p-adic valuation|{{mvar|p}}-adic valuation]] of the factorial.<ref name=padic/> Applying Legendre's formula to the product formula for [[binomial coefficient]]s produces [[Kummer's theorem]], a similar result on the exponent of each prime in the factorization of a binomial coefficient.<ref>{{cite book | last1 = Peitgen | author1-link=Heinz-Otto Peitgen | first1 = Heinz-Otto | last2 = Jürgens | first2 = Hartmut | author2-link = Hartmut Jürgens | last3 = Saupe | first3 = Dietmar | author3-link = Dietmar Saupe | contribution = Kummer's result and Legendre's identity | doi = 10.1007/b97624 | location = New York | pages = 399–400 | publisher = Springer | title = Chaos and Fractals: New Frontiers of Science | year = 2004| isbn=978-1-4684-9396-2 }}</ref> Grouping the prime factors of the factorial into [[prime power]]s in different ways produces the [[multiplicative partitions of factorials]].<ref>{{Cite journal|last1=Alladi|first1=Krishnaswami|last2=Grinstead|first2=Charles|authorlink1=Krishnaswami Alladi |title=On the decomposition of n! into prime powers|journal=[[Journal of Number Theory]]|year=1977 |language=en|volume=9|issue=4|pages=452–458|doi=10.1016/0022-314x(77)90006-3|doi-access=free}}</ref>

The special case of Legendre's formula for <math>p=5</math> gives the number of [[trailing zero#Factorial|trailing zeros]] in the decimal representation of the factorials.<ref name=koshy/> According to this formula, the number of zeros can be obtained by subtracting the base-5 digits of <math>n</math> from <math>n</math>, and dividing the result by four.<ref>{{cite OEIS|A027868|Number of trailing zeros in n!; highest power of 5 dividing n!}}</ref> Legendre's formula implies that the exponent of the prime <math>p=2</math> is always larger than the exponent for {{nowrap|<math>p=5</math>,}} so each factor of five can be paired with a factor of two to produce one of these trailing zeros.<ref name=koshy>{{cite book|title=Elementary Number Theory with Applications|first=Thomas|last=Koshy|edition=2nd|publisher=Elsevier|year=2007|isbn=978-0-08-054709-1|contribution=Example 3.12|page=178|contribution-url=https://books.google.com/books?id=d5Z5I3gnFh0C&pg=PA178}}</ref> The leading digits of the factorials are distributed according to [[Benford's law]].<ref>{{cite journal | last = Diaconis | first = Persi | author-link = Persi Diaconis | doi = 10.1214/aop/1176995891 | issue = 1 | journal = [[Annals of Probability]] | mr = 422186 | pages = 72–81 | title = The distribution of leading digits and uniform distribution mod 1 | volume = 5 | year = 1977| doi-access = free }}</ref> Every sequence of digits, in any base, is the sequence of initial digits of some factorial number in that base.<ref>{{cite journal|last=Bird|first=R. S.|author-link=Richard Bird (computer scientist)|doi=10.1080/00029890.1972.11993051|journal=[[The American Mathematical Monthly]]|jstor=2978087|mr=302553|pages=367–370|title=Integers with given initial digits|volume=79|year=1972|issue=4}}</ref>

Another result on divisibility of factorials, [[Wilson's theorem]], states that <math>(n-1)!+1</math> is divisible by <math>n</math> if and only if <math>n</math> is a [[prime number]].<ref name=beiler/> For any given {{nowrap|integer <math>x</math>,}} the [[Kempner function]] of <math>x</math> is given by the smallest <math>n</math> for which <math>x</math> divides {{nowrap|<math>n!</math>.<ref>{{cite journal | jstor = 2972639 | first = A. J. | last = Kempner | title = Miscellanea | journal = [[The American Mathematical Monthly]] | volume = 25 | pages = 201–210 | year = 1918 | doi = 10.2307/2972639 | issue = 5}}</ref>}} For almost all numbers (all but a subset of exceptions with [[asymptotic density]] zero), it coincides with the largest prime factor {{nowrap|of <math>x</math>.<ref>{{cite journal|title=The smallest factorial that is a multiple of {{mvar|n}} (solution to problem 6674)|journal=[[The American Mathematical Monthly]]|volume=101|year=1994|page=179|url=http://www-fourier.ujf-grenoble.fr/~marin/une_autre_crypto/articles_et_extraits_livres/irationalite/Erdos_P._Kastanas_I.The_smallest_factorial...-.pdf|first1=Paul|last1=Erdős|author1-link=Paul Erdős|first2=Ilias|last2=Kastanas|doi=10.2307/2324376|jstor=2324376}}.</ref>}}

The product of two factorials, {{nowrap|<math>m!\cdot n!</math>,}} always evenly divides {{nowrap|<math>(m+n)!</math>.<ref name=bhargava/>}} There are infinitely many factorials that equal the product of other factorials: if <math>n</math> is itself any product of factorials, then <math>n!</math> equals that same product multiplied by one more factorial, {{nowrap|<math>(n-1)!</math>.}} The only known examples of factorials that are products of other factorials but are not of this "trivial" form are {{nowrap|<math>9!=7!\cdot 3!\cdot 3!\cdot 2!</math>,}} {{nowrap|<math>10!=7!\cdot 6!=7!\cdot 5!\cdot 3!</math>,}} and {{nowrap|<math>16!=14!\cdot 5!\cdot 2!</math>.<ref>{{harvnb|Guy|2004}}. "B23: Equal products of factorials". p. 123.</ref>}} It would follow from the [[abc conjecture|{{mvar|abc}} conjecture]] that there are only finitely many nontrivial examples.<ref>{{cite journal | last = Luca | first = Florian | author-link = Florian Luca | doi = 10.1017/S0305004107000308 | issue = 3 | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]] | mr = 2373957 | pages = 533–542 | title = On factorials which are products of factorials | volume = 143 | year = 2007| bibcode = 2007MPCPS.143..533L | s2cid = 120875316 }}</ref>

The [[greatest common divisor]] of the values of a [[Primitive part and content|primitive polynomial]] of degree <math>d</math> over the integers evenly divides {{nowrap|<math>d!</math>.<ref name=bhargava>{{cite journal | last = Bhargava | first = Manjul | author-link = Manjul Bhargava | url = https://scholar.archive.org/work/dk6exbnlyrhp3bai62vnokou2q | title = The factorial function and generalizations | journal = [[The American Mathematical Monthly]] | volume = 107 | year = 2000 | pages = 783–799 | doi = 10.2307/2695734 | issue = 9 | jstor = 2695734 | citeseerx = 10.1.1.585.2265 }}</ref>}}