:Factorial

The concept of factorials has arisen independently in many cultures:

• In Indian mathematics, one of the earliest known descriptions of factorials comes from the Anuyogadvāra-sūtra, one of the canonical works of Jain literature, which has been assigned dates varying from 300 BCE to 400 CE.{{cite journal | last = Jadhav | first = Dipak | date = August 2021 | doi = 10.18732/hssa67 | journal = History of Science in South Asia | pages = 209–231 | publisher = University of Alberta Libraries | title = Jaina Thoughts on Unity Not Being a Number | volume = 9| s2cid = 238656716 | doi-access = free }}. See discussion of dating on p. 211. It separates out the sorted and reversed order of a set of items from the other ("mixed") orders, evaluating the number of mixed orders by subtracting two from the usual product formula for the factorial. The product rule for permutations was also described by 6th-century CE Jain monk Jinabhadra.{{cite book | last1 = Datta | first1 = Bibhutibhusan | author1-link = Bibhutibhushan Datta | last2 = Singh | first2 = Awadhesh Narayan | editor1-last = Kolachana | editor1-first = Aditya | editor2-last = Mahesh | editor2-first = K. | editor3-last = Ramasubramanian | editor3-first = K. | contribution = Use of permutations and combinations in India | doi = 10.1007/978-981-13-7326-8_18 | pages = 356–376 | publisher = Springer Singapore | series = Sources and Studies in the History of Mathematics and Physical Sciences | title = Studies in Indian Mathematics and Astronomy: Selected Articles of Kripa Shankar Shukla | year = 2019| isbn = 978-981-13-7325-1 | s2cid = 191141516 }}. Revised by K. S. Shukla from a paper in Indian Journal of History of Science 27 (3): 231–249, 1992, {{MR|1189487}}. See p. 363. Hindu scholars have been using factorial formulas since at least 1150, when Bhāskara II mentioned factorials in his work Līlāvatī, in connection with a problem of how many ways Vishnu could hold his four characteristic objects (a conch shell, discus, mace, and lotus flower) in his four hands, and a similar problem for a ten-handed god.{{Cite journal |last=Biggs |first=Norman L. |author-link=Norman L. Biggs |date=May 1979 |title=The roots of combinatorics |journal=Historia Mathematica |volume=6 |issue=2 |pages=109–136 |doi=10.1016/0315-0860(79)90074-0 |doi-access= | mr = 0530622 }}
• In the mathematics of the Middle East, the Hebrew mystic book of creation Sefer Yetzirah, from the Talmudic period (200 to 500 CE), lists factorials up to 7! as part of an investigation into the number of words that can be formed from the Hebrew alphabet.{{cite journal | last = Katz | first = Victor J. | author-link = Victor J. Katz | date = June 1994 | issue = 2 | journal = For the Learning of Mathematics | jstor = 40248112 | pages = 26–30 | title = Ethnomathematics in the classroom | volume = 14}}[https://en.wikisource.org/wiki/Sefer_Yetzirah#CHAPTER_IV Sefer Yetzirah at Wikisource], Chapter IV, Section 4 Factorials were also studied for similar reasons by 8th-century Arab grammarian Al-Khalil ibn Ahmad al-Farahidi. Arab mathematician Ibn al-Haytham (also known as Alhazen, c. 965 – c. 1040) was the first to formulate Wilson's theorem connecting the factorials with the prime numbers.{{cite journal | last = Rashed | first = Roshdi | author-link = Roshdi Rashed | doi = 10.1007/BF00717654 | issue = 4 | journal = Archive for History of Exact Sciences | language = fr | mr = 595903 | pages = 305–321 | title = Ibn al-Haytham et le théorème de Wilson | volume = 22 | year = 1980| s2cid = 120885025 }}
• In Europe, although Greek mathematics included some combinatorics, and Plato famously used 5,040 (a factorial) as the population of an ideal community, in part because of its divisibility properties,{{cite journal | last = Acerbi | first = F. | doi = 10.1007/s00407-003-0067-0 | issue = 6 | journal = Archive for History of Exact Sciences | jstor = 41134173 | mr = 2004966 | pages = 465–502 | title = On the shoulders of Hipparchus: a reappraisal of ancient Greek combinatorics | volume = 57 | year = 2003| s2cid = 122758966 }} there is no direct evidence of ancient Greek study of factorials. Instead, the first work on factorials in Europe was by Jewish scholars such as Shabbethai Donnolo, explicating the Sefer Yetzirah passage.{{cite book|editor1-last=Wilson|editor1-first=Robin|editor2-last=Watkins|editor2-first=John J.|title=Combinatorics: Ancient & Modern|publisher=Oxford University Press|date=2013|isbn=978-0-19-965659-2|first=Victor J.|last=Katz|author-link=Victor J. Katz|contribution=Chapter 4: Jewish combinatorics|pages=109–121}} See p. 111. In 1677, British author Fabian Stedman described the application of factorials to change ringing, a musical art involving the ringing of several tuned bells.{{cite journal | last = Hunt | first = Katherine | date = May 2018 | doi = 10.1215/10829636-4403136 | issue = 2 | journal = Journal of Medieval and Early Modern Studies | pages = 387–412 | title = The Art of Changes: Bell-Ringing, Anagrams, and the Culture of Combination in Seventeenth-Century England | volume = 48| url = https://ueaeprints.uea.ac.uk/id/eprint/83227/1/Accepted_Mnauscript.pdf }}{{cite book|last=Stedman|first=Fabian|author-link=Fabian Stedman|title=Campanalogia|year=1677|place=London|pages=6–9}} The publisher is given as "W.S." who may have been William Smith, possibly acting as agent for the Society of College Youths, to which society the "Dedicatory" is addressed.

From the late 15th century onward, factorials became the subject of study by Western mathematicians. In a 1494 treatise, Italian mathematician Luca Pacioli calculated factorials up to 11!, in connection with a problem of dining table arrangements.{{cite book|editor1-last=Wilson|editor1-first=Robin|editor2-last=Watkins|editor2-first=John J.|title=Combinatorics: Ancient & Modern|publisher=Oxford University Press|date=2013|isbn=978-0-19-965659-2|first=Eberhard|last=Knobloch|author-link=Eberhard Knobloch|contribution=Chapter 5: Renaissance combinatorics|pages=123–145}} See p. 126. Christopher Clavius discussed factorials in a 1603 commentary on the work of Johannes de Sacrobosco, and in the 1640s, French polymath Marin Mersenne published large (but not entirely correct) tables of factorials, up to 64!, based on the work of Clavius.{{sfn|Knobloch|2013|pages=130–133}} The power series for the exponential function, with the reciprocals of factorials for its coefficients, was first formulated in 1676 by Isaac Newton in a letter to Gottfried Wilhelm Leibniz.{{cite book | last1 = Ebbinghaus | first1 = H.-D. | author1-link = Heinz-Dieter Ebbinghaus | last2 = Hermes | first2 = H. | author2-link = Hans Hermes | last3 = Hirzebruch | first3 = F. | author3-link = Friedrich Hirzebruch | last4 = Koecher | first4 = M. | author4-link = Max Koecher | last5 = Mainzer | first5 = K. | author5-link = Klaus Mainzer | last6 = Neukirch | first6 = J. | author6-link = Jürgen Neukirch | last7 = Prestel | first7 = A. | last8 = Remmert | first8 = R. | author8-link = Reinhold Remmert | doi = 10.1007/978-1-4612-1005-4 | isbn = 0-387-97202-1 | mr = 1066206 | page = 131 | publisher = Springer-Verlag | location = New York | series = Graduate Texts in Mathematics | title = Numbers | url = https://books.google.com/books?id=Z53SBwAAQBAJ&pg=PA131 | volume = 123 | year = 1990}} Other important works of early European mathematics on factorials include extensive coverage in a 1685 treatise by John Wallis, a study of their approximate values for large values of $n$ by Abraham de Moivre in 1721, a 1729 letter from James Stirling to de Moivre stating what became known as Stirling's approximation, and work at the same time by Daniel Bernoulli and Leonhard Euler formulating the continuous extension of the factorial function to the gamma function.{{cite journal | last = Dutka | first = Jacques | doi = 10.1007/BF00389433 | issue = 3 | journal = Archive for History of Exact Sciences | jstor = 41133918 | mr = 1171521 | pages = 225–249 | title = The early history of the factorial function | volume = 43 | year = 1991| s2cid = 122237769 }} Adrien-Marie Legendre included Legendre's formula, describing the exponents in the factorization of factorials into prime powers, in an 1808 text on number theory.{{cite book|first=Leonard E.|last=Dickson|author-link=Leonard Eugene Dickson|title=History of the Theory of Numbers|title-link=History of the Theory of Numbers|volume=1|publisher=Carnegie Institution of Washington|year=1919|contribution=Chapter IX: Divisibility of factorials and multinomial coefficients|pages=263–278|contribution-url=https://archive.org/details/historyoftheoryo01dick/page/262}} See in particular p. 263.

The notation $n!$ for factorials was introduced by the French mathematician Christian Kramp in 1808. Many other notations have also been used. Another later notation $\backslash vert\backslash !\backslash underline\{\backslash ,n\}$, in which the argument of the factorial was half-enclosed by the left and bottom sides of a box, was popular for some time in Britain and America but fell out of use, perhaps because it is difficult to typeset.{{cite book | last = Cajori | first = Florian | author-link = Florian Cajori | contribution = 448–449. Factorial "{{mvar|n}}" | contribution-url = https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n93 | pages = 71–77 | publisher = The Open Court Publishing Company | title = A History of Mathematical Notations, Volume II: Notations Mainly in Higher Mathematics | title-link = A History of Mathematical Notations | year = 1929}} The word "factorial" (originally French: factorielle) was first used in 1800 by Louis François Antoine Arbogast,{{cite web|url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/f/|title=Earliest Known Uses of Some of the Words of Mathematics (F)|work=MacTutor History of Mathematics archive|publisher=University of St Andrews|first=Jeff|last=Miller}} in the first work on Faà di Bruno's formula,{{cite journal | last = Craik | first = Alex D. D. | doi = 10.1080/00029890.2005.11920176 | issue = 2 | journal = The American Mathematical Monthly | jstor = 30037410 | mr = 2121322 | pages = 119–130 | title = Prehistory of Faà di Bruno's formula | volume = 112 | year = 2005| s2cid = 45380805 }} but referring to a more general concept of products of arithmetic progressions. The "factors" that this name refers to are the terms of the product formula for the factorial.{{cite book|title=Du calcul des dérivations|last=Arbogast|first=Louis François Antoine|author-link=Louis François Antoine Arbogast|publisher=L'imprimerie de Levrault, frères|location=Strasbourg|year=1800|pages=364–365|url=https://archive.org/details/ducalculdesdri00arbouoft/page/364|language=fr}}

The factorial function of a positive integer $n$ is defined by the product of all positive integers not greater than $n$

$$n!\; =\; 1\; \backslash cdot\; 2\; \backslash cdot\; 3\; \backslash cdots\; (n-2)\; \backslash cdot\; (n-1)\; \backslash cdot\; n.$$

This may be written more concisely in product notation as

$$n!\; =\; \backslash prod\_\{i\; =\; 1\}^n\; i.$$

If this product formula is changed to keep all but the last term, it would define a product of the same form, for a smaller factorial. This leads to a recurrence relation, according to which each value of the factorial function can be obtained by multiplying the previous value {{nowrap|by $n$:}}

$$n!\; =\; n\backslash cdot\; (n-1)!.$$

For example, {{nowrap|$5!\; =\; 5\backslash cdot\; 4!=5\backslash cdot\; 24=120$.}}

The factorial {{nowrap|of $0$}} {{nowrap|is $1$,}} or in symbols, {{nowrap|$0!=1$.}} There are several motivations for this definition:

• For {{nowrap|$n=0$,}} the definition of $n!$ as a product involves the product of no numbers at all, and so is an example of the broader convention that the empty product, a product of no factors, is equal to the multiplicative identity.{{cite book|title=CRC Handbook of Engineering Tables|first=Richard C.|last=Dorf|publisher=CRC Press|year=2003|page=5-5|contribution=Factorials|contribution-url=https://books.google.com/books?id=TCLOBgAAQBAJ&pg=SA5-PA5|isbn=978-0-203-00922-2}}
• There is exactly one permutation of zero objects: with nothing to permute, the only rearrangement is to do nothing.{{cite book | last = Hamkins | first = Joel David | author-link = Joel David Hamkins | isbn = 978-0-262-53979-1 | location = Cambridge, Massachusetts | mr = 4205951 | page = 50 | publisher = MIT Press | title = Proof and the Art of Mathematics | url = https://books.google.com/books?id=Ns_tDwAAQBAJ&pg=PA50 | year = 2020}}
• This convention makes many identities in combinatorics valid for all valid choices of their parameters. For instance, the number of ways to choose all $n$ elements from a set of $n$ is $\backslash tbinom\{n\}\{n\}\; =\; \backslash tfrac\{n!\}\{n!0!\}\; =\; 1,$ a binomial coefficient identity that would only be valid {{nowrap|with $0!=1$.{{cite journal | last1 = Goldenberg | first1 = E. Paul | last2 = Carter | first2 = Cynthia J. | date = October 2017 | doi = 10.5951/mathteacher.111.2.0104 | issue = 2 | journal = The Mathematics Teacher | jstor = 10.5951/mathteacher.111.2.0104 | pages = 104–110 | title = A student asks about (−5)! | volume = 111}}}}
• With {{nowrap|$0!=1$,}} the recurrence relation for the factorial remains valid {{nowrap|at $n=1$.}} Therefore, with this convention, a recursive computation of the factorial needs to have only the value for zero as a base case, simplifying the computation and avoiding the need for additional special cases.{{cite conference | last1 = Haberman | first1 = Bruria | last2 = Averbuch | first2 = Haim | editor1-last = Caspersen | editor1-first = Michael E. | editor2-last = Joyce | editor2-first = Daniel T. | editor3-last = Goelman | editor3-first = Don | editor4-last = Utting | editor4-first = Ian | contribution = The case of base cases: Why are they so difficult to recognize? Student difficulties with recursion | doi = 10.1145/544414.544441 | pages = 84–88 | publisher = Association for Computing Machinery | title = Proceedings of the 7th Annual SIGCSE Conference on Innovation and Technology in Computer Science Education, ITiCSE 2002, Aarhus, Denmark, June 24-28, 2002 | year = 2002}}
• Setting $0!=1$ allows for the compact expression of many formulae, such as the exponential function, as a power series: {{nowrap|$e^x\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; \backslash frac\{x^n\}\{n!\}.$}}
• This choice matches the gamma function {{nowrap|$0!\; =\; \backslash Gamma(0+1)\; =\; 1$,}} and the gamma function must have this value to be a continuous function.{{cite book|title=Solved Problems in Analysis: As Applied to Gamma, Beta, Legendre and Bessel Functions|series=Dover Books on Mathematics|first1=Orin J.|last1=Farrell|first2=Bertram|last2=Ross|publisher=Courier Corporation|year=1971|isbn=978-0-486-78308-6|page=10|url=https://books.google.com/books?id=fXPDAgAAQBAJ&pg=PA10}}

The earliest uses of the factorial function involve counting permutations: there are $n!$ different ways of arranging $n$ distinct objects into a sequence.{{Cite book |title=The Book of Numbers |title-link=The Book of Numbers (math book) |last1=Conway |first1=John H. |last2=Guy |first2=Richard |year=1998 |publisher=Springer Science & Business Media |isbn=978-0-387-97993-9 |language=en |author-link=John Horton Conway |author-link2=Richard K. Guy |pages=55–56|contribution=Factorial numbers}} Factorials appear more broadly in many formulas in combinatorics, to account for different orderings of objects. For instance the binomial coefficients $\backslash tbinom\{n\}\{k\}$ count the {{nowrap|$k$-element}} combinations (subsets of {{nowrap|$k$ elements)}} from a set with {{nowrap|$n$ elements,}} and can be computed from factorials using the formula{{sfn|Graham|Knuth|Patashnik|1988|p=156}} $$\backslash binom\{n\}\{k\}=\backslash frac\{n!\}\{k!(n-k)!\}.$$ The Stirling numbers of the first kind sum to the factorials, and count the permutations {{nowrap|of $n$}} grouped into subsets with the same numbers of cycles.{{cite book | last = Riordan | first = John | author-link = John Riordan (mathematician) | mr = 0096594 | page = 76 | publisher = Chapman & Hall | series = Wiley Publications in Mathematical Statistics | title = An Introduction to Combinatorial Analysis | year = 1958}} [https://books.google.com/books?id=Sbb_AwAAQBAJ&pg=PA76 Reprinted], Princeton Legacy Library, Princeton University Press, 2014, {{isbn|9781400854332}}. Another combinatorial application is in counting derangements, permutations that do not leave any element in its original position; the number of derangements of $n$ items is the nearest integer {{nowrap|to $n!/e$.{{sfn|Graham|Knuth|Patashnik|1988|p=195}}}}

In algebra, the factorials arise through the binomial theorem, which uses binomial coefficients to expand powers of sums.{{sfn|Graham|Knuth|Patashnik|1988|p=162}} They also occur in the coefficients used to relate certain families of polynomials to each other, for instance in Newton's identities for symmetric polynomials.{{cite journal | last = Randić | first = Milan | doi = 10.1007/BF01205340 | issue = 1 | journal = Journal of Mathematical Chemistry | mr = 895533 | pages = 145–152 | title = On the evaluation of the characteristic polynomial via symmetric function theory | volume = 1 | year = 1987| s2cid = 121752631 }} Their use in counting permutations can also be restated algebraically: the factorials are the orders of finite symmetric groups.{{cite book|title=Groups and Characters|first=Victor E.|last=Hill|publisher=Chapman & Hall|year=2000|mr=1739394|isbn=978-1-351-44381-4|page=70|contribution=8.1 Proposition: Symmetric group {{math|S_n}}|contribution-url=https://books.google.com/books?id=yjL3DwAAQBAJ&pg=PA70}} In calculus, factorials occur in Faà di Bruno's formula for chaining higher derivatives. In mathematical analysis, factorials frequently appear in the denominators of power series, most notably in the series for the exponential function, $$e^x=1+\backslash frac\{x\}\{1\}+\backslash frac\{x^2\}\{2\}+\backslash frac\{x^3\}\{6\}+\backslash cdots=\backslash sum\_\{i=0\}^\{\backslash infty\}\backslash frac\{x^i\}\{i!\},$$

and in the coefficients of other Taylor series (in particular those of the trigonometric and hyperbolic functions), where they cancel factors of $n!$ coming from the {{nowrap|$n$th derivative}} {{nowrap|of $x^n$.{{cite book|title=Complexity and Criticality|series=Advanced physics texts|volume=1|first1=Kim|last1=Christensen|first2=Nicholas R.|last2=Moloney|publisher=Imperial College Press|year=2005|isbn=978-1-86094-504-5|contribution=Appendix A: Taylor expansion|page=341|contribution-url=https://books.google.com/books?id=bAIM1_EoQu0C&pg=PA341}}}} This usage of factorials in power series connects back to analytic combinatorics through the exponential generating function, which for a combinatorial class with $n\_i$ elements of {{nowrap|size $i$}} is defined as the power series{{cite book | last = Wilf | first = Herbert S. | author-link = Herbert Wilf | edition = 3rd | isbn = 978-1-56881-279-3 | mr = 2172781 | page = 22 | publisher = A K Peters | location = Wellesley, Massachusetts | title = generatingfunctionology | url = https://books.google.com/books?id=XOPMBQAAQBAJ&pg=PA22 | year = 2006}} $$\backslash sum\_\{i=0\}^\{\backslash infty\}\; \backslash frac\{x^i\; n\_i\}\{i!\}.$$

In number theory, the most salient property of factorials is the divisibility of $n!$ by all positive integers up {{nowrap|to $n$,}} described more precisely for prime factors by Legendre's formula. It follows that arbitrarily large prime numbers can be found as the prime factors of the numbers

$n!\backslash pm\; 1$, leading to a proof of Euclid's theorem that the number of primes is infinite.{{cite book | last = Ore | first = Øystein | author-link = Øystein Ore | location = New York | mr = 0026059 | page = 66 | publisher = McGraw-Hill | title = Number Theory and Its History | url = https://books.google.com/books?id=Sl_6BPp7S0AC&pg=PA66 | year = 1948}} Reprinted, Courier Dover Publications, 1988, {{isbn|9780486656205}}. When $n!\backslash pm\; 1$ is itself prime it is called a factorial prime;{{cite journal | last1 = Caldwell | first1 = Chris K. | last2 = Gallot | first2 = Yves | doi = 10.1090/S0025-5718-01-01315-1 | issue = 237 | journal = Mathematics of Computation | mr = 1863013 | pages = 441–448 | title = On the primality of $n!\backslash pm1$ and $2\backslash times3\backslash times5\backslash times\backslash dots\backslash times\; p\backslash pm1$ | volume = 71 | year = 2002| doi-access = free }} relatedly, Brocard's problem, also posed by Srinivasa Ramanujan, concerns the existence of square numbers of the form {{nowrap|$n!+1$.{{cite book | last = Guy | first = Richard K. | author-link = Richard K. Guy | contribution = D25: Equations involving factorial $n$ | doi = 10.1007/978-0-387-26677-0 | edition = 3rd | isbn = 0-387-20860-7 | mr = 2076335 | pages = 301–302 | publisher = Springer-Verlag | location = New York | series = Problem Books in Mathematics | title = Unsolved Problems in Number Theory | year = 2004| volume = 1 }}}} In contrast, the numbers $n!+2,n!+3,\backslash dots\; n!+n$ must all be composite, proving the existence of arbitrarily large prime gaps.{{cite book|title=Closing the Gap: The Quest to Understand Prime Numbers|first=Vicky|last=Neale|author-link=Vicky Neale|publisher=Oxford University Press|year=2017|isbn=978-0-19-878828-7|pages=146–147|url=https://books.google.com/books?id=T7Q1DwAAQBAJ&pg=PA146}} An elementary proof of Bertrand's postulate on the existence of a prime in any interval of the {{nowrap|form $[n,2n]$,}} one of the first results of Paul Erdős, was based on the divisibility properties of factorials.{{cite journal | last = Erdős | first = Pál | author-link = Paul Erdős | journal = Acta Litt. Sci. Szeged | language = de | pages = 194–198 | title = Beweis eines Satzes von Tschebyschef | trans-title = Proof of a theorem of Chebyshev | url = https://users.renyi.hu/~p_erdos/1932-01.pdf | volume = 5 | year = 1932 | zbl = 0004.10103}}{{cite book | last = Chvátal | first = Vašek | author-link = Václav Chvátal | contribution = 1.5: Erdős's proof of Bertrand's postulate | contribution-url = https://books.google.com/books?id=_gVDEAAAQBAJ&pg=PA7 | doi = 10.1017/9781108912181 | isbn = 978-1-108-83183-3 | mr = 4282416 | pages = 7–10 | publisher = Cambridge University Press | location = Cambridge, England | title = The Discrete Mathematical Charms of Paul Erdős: A Simple Introduction | year = 2021| s2cid = 242637862 }} The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials.{{cite journal | last = Fraenkel | first = Aviezri S. | author-link = Aviezri Fraenkel | doi = 10.1080/00029890.1985.11971550 | issue = 2 | journal = The American Mathematical Monthly | jstor = 2322638 | mr = 777556 | pages = 105–114 | title = Systems of numeration | volume = 92 | year = 1985}}

Factorials are used extensively in probability theory, for instance in the Poisson distribution{{cite book | last = Pitman | first = Jim | contribution = 3.5: The Poisson distribution | doi = 10.1007/978-1-4612-4374-8 | pages = 222–236 | publisher = Springer | location = New York | title = Probability | year = 1993| isbn = 978-0-387-94594-1 }} and in the probabilities of random permutations.{{sfn|Pitman|1993|p=153}} In computer science, beyond appearing in the analysis of brute-force searches over permutations,{{cite book|title=Algorithm Design|first1=Jon|last1=Kleinberg|author1-link=Jon Kleinberg|first2=Éva|last2=Tardos|author2-link=Éva Tardos|publisher=Addison-Wesley|year=2006|page=55}} factorials arise in the lower bound of $\backslash log\_2\; n!=n\backslash log\_2n-O(n)$ on the number of comparisons needed to comparison sort a set of $n$ items, and in the analysis of chained hash tables, where the distribution of keys per cell can be accurately approximated by a Poisson distribution.{{cite book|title=Algorithms|edition=4th|publisher=Addison-Wesley|first1=Robert|last1=Sedgewick|author1-link=Robert Sedgewick (computer scientist)|first2=Kevin|last2=Wayne|year=2011|isbn=978-0-13-276256-4|page=466|url=https://books.google.com/books?id=idUdqdDXqnAC&pg=PA466}} Moreover, factorials naturally appear in formulae from quantum and statistical physics, where one often considers all the possible permutations of a set of particles. In statistical mechanics, calculations of entropy such as Boltzmann's entropy formula or the Sackur–Tetrode equation must correct the count of microstates by dividing by the factorials of the numbers of each type of indistinguishable particle to avoid the Gibbs paradox. Quantum physics provides the underlying reason for why these corrections are necessary.{{cite book|first=Mehran |last=Kardar |author-link=Mehran Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher=Cambridge University Press |isbn=978-0-521-87342-0 |oclc=860391091 |pages=107–110, 181–184}}

File:Mplwp factorial stirling loglog2.svg

File:Stirling series relative error.svg in a truncated Stirling series vs. number of terms]]

{{main|Stirling's approximation}}

As a function {{nowrap|of $n$,}} the factorial has faster than exponential growth, but grows more slowly than a double exponential function.{{cite book | last = Cameron | first = Peter J. | author-link = Peter Cameron (mathematician) | contribution = 2.4: Orders of magnitude | isbn = 978-0-521-45133-8 | pages = 12–14 | publisher = Cambridge University Press | title = Combinatorics: Topics, Techniques, Algorithms | year = 1994}} Its growth rate is similar {{nowrap|to $n^n$,}} but slower by an exponential factor. One way of approaching this result is by taking the natural logarithm of the factorial, which turns its product formula into a sum, and then estimating the sum by an integral:

$$\backslash ln\; n!\; =\; \backslash sum\_\{x=1\}^n\; \backslash ln\; x\; \backslash approx\; \backslash int\_1^n\backslash ln\; x\backslash ,\; dx=n\backslash ln\; n-n+1.$$

Exponentiating the result (and ignoring the negligible $+1$ term) approximates $n!$ as {{nowrap|$(n/e)^n$.{{cite book | last = Magnus | first = Robert | contribution = 11.10: Stirling's approximation | contribution-url = https://books.google.com/books?id=5hvxDwAAQBAJ&pg=PA391 | doi = 10.1007/978-3-030-46321-2 | isbn = 978-3-030-46321-2 | location = Cham | mr = 4178171 | page = 391 | publisher = Springer | series = Springer Undergraduate Mathematics Series | title = Fundamental Mathematical Analysis | year = 2020| s2cid = 226465639 }}}}

More carefully bounding the sum both above and below by an integral, using the trapezoid rule, shows that this estimate needs a correction factor proportional {{nowrap|to $\backslash sqrt\; n$.}} The constant of proportionality for this correction can be found from the Wallis product, which expresses $\backslash pi$ as a limiting ratio of factorials and powers of two. The result of these corrections is Stirling's approximation:{{cite book | last = Palmer | first = Edgar M. | contribution = Appendix II: Stirling's formula | isbn = 0-471-81577-2 | location = Chichester | mr = 795795 | pages = 127–128 | publisher = John Wiley & Sons | series = Wiley-Interscience Series in Discrete Mathematics | title = Graphical Evolution: An introduction to the theory of random graphs | year = 1985}}

$$n!\backslash sim\backslash sqrt\{2\backslash pi\; n\}\backslash left(\backslash frac\{n\}\{e\}\backslash right)^n\backslash ,.$$

Here, the $\backslash sim$ symbol means that, as $n$ goes to infinity, the ratio between the left and right sides approaches one in the limit.

Stirling's formula provides the first term in an asymptotic series that becomes even more accurate when taken to greater numbers of terms:{{cite journal | last1 = Chen | first1 = Chao-Ping | last2 = Lin | first2 = Long | doi = 10.1016/j.aml.2012.06.025 | issue = 12 | journal = Applied Mathematics Letters | mr = 2967837 | pages = 2322–2326 | title = Remarks on asymptotic expansions for the gamma function | volume = 25 | year = 2012| doi-access = free }}

n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \left(1 +\frac{1}{12n}+\frac{1}{288n^2} - \frac{139}{51840n^3} -\frac{571}{2488320n^4}+ \cdots \right).

An alternative version uses only odd exponents in the correction terms:

n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \exp\left(\frac{1}{12n} - \frac{1}{360n^3} + \frac{1}{1260n^5} -\frac{1}{1680n^7}+ \cdots \right).

Many other variations of these formulas have also been developed, by Srinivasa Ramanujan, Bill Gosper, and others.

The binary logarithm of the factorial, used to analyze comparison sorting, can be very accurately estimated using Stirling's approximation. In the formula below, the $O(1)$ term invokes big O notation.{{cite book|title=The Art of Computer Programming, Volume 3: Sorting and Searching|first=Donald E.|last=Knuth|author-link=Donald Knuth|edition=2nd|publisher=Addison-Wesley|year=1998|isbn=978-0-321-63578-5|page=182|url=https://books.google.com/books?id=cYULBAAAQBAJ&pg=PA182}}

$$\backslash log\_2\; n!\; =\; n\backslash log\_2\; n-(\backslash log\_2\; e)n\; +\; \backslash frac12\backslash log\_2\; n\; +\; O(1).$$

{{main|Legendre's formula}}

The product formula for the factorial implies that $n!$ is divisible by all prime numbers that are at {{nowrap|most $n$,}} and by no larger prime numbers.{{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}} More precise information about its divisibility is given by Legendre's formula, which gives the exponent of each prime $p$ in the prime factorization of $n!$ as{{harvnb|Chvátal|2021}}. "1.4: Legendre's formula". pp. 6–7.{{cite book | last = Robert | first = Alain M. | author-link = Alain M. Robert | contribution = 3.1: The {{nowrap|$p$-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|$p$-adic}} Analysis | volume = 198 | year = 2000}}

$$\backslash sum\_\{i=1\}^\backslash infty\; \backslash left\; \backslash lfloor\; \backslash frac\; n\; \{p^i\}\; \backslash right\; \backslash rfloor=\backslash frac\{n\; -\; s\_p(n)\}\{p\; -\; 1\}.$$

Here $s\_p(n)$ denotes the sum of the {{nowrap|base-$p$}} digits {{nowrap|of $n$,}} and the exponent given by this formula can also be interpreted in advanced mathematics as the p-adic valuation of the factorial. Applying Legendre's formula to the product formula for binomial coefficients produces Kummer's theorem, a similar result on the exponent of each prime in the factorization of a binomial coefficient.{{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 }} Grouping the prime factors of the factorial into prime powers in different ways produces the multiplicative partitions of factorials.{{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}}

The special case of Legendre's formula for $p=5$ gives the number of trailing zeros in the decimal representation of the factorials. According to this formula, the number of zeros can be obtained by subtracting the base-5 digits of $n$ from $n$, and dividing the result by four.{{cite OEIS|A027868|Number of trailing zeros in n!; highest power of 5 dividing n!}} Legendre's formula implies that the exponent of the prime $p=2$ is always larger than the exponent for {{nowrap|$p=5$,}} so each factor of five can be paired with a factor of two to produce one of these trailing zeros.{{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}} The leading digits of the factorials are distributed according to Benford's law.{{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 }} Every sequence of digits, in any base, is the sequence of initial digits of some factorial number in that base.{{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}}

Another result on divisibility of factorials, Wilson's theorem, states that $(n-1)!+1$ is divisible by $n$ if and only if $n$ is a prime number. For any given {{nowrap|integer $x$,}} the Kempner function of $x$ is given by the smallest $n$ for which $x$ divides {{nowrap|$n!$.{{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}}}} For almost all numbers (all but a subset of exceptions with asymptotic density zero), it coincides with the largest prime factor {{nowrap|of $x$.{{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}}.}}

The product of two factorials, {{nowrap|$m!\backslash cdot\; n!$,}} always evenly divides {{nowrap|$(m+n)!$.}} There are infinitely many factorials that equal the product of other factorials: if $n$ is itself any product of factorials, then $n!$ equals that same product multiplied by one more factorial, {{nowrap|$(n-1)!$.}} The only known examples of factorials that are products of other factorials but are not of this "trivial" form are {{nowrap|$9!=7!\backslash cdot\; 3!\backslash cdot\; 3!\backslash cdot\; 2!$,}} {{nowrap|$10!=7!\backslash cdot\; 6!=7!\backslash cdot\; 5!\backslash cdot\; 3!$,}} and {{nowrap|$16!=14!\backslash cdot\; 5!\backslash cdot\; 2!$.{{harvnb|Guy|2004}}. "B23: Equal products of factorials". p. 123.}} It would follow from the abc conjecture that there are only finitely many nontrivial examples.{{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 }}

The greatest common divisor of the values of a primitive polynomial of degree $d$ over the integers evenly divides {{nowrap|$d!$.{{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 }}}}

File:Generalized factorial function more infos.svg

File:Gamma abs 3D.png

{{Main|Gamma function}}

There are infinitely many ways to extend the factorials to a continuous function. The most widely used of these uses the gamma function, which can be defined for positive real numbers as the integral

$$\backslash Gamma(z)\; =\; \backslash int\_0^\backslash infty\; x^\{z-1\}\; e^\{-x\}\backslash ,dx.$$

The resulting function is related to the factorial of a non-negative integer $n$ by the equation

$$n!=\backslash Gamma(n+1),$$

which can be used as a definition of the factorial for non-integer arguments.

At all values $x$ for which both $\backslash Gamma(x)$ and $\backslash Gamma(x-1)$ are defined, the gamma function obeys the functional equation

$$\backslash Gamma(n)=(n-1)\backslash Gamma(n-1),$$

generalizing the recurrence relation for the factorials.{{cite journal | last = Davis | first = Philip J. | author-link = Philip J. Davis | doi = 10.1080/00029890.1959.11989422 | journal = The American Mathematical Monthly | jstor = 2309786 | mr = 106810 | pages = 849–869 | title = Leonhard Euler's integral: A historical profile of the gamma function | url = https://www.maa.org/programs/maa-awards/writing-awards/leonhard-eulers-integral-an-historical-profile-of-the-gamma-function | volume = 66 | year = 1959 | issue = 10 | access-date = 2021-12-20 | archive-date = 2023-01-01 | archive-url = https://web.archive.org/web/20230101190952/https://www.maa.org/programs/maa-awards/writing-awards/leonhard-eulers-integral-an-historical-profile-of-the-gamma-function | url-status = dead }}

The same integral converges more generally for any complex number $z$ whose real part is positive. It can be extended to the non-integer points in the rest of the complex plane by solving for Euler's reflection formula

$$\backslash Gamma(z)\backslash Gamma(1-z)=\backslash frac\{\backslash pi\}\{\backslash sin\backslash pi\; z\}.$$

However, this formula cannot be used at integers because, for them, the $\backslash sin\backslash pi\; z$ term would produce a division by zero. The result of this extension process is an analytic function, the analytic continuation of the integral formula for the gamma function. It has a nonzero value at all complex numbers, except for the non-positive integers where it has simple poles. Correspondingly, this provides a definition for the factorial at all complex numbers other than the negative integers.{{cite journal | last1 = Borwein | first1 = Jonathan M. | author1-link = Jonathan Borwein | last2 = Corless | first2 = Robert M. | doi = 10.1080/00029890.2018.1420983 | issue = 5 | journal = The American Mathematical Monthly | mr = 3785875 | pages = 400–424 | title = Gamma and factorial in the Monthly | volume = 125 | year = 2018| arxiv = 1703.05349 | s2cid = 119324101 }}

One property of the gamma function, distinguishing it from other continuous interpolations of the factorials, is given by the Bohr–Mollerup theorem, which states that the gamma function (offset by one) is the only log-convex function on the positive real numbers that interpolates the factorials and obeys the same functional equation. A related uniqueness theorem of Helmut Wielandt states that the complex gamma function and its scalar multiples are the only holomorphic functions on the positive complex half-plane that obey the functional equation and remain bounded for complex numbers with real part between 1 and 2.{{cite journal | last = Remmert | first = Reinhold | author-link = Reinhold Remmert | doi = 10.1080/00029890.1996.12004726 | issue = 3 | journal = The American Mathematical Monthly | jstor = 2975370 | mr = 1376175 | pages = 214–220 | title = Wielandt's theorem about the {{nowrap|$\backslash Gamma$-function}} | volume = 103 | year = 1996}}

Other complex functions that interpolate the factorial values include Hadamard's gamma function, which is an entire function over all the complex numbers, including the non-positive integers.{{cite book|first=J.|last=Hadamard|author-link=Jacques Hadamard|chapter=Sur l'expression du produit {{math|1·2·3· · · · ·(n−1)}} par une fonction entière|title=Œuvres de Jacques Hadamard|publisher=Centre National de la Recherche Scientifiques|location=Paris|date=1968|chapter-url=http://www.luschny.de/math/factorial/hadamard/HadamardFactorial.pdf|orig-date=1894|language=fr}}

{{cite journal | last = Alzer | first = Horst | doi = 10.1007/s12188-008-0009-5 | issue = 1 | journal = Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | mr = 2541340 | pages = 11–23 | title = A superadditive property of Hadamard's gamma function | volume = 79 | year = 2009| s2cid = 123691692 }} In the p-adic numbers, it is not possible to continuously interpolate the factorial function directly, because the factorials of large integers (a dense subset of the {{mvar|p}}-adics) converge to zero according to Legendre's formula, forcing any continuous function that is close to their values to be zero everywhere. Instead, the p-adic gamma function provides a continuous interpolation of a modified form of the factorial, omitting the factors in the factorial that are divisible by {{mvar|p}}.{{harvnb|Robert|2000}}. "7.1: The gamma function {{nowrap|$\backslash Gamma\_p$".}} pp. 366–385.

The digamma function is the logarithmic derivative of the gamma function. Just as the gamma function provides a continuous interpolation of the factorials, offset by one, the digamma function provides a continuous interpolation of the harmonic numbers, offset by the Euler–Mascheroni constant.{{cite journal | last = Ross | first = Bertram | doi = 10.1080/0025570X.1978.11976704 | issue = 3 | journal = Mathematics Magazine | jstor = 2689999 | mr = 1572267 | pages = 176–179 | title = The psi function | volume = 51 | year = 1978}}

File:Vintage Texas Instruments Model SR-50A Handheld LED Electronic Calculator, Made in the USA, Price Was $109.50 in 1975 (8715012843).jpg, a 1975 calculator with a factorial key (third row, center right)]]

The factorial function is a common feature in scientific calculators.{{cite book|title=Understandable Statistics: Concepts and Methods|first1=Charles Henry|last1=Brase|first2=Corrinne Pellillo|last2=Brase|edition=11th|publisher=Cengage Learning|year=2014|isbn=978-1-305-14290-9|page=182|url=https://books.google.com/books?id=a8OiAgAAQBAJ&pg=PA182}} It is also included in scientific programming libraries such as the Python mathematical functions module{{cite web|url=https://docs.python.org/3/library/math.html|title=math — Mathematical functions|work=Python 3 Documentation: The Python Standard Library|access-date=2021-12-21}} and the Boost C++ library.{{cite web|url=https://www.boost.org/doc/libs/1_78_0/libs/math/doc/html/math_toolkit/factorials/sf_factorial.html| title=Factorial|work=Boost 1.78.0 Documentation: Math Special Functions|access-date=2021-12-21}} If efficiency is not a concern, computing factorials is trivial: just successively multiply a variable initialized {{nowrap|to $1$}} by the integers up {{nowrap|to $n$.}} The simplicity of this computation makes it a common example in the use of different computer programming styles and methods.{{cite book|title=Drawing Programs: The Theory and Practice of Schematic Functional Programming|first1=Tom|last1=Addis|first2=Jan|last2=Addis|publisher=Springer| year=2009| isbn=978-1-84882-618-2| pages=149–150|url=https://books.google.com/books?id=cWM7ZBfEl_0C&pg=PA149}}

The computation of $n!$ can be expressed in pseudocode using iteration{{cite book|title=MATLAB Programming for Engineers|first=Stephen J.|last=Chapman|edition=6th|publisher=Cengage Learning|year=2019| isbn=978-0-357-03052-3| page=215|contribution=Example 5.2: The factorial function|contribution-url=https://books.google.com/books?id=jVEzEAAAQBAJ&pg=PA215}} as

define factorial(n):

f := 1

for i := 1, 2, 3, ..., n:

f := f * i

return f

or using recursion{{cite book|title=The Computing Universe: A Journey through a Revolution|first1=Tony|last1=Hey|first2=Gyuri|last2=Pápay|publisher=Cambridge University Press|year=2014|isbn=9781316123225|page=64|url=https://books.google.com/books?id=q4FIBQAAQBAJ&pg=PA64}} based on its recurrence relation as

define factorial(n):

if (n = 0) return 1

return n * factorial(n − 1)

Other methods suitable for its computation include memoization,{{cite book|title=Hands-On Functional Programming with C++: An effective guide to writing accelerated functional code using C++17 and C++20| first=Alexandru|last=Bolboaca | publisher=Packt Publishing|year=2019|isbn=978-1-78980-921-3|page=188|url=https://books.google.com/books?id=GwSgDwAAQBAJ&pg=PA188}} dynamic programming,{{cite book|title=Mastering Mathematica: Programming Methods and Applications| first=John W.|last=Gray|publisher=Academic Press|year=2014|isbn=978-1-4832-1403-0|pages=233–234| url=https://books.google.com/books?id=a4riBQAAQBAJ&pg=PA233}} and functional programming.{{cite book|title=Scala From a Functional Programming Perspective: An Introduction to the Programming Language|volume=9980|series=Lecture Notes in Computer Science| first=Vicenç| last=Torra| publisher=Springer|year=2016|isbn=978-3-319-46481-7|page=96|url=https://books.google.com/books?id=eMwcDQAAQBAJ&pg=PA96}} The computational complexity of these algorithms may be analyzed using the unit-cost random-access machine model of computation, in which each arithmetic operation takes constant time and each number uses a constant amount of storage space. In this model, these methods can compute $n!$ in time {{nowrap|$O(n)$,}} and the iterative version uses space {{nowrap|$O(1)$.}} Unless optimized for tail recursion, the recursive version takes linear space to store its call stack.{{cite book|title=Functional Programming and Its Applications: An Advanced Course| publisher=Cambridge University Press|series=CREST Advanced Courses|contribution=LISP, programming, and implementation| first=Gerald Jay|last=Sussman|author-link=Gerald Jay Sussman|year=1982|pages=29–72|isbn=978-0-521-24503-6}} See in particular [https://books.google.com/books?id=O_M8AAAAIAAJ&pg=PA34 p. 34]. However, this model of computation is only suitable when $n$ is small enough to allow $n!$ to fit into a machine word.{{cite journal | last = Chaudhuri | first = Ranjan | date = June 2003 | doi = 10.1145/782941.782977 | issue = 2 | journal = ACM SIGCSE Bulletin | pages = 43–44 | publisher = Association for Computing Machinery | title = Do the arithmetic operations really execute in constant time? | volume = 35| s2cid = 13629142 }} The values 12! and 20! are the largest factorials that can be stored in, respectively, the 32-bit and 64-bit integers.{{cite journal | last1 = Winkler | first1 = Jürgen F. H. | last2 = Kauer | first2 = Stefan | date = March 1997 | doi = 10.1145/251634.251638 | issue = 3 | journal = ACM SIGPLAN Notices | pages = 38–41 | publisher = Association for Computing Machinery | title = Proving assertions is also useful | volume = 32| s2cid = 17347501 | doi-access = free }} Floating point can represent larger factorials, but approximately rather than exactly, and will still overflow for factorials larger than {{nowrap|$170!$.{{cite web| url=http://people.eecs.berkeley.edu/~fateman/papers/factorial.pdf|title=Comments on Factorial Programs|date=April 11, 2006| publisher=University of California, Berkeley|first=Richard J.|last=Fateman|author-link=Richard Fateman}}}}

The exact computation of larger factorials involves arbitrary-precision arithmetic, because of fast growth and integer overflow. Time of computation can be analyzed as a function of the number of digits or bits in the result. By Stirling's formula, $n!$ has $b\; =\; O(n\backslash log\; n)$ bits. The Schönhage–Strassen algorithm can produce a {{nowrap|$b$-bit}} product in time {{nowrap|$O(b\backslash log\; b\backslash log\backslash log\; b)$,}} and faster multiplication algorithms taking time $O(b\backslash log\; b)$ are known.{{cite journal | last1 = Harvey | first1 = David | last2 = van der Hoeven | first2 = Joris | author2-link = Joris van der Hoeven | doi = 10.4007/annals.2021.193.2.4 | issue = 2 | journal = Annals of Mathematics | mr = 4224716 | pages = 563–617 | series = Second Series | title = Integer multiplication in time $O(n\; \backslash log\; n)$| volume = 193 | year = 2021| s2cid = 109934776 | url = https://hal.archives-ouvertes.fr/hal-02070778/file/nlogn.pdf }} However, computing the factorial involves repeated products, rather than a single multiplication, so these time bounds do not apply directly. In this setting, computing $n!$ by multiplying the numbers from 1 {{nowrap|to $n$}} in sequence is inefficient, because it involves $n$ multiplications, a constant fraction of which take time $O(n\backslash log^2\; n)$ each, giving total time {{nowrap|$O(n^2\backslash log^2\; n)$.}} A better approach is to perform the multiplications as a divide-and-conquer algorithm that multiplies a sequence of $i$ numbers by splitting it into two subsequences of $i/2$ numbers, multiplies each subsequence, and combines the results with one last multiplication. This approach to the factorial takes total time {{nowrap|$O(n\backslash log^3\; n)$:}} one logarithm comes from the number of bits in the factorial, a second comes from the multiplication algorithm, and a third comes from the divide and conquer.{{cite book|last=Arndt|first=Jörg| title=Matters Computational: Ideas, Algorithms, Source Code|publisher=Springer|year=2011|url=http://jjj.de/fxt/fxtbook.pdf| contribution=34.1.1.1: Computation of the factorial|pages=651–652}} See also "34.1.5: Performance", pp. 655–656.

Even better efficiency is obtained by computing {{math|n!}} from its prime factorization, based on the principle that exponentiation by squaring is faster than expanding an exponent into a product.{{cite journal | last = Borwein | first = Peter B. | author-link = Peter Borwein | doi = 10.1016/0196-6774(85)90006-9 | issue = 3 | journal = Journal of Algorithms | mr = 800727 | pages = 376–380 | title = On the complexity of calculating factorials | volume = 6 | year = 1985}}{{cite book|first=Arnold|last=Schönhage|year=1994|title=Fast algorithms: a multitape Turing machine implementation|publisher=B.I. Wissenschaftsverlag|page=226}} An algorithm for this by Arnold Schönhage begins by finding the list of the primes up {{nowrap|to $n$,}} for instance using the sieve of Eratosthenes, and uses Legendre's formula to compute the exponent for each prime. Then it computes the product of the prime powers with these exponents, using a recursive algorithm, as follows:

• Use divide and conquer to compute the product of the primes whose exponents are odd
• Divide all of the exponents by two (rounding down to an integer), recursively compute the product of the prime powers with these smaller exponents, and square the result
• Multiply together the results of the two previous steps

The product of all primes up to $n$ is an $O(n)$-bit number, by the prime number theorem, so the time for the first step is $O(n\backslash log^2\; n)$, with one logarithm coming from the divide and conquer and another coming from the multiplication algorithm. In the recursive calls to the algorithm, the prime number theorem can again be invoked to prove that the numbers of bits in the corresponding products decrease by a constant factor at each level of recursion, so the total time for these steps at all levels of recursion adds in a geometric series {{nowrap|to $O(n\backslash log^2\; n)$.}} The time for the squaring in the second step and the multiplication in the third step are again {{nowrap|$O(n\backslash log^2\; n)$,}} because each is a single multiplication of a number with $O(n\backslash log\; n)$ bits. Again, at each level of recursion the numbers involved have a constant fraction as many bits (because otherwise repeatedly squaring them would produce too large a final result) so again the amounts of time for these steps in the recursive calls add in a geometric series {{nowrap|to $O(n\backslash log^2\; n)$.}} Consequentially, the whole algorithm takes {{nowrap|time $O(n\backslash log^2\; n)$,}} proportional to a single multiplication with the same number of bits in its result.

{{main|List of factorial and binomial topics}}

Several other integer sequences are similar to or related to the factorials:

;Alternating factorial

:The alternating factorial is the absolute value of the alternating sum of the first $n$ factorials, {{nowrap|$\backslash sum\_\{i\; =\; 1\}^n\; (-1)^\{n\; -\; i\}i!$.}} These have mainly been studied in connection with their primality; only finitely many of them can be prime, but a complete list of primes of this form is not known.{{harvnb|Guy|2004}}. "B43: Alternating sums of factorials". pp. 152–153.

;Bhargava factorial

:The Bhargava factorials are a family of integer sequences defined by Manjul Bhargava with similar number-theoretic properties to the factorials, including the factorials themselves as a special case.

;Double factorial

:The product of all the odd integers up to some odd positive {{nowrap|integer $n$}} is called the double factorial {{nowrap|of $n$,}} and denoted by {{nowrap|$n!!$.{{cite arXiv|title=A combinatorial survey of identities for the double factorial|first=David|last=Callan|eprint=0906.1317|year=2009|class=math.CO}}}} That is, $$(2k-1)!!\; =\; \backslash prod\_\{i=1\}^k\; (2i-1)\; =\; \backslash frac\{(2k)!\}\{2^k\; k!\}.$$ For example, {{nowrap|1=9!! = 1 × 3 × 5 × 7 × 9 = 945}}. Double factorials are used in trigonometric integrals,{{cite journal

| last = Meserve | first = B. E.

| doi = 10.2307/2306136

| issue = 7

| journal = The American Mathematical Monthly

| mr = 1527019

| pages = 425–426

| title = Classroom Notes: Double Factorials

| volume = 55

| year = 1948| jstor = 2306136

}} in expressions for the gamma function at half-integers and the volumes of hyperspheres,{{cite journal|title=Some dimension problems in molecular databases|first=Paul G.|last=Mezey|year=2009|journal=Journal of Mathematical Chemistry|volume=45|issue=1|pages=1–6|doi=10.1007/s10910-008-9365-8|s2cid=120103389}}. and in counting binary trees and perfect matchings.{{cite journal

| last1 = Dale | first1 = M. R. T.

| last2 = Moon | first2 = J. W.

| doi = 10.1016/0378-3758(93)90035-5

| issue = 1

| journal = Journal of Statistical Planning and Inference

| mr = 1209991

| pages = 75–87

| title = The permuted analogues of three Catalan sets

| volume = 34

| year = 1993}}.

;Exponential factorial

:Just as triangular numbers sum the numbers from $1$ {{nowrap|to $n$,}} and factorials take their product, the exponential factorial exponentiates. The exponential factorial is defined recursively {{nowrap|as $a\_0\; =\; 1,\backslash \; a\_n\; =\; n^\{a\_\{n\; -\; 1\}\}$.}} For example, the exponential factorial of 4 is $$4^\{3^\{2^\{1\}\}\}=262144.$$ These numbers grow much more quickly than regular factorials.{{cite journal | last1 = Luca | first1 = Florian | author1-link = Florian Luca | last2 = Marques | first2 = Diego | issue = 3 | journal = Journal de Théorie des Nombres de Bordeaux | mr = 2769339 | pages = 703–718 | title = Perfect powers in the summatory function of the power tower | url = http://jtnb.cedram.org/item?id=JTNB_2010__22_3_703_0 | volume = 22 | year = 2010| doi = 10.5802/jtnb.740 | doi-access = free }}

;Falling factorial

:The notations $(x)\_\{n\}$ or $x^\{\backslash underline\; n\}$ are sometimes used to represent the product of the greatest $n$ integers counting up to and {{nowrap|including $x$,}} equal to {{nowrap|$x!/(x-n)!$.}} This is also known as a falling factorial or backward factorial, and the $(x)\_\{n\}$ notation is a Pochhammer symbol.{{sfn|Graham|Knuth|Patashnik|1988|pp=x, 47–48}} Falling factorials count the number of different sequences of $n$ distinct items that can be drawn from a universe of $x$ items.{{cite book | last = Sagan | first = Bruce E. | author-link = Bruce Sagan | contribution = Theorem 1.2.1 | contribution-url = https://books.google.com/books?id=DYgEEAAAQBAJ&pg=PA5 | isbn = 978-1-4704-6032-7 | location = Providence, Rhode Island | mr = 4249619 | page = 5 | publisher = American Mathematical Society | series = Graduate Studies in Mathematics | title = Combinatorics: the Art of Counting | volume = 210 | year = 2020}} They occur as coefficients in the higher derivatives of polynomials,{{cite book|first=G. H.|last=Hardy|author-link=G. H. Hardy|title=A Course of Pure Mathematics|title-link=A Course of Pure Mathematics|edition=3rd|publisher=Cambridge University Press|year=1921|contribution=Examples XLV|page=215|contribution-url=https://archive.org/details/coursepuremath00hardrich/page/n229}} and in the factorial moments of random variables.{{cite book | last1 = Daley | first1 = D. J. | last2 = Vere-Jones | first2 = D. | contribution = 5.2: Factorial moments, cumulants, and generating function relations for discrete distributions | contribution-url = https://books.google.com/books?id=Af7lBwAAQBAJ&pg=PA112 | isbn = 0-387-96666-8 | location = New York | mr = 950166 | page = 112 | publisher = Springer-Verlag | series = Springer Series in Statistics | title = An Introduction to the Theory of Point Processes | year = 1988}}

;Hyperfactorials

:The hyperfactorial of $n$ is the product $1^1\backslash cdot\; 2^2\backslash cdots\; n^n$. These numbers form the discriminants of Hermite polynomials.{{cite OEIS | 1=A002109 | 2=Hyperfactorials: Product_{k = 1..n} k^k}} They can be continuously interpolated by the K-function,{{cite journal | last = Kinkelin | first = H. | author-link = Hermann Kinkelin | doi = 10.1515/crll.1860.57.122 | journal = Journal für die reine und angewandte Mathematik | language = de | pages = 122–138 | title = Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechung | trans-title = On a transcendental variation of the gamma function and its application to the integral calculus | volume = 1860 | year = 1860| issue = 57 | s2cid = 120627417 }} and obey analogues to Stirling's formula{{cite journal | last = Glaisher | first = J. W. L. | author-link = James Whitbread Lee Glaisher | journal = Messenger of Mathematics | pages = 43–47 | title = On the product {{math|1¹.2².3³...nⁿ}} | url = https://archive.org/details/messengermathem01glaigoog/page/n56 | volume = 7 | year = 1877}} and Wilson's theorem.{{cite journal | last1 = Aebi | first1 = Christian | last2 = Cairns | first2 = Grant | doi = 10.4169/amer.math.monthly.122.5.433 | issue = 5 | journal = The American Mathematical Monthly | jstor = 10.4169/amer.math.monthly.122.5.433 | mr = 3352802 | pages = 433–443 | title = Generalizations of Wilson's theorem for double-, hyper-, sub- and superfactorials | volume = 122 | year = 2015| s2cid = 207521192 }}

;Jordan–Pólya numbers

:The Jordan–Pólya numbers are the products of factorials, allowing repetitions. Every tree has a symmetry group whose number of symmetries is a Jordan–Pólya number, and every Jordan–Pólya number counts the symmetries of some tree.{{cite OEIS|A001013|Jordan-Polya numbers: products of factorial numbers}}

;Primorial

:The primorial $n\backslash \#$ is the product of prime numbers less than or equal {{nowrap|to $n$;}} this construction gives them some similar divisibility properties to factorials, but unlike factorials they are squarefree.{{cite book | last = Nelson | first = Randolph | doi = 10.1007/978-3-030-37861-5 | isbn = 978-3-030-37861-5 | location = Cham | mr = 4297795 | page = 127 | publisher = Springer | title = A Brief Journey in Discrete Mathematics | url = https://books.google.com/books?id=m8PPDwAAQBAJ&pg=PA127 | year = 2020| s2cid = 213895324 }} As with the factorial primes {{nowrap|$n!\backslash pm\; 1$,}} researchers have studied primorial primes {{nowrap|$n\backslash \#\backslash pm\; 1$.}}

;Subfactorial

:The subfactorial yields the number of derangements of a set of $n$ objects. It is sometimes denoted $!n$, and equals the closest integer {{nowrap|to $n!/e$.{{sfn|Graham|Knuth|Patashnik|1988|p=195}}}}

;Superfactorial

:The superfactorial of $n$ is the product of the first $n$ factorials. The superfactorials are continuously interpolated by the Barnes G-function.{{cite journal|last=Barnes|first=E. W.|author-link=Ernest Barnes|jfm=30.0389.02|journal=The Quarterly Journal of Pure and Applied Mathematics|pages=264–314|title=The theory of the {{mvar|G}}-function|url=https://gdz.sub.uni-goettingen.de/id/PPN600494829_0031?tify={%22pages%22:[268],%22view%22:%22toc%22}|volume=31|year=1900}}

{{Reflist}}

{{Portal|Arithmetic|Mathematics}}

{{Commons category|Factorial (function)}}

• {{OEIS el|A000142|Factorial numbers}}
• {{springer|title=Factorial|id=p/f038080|mode=cs1}}
• {{MathWorld|urlname=Factorial|title=Factorial}}

{{Calculus topics}}

{{Series (mathematics)}}

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Category:Combinatorics

Category:Gamma and related functions

Category:Factorial and binomial topics

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