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Factorial

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Product of numbers from 1 to n

This article is about a mathematical function. For the game, seeFactorio. For other uses, seeFactorial (disambiguation).

Selected factorials; values in scientific notation are rounded
$n {\displaystyle n}$	$n! {\displaystyle n!}$
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5040
8	40320
9	362880
10	3628800
11	39916800
12	479001600
13	6227020800
14	87178291200
15	1307674368000
16	20922789888000
17	355687428096000
18	6402373705728000
19	121645100408832000
20	2432902008176640000
25	1.551121004×10²⁵
50	3.041409320×10⁶⁴
70	1.197857167×10¹⁰⁰
100	9.332621544×10¹⁵⁷
450	1.733368733×10¹⁰⁰⁰
1000	4.023872601×10²⁵⁶⁷
3249	6.412337688×10¹⁰⁰⁰⁰
10000	2.846259681×10³⁵⁶⁵⁹
25206	1.205703438×10¹⁰⁰⁰⁰⁰
100000	2.824229408×10⁴⁵⁶⁵⁷³
205023	2.503898932×10^1000004
1000000	8.263931688×10^5565708
10¹⁰⁰	10^{10^{101.9981097754820}}

Inmathematics, thefactorial of a non-negativeinteger $n {\displaystyle n}$ , denotedby $n! {\displaystyle n!}$ , is theproduct of all positive integers less than or equalto $n {\displaystyle n}$ . The factorialof $n {\displaystyle n}$ also equals the product of $n {\displaystyle n}$ with the next smaller factorial: ${\begin{aligned}n!&=n\times (n-1)\times (n-2)\times (n-3)\times \cdots \times 3\times 2\times 1\\&=n\times (n-1)!\\\end{aligned}}$ For example, $5!=5\times 4!=5\times 4\times 3\times 2\times 1=120.$ The value of 0! is 1, according to the convention for anempty product.^[1]

Factorials have been discovered in several ancient cultures, notably inIndian mathematics in the canonical works ofJain literature, and by Jewish mystics in the Talmudic bookSefer Yetzirah. The factorial operation is encountered in many areas of mathematics, notably incombinatorics, where its most basic use counts the possible distinctsequences – thepermutations – of $n {\displaystyle n}$ distinct objects: thereare $n! {\displaystyle n!}$ . Inmathematical analysis, factorials are used inpower series for theexponential function and other functions, and they also have applications inalgebra,number theory,probability theory, andcomputer science.

Much of the mathematics of the factorial function was developed beginning in the late 18th and early 19th centuries.Stirling's approximation provides an accurate approximation to the factorial of large numbers, showing that it grows more quickly thanexponential growth.Legendre's formula describes the exponents of the prime numbers in aprime factorization of the factorials, and can be used to count the trailing zeros of the factorials.Daniel Bernoulli andLeonhard Euler interpolated the factorial function to a continuous function ofcomplex numbers, except at the negative integers, the (offset)gamma function.

Many other notable functions and number sequences are closely related to the factorials, including thebinomial coefficients,double factorials,falling factorials,primorials, andsubfactorials. Implementations of the factorial function are commonly used as an example of differentcomputer programming styles, and are included inscientific calculators and scientific computing software libraries. Although directly computing large factorials using the product formula or recurrence is not efficient, faster algorithms are known, matching to within a constant factor the time for fastmultiplication algorithms for numbers with the same number of digits.

History

[edit]

The concept of factorials has arisen independently in many cultures:

InIndian mathematics, one of the earliest known descriptions of factorials comes from the Anuyogadvāra-sūtra,^[2] one of the canonical works ofJain literature, which has been assigned dates varying from 300 BCE to 400 CE.^[3] 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 monkJinabhadra.^[2] Hindu scholars have been using factorial formulas since at least 1150, whenBhāskara II mentioned factorials in his workLīlāvatī, in connection with a problem of how many waysVishnu could hold his four characteristic objects (aconch shell,discus,mace, andlotus flower) in his four hands, and a similar problem for a ten-handed god.^[4]
In the mathematics of the Middle East, the Hebrew mystic book of creationSefer Yetzirah, from theTalmudic 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 theHebrew alphabet.^[5]^[6] Factorials were also studied for similar reasons by 8th-century Arab grammarianAl-Khalil ibn Ahmad al-Farahidi.^[5] Arab mathematicianIbn al-Haytham (also known as Alhazen, c. 965 – c. 1040) was the first to formulateWilson's theorem connecting the factorials with theprime numbers.^[7]
In Europe, althoughGreek mathematics included some combinatorics, andPlato famously used 5,040 (a factorial) as the population of an ideal community, in part because of its divisibility properties,^[8] there is no direct evidence of ancient Greek study of factorials. Instead, the first work on factorials in Europe was by Jewish scholars such asShabbethai Donnolo, explicating the Sefer Yetzirah passage.^[9] In 1677, British authorFabian Stedman described the application of factorials tochange ringing, a musical art involving the ringing of several tuned bells.^[10]^[11]

From the late 15th century onward, factorials became the subject of study by Western mathematicians. In a 1494 treatise, Italian mathematicianLuca Pacioli calculated factorials up to 11!, in connection with a problem of dining table arrangements.^[12]Christopher Clavius discussed factorials in a 1603 commentary on the work ofJohannes de Sacrobosco, and in the 1640s, French polymathMarin Mersenne published large (but not entirely correct) tables of factorials, up to 64!, based on the work of Clavius.^[13] Thepower series for theexponential function, with the reciprocals of factorials for its coefficients, was first formulated in 1676 byIsaac Newton in a letter toGottfried Wilhelm Leibniz.^[14] Other important works of early European mathematics on factorials include extensive coverage in a 1685 treatise byJohn Wallis, a study of their approximate values for large values of $n {\displaystyle n}$ byAbraham de Moivre in 1721, a 1729 letter fromJames Stirling to de Moivre stating what became known asStirling's approximation, and work at the same time byDaniel Bernoulli andLeonhard Euler formulating the continuous extension of the factorial function to thegamma function.^[15]Adrien-Marie Legendre includedLegendre's formula, describing the exponents in thefactorization of factorials intoprime powers, in an 1808 text onnumber theory.^[16]

The notation $n! {\displaystyle n!}$ for factorials was introduced by the French mathematicianChristian Kramp in 1808.^[17] Many other notations have also been used. Another later notation $\vert \!{\underline {\,n}}$ , 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.^[17] The word "factorial" (originally French:factorielle) was first used in 1800 byLouis François Antoine Arbogast,^[18] in the first work onFaà di Bruno's formula,^[19] but referring to a more general concept of products ofarithmetic progressions. The "factors" that this name refers to are the terms of the product formula for the factorial.^[20]

Definition

[edit]

The factorial function of a positive integer $n {\displaystyle n}$ is defined by the product of all positive integers not greater than $n {\displaystyle n}$ ^[1] $n!=1\cdot 2\cdot 3\cdots (n-2)\cdot (n-1)\cdot n.$ This may be written more concisely inproduct notation as^[1] $n!=\prod _{i=1}^{n}i.$

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 arecurrence relation, according to which each value of the factorial function can be obtained by multiplying the previous valueby $n {\displaystyle n}$ :^[21] $n!=n\cdot (n-1)!.$ For example, $5!=5\cdot 4!=5\cdot 24=120$ .

Factorial of zero

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The factorialof $0 {\displaystyle 0}$ is $1 {\displaystyle 1}$ , or in symbols, $0!=1$ . There are several motivations for this definition:

For $n=0$ , the definition of $n! {\displaystyle n!}$ as a product involves the product of no numbers at all, and so is an example of the broader convention that theempty product, a product of no factors, is equal to the multiplicative identity.^[22]
There is exactly one permutation of zero objects: with nothing to permute, the only rearrangement is to do nothing.^[21]
This convention makes many identities incombinatorics valid for all valid choices of their parameters. For instance, the number of ways to choose all $n {\displaystyle n}$ elements from a set of $n {\displaystyle n}$ is ${\textstyle {\tbinom {n}{n}}={\tfrac {n!}{n!0!}}=1,}$ abinomial coefficient identity that would only be validwith $0!=1$ .^[23]
With $0!=1$ , the recurrence relation for the factorial remains validat $n=1$ . Therefore, with this convention, arecursive computation of the factorial needs to have only the value for zero as abase case, simplifying the computation and avoiding the need for additional special cases.^[24]
Setting $0!=1$ allows for the compact expression of many formulae, such as theexponential function, as apower series: ${\textstyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}$ ^[14]
This choice matches thegamma function $0!=\Gamma (0+1)=1$ , and the gamma function must have this value to be acontinuous function.^[25]

Applications

[edit]

The earliest uses of the factorial function involve countingpermutations: there are $n! {\displaystyle n!}$ different ways of arranging $n {\displaystyle n}$ distinct objects into a sequence.^[26] Factorials appear more broadly in many formulas incombinatorics, to account for different orderings of objects. For instance thebinomial coefficients ${\tbinom {n}{k}}$ count the $k {\displaystyle k}$ -elementcombinations (subsets of $k {\displaystyle k}$ elements) from a set with $n {\displaystyle n}$ elements, and can be computed from factorials using the formula^[27] ${\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.$ TheStirling numbers of the first kind sum to the factorials, and count the permutationsof $n {\displaystyle n}$ grouped into subsets with the same numbers of cycles.^[28] Another combinatorial application is in countingderangements, permutations that do not leave any element in its original position; the number of derangements of $n {\displaystyle n}$ items is thenearest integerto $n!/e$ .^[29]

Inalgebra, the factorials arise through thebinomial theorem, which uses binomial coefficients to expand powers of sums.^[30] They also occur in the coefficients used to relate certain families of polynomials to each other, for instance inNewton's identities forsymmetric polynomials.^[31] Their use in counting permutations can also be restated algebraically: the factorials are theorders of finitesymmetric groups.^[32] Incalculus, factorials occur inFaà di Bruno's formula for chaining higher derivatives.^[19] Inmathematical analysis, factorials frequently appear in the denominators ofpower series, most notably in the series for theexponential function,^[14] $e^{x}=1+{\frac {x}{1}}+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+\cdots =\sum _{i=0}^{\infty }{\frac {x^{i}}{i!}},$ and in the coefficients of otherTaylor series (in particular those of thetrigonometric andhyperbolic functions), where they cancel factors of $n! {\displaystyle n!}$ coming from the $n {\displaystyle n}$ th derivativeof $x^{n}$ .^[33] This usage of factorials in power series connects back toanalytic combinatorics through theexponential generating function, which for acombinatorial class with $n_{i}$ elements ofsize $i {\displaystyle i}$ is defined as the power series^[34] $\sum _{i=0}^{\infty }{\frac {x^{i}n_{i}}{i!}}.$

Innumber theory, the most salient property of factorials is thedivisibility of $n! {\displaystyle n!}$ by all positive integers upto $n {\displaystyle n}$ , described more precisely for prime factors byLegendre's formula. It follows that arbitrarily largeprime numbers can be found as the prime factors of the numbers $n!\pm 1$ , leading to a proof ofEuclid's theorem that the number of primes is infinite.^[35] When $n!\pm 1$ is itself prime it is called afactorial prime;^[36] relatedly,Brocard's problem, also posed bySrinivasa Ramanujan, concerns the existence ofsquare numbers of the form $n!+1$ .^[37] In contrast, the numbers $n!+2,n!+3,\dots n!+n$ must all be composite, proving the existence of arbitrarily largeprime gaps.^[38] An elementaryproof of Bertrand's postulate on the existence of a prime in any interval of theform $[n,2n]$ , one of the first results ofPaul Erdős, was based on the divisibility properties of factorials.^[39]^[40] Thefactorial number system is amixed radix notation for numbers in which the place values of each digit are factorials.^[41]

Factorials are used extensively inprobability theory, for instance in thePoisson distribution^[42] and in the probabilities ofrandom permutations.^[43] Incomputer science, beyond appearing in the analysis ofbrute-force searches over permutations,^[44] factorials arise in thelower bound of $\log _{2}n!=n\log _{2}n-O(n)$ on the number of comparisons needed tocomparison sort a set of $n {\displaystyle n}$ items,^[45] and in the analysis of chainedhash tables, where the distribution of keys per cell can be accurately approximated by a Poisson distribution.^[46] Moreover, factorials naturally appear in formulae fromquantum andstatistical physics, where one often considers all the possible permutations of a set of particles. Instatistical mechanics, calculations ofentropy such asBoltzmann's entropy formula or theSackur–Tetrode equation must correct the count ofmicrostates by dividing by the factorials of the numbers of each type ofindistinguishable particle to avoid theGibbs paradox. Quantum physics provides the underlying reason for why these corrections are necessary.^[47]

Properties

[edit]

Growth and approximation

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Comparison of the factorial, Stirling's approximation, and the simpler approximation $(n/e)^{n}$ , on a doubly logarithmic scale

{\displaystyle (n/e)^{n}} — Comparison of the factorial, Stirling's approximation, and the simpler approximation $(n/e)^{n}$ , on a doubly logarithmic scale

Relative error in a truncated Stirling series vs. number of terms

Main article:Stirling's approximation

As a functionof $n {\displaystyle n}$ , the factorial has faster thanexponential growth, but grows more slowly than adouble exponential function.^[48] Its growth rate is similarto $n^{n}$ , but slower by an exponential factor. One way of approaching this result is by taking thenatural logarithm of the factorial, which turns its product formula into a sum, and then estimating the sum by an integral: $\ln n!=\sum _{x=1}^{n}\ln x\approx \int _{1}^{n}\ln x\,dx=n\ln n-n+1.$ Exponentiating the result (and ignoring the negligible $+1$ term) approximates $n! {\displaystyle n!}$ as $(n/e)^{n}$ .^[49]More carefully bounding the sum both above and below by an integral, using thetrapezoid rule, shows that this estimate needs a correction factor proportionalto ${\sqrt {n}}$ . The constant of proportionality for this correction can be found from theWallis product, which expresses $\pi$ as a limiting ratio of factorials and powers of two. The result of these corrections isStirling's approximation:^[50] $n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\,.$ Here, the $\sim$ symbol means that, as $n {\displaystyle n}$ goes to infinity, the ratio between the left and right sides approaches one in thelimit.Stirling's formula provides the first term in anasymptotic series that becomes even more accurate when taken to greater numbers of terms:^[51] $n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+{\frac {1}{288n^{2}}}-{\frac {139}{51840n^{3}}}-{\frac {571}{2488320n^{4}}}+\cdots \right).$ An alternative version uses only odd exponents in the correction terms:^[51] $n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\exp \left({\frac {1}{12n}}-{\frac {1}{360n^{3}}}+{\frac {1}{1260n^{5}}}-{\frac {1}{1680n^{7}}}+\cdots \right).$ Many other variations of these formulas have also been developed, bySrinivasa Ramanujan,Bill Gosper, and others.^[51]

Thebinary logarithm of the factorial, used to analyzecomparison sorting, can be very accurately estimated using Stirling's approximation. In the formula below, the $O(1)$ term invokesbig O notation.^[45] $\log _{2}n!=n\log _{2}n-(\log _{2}e)n+{\frac {1}{2}}\log _{2}n+O(1).$

Divisibility and digits

[edit]

Main article:Legendre's formula

The product formula for the factorial implies that $n! {\displaystyle n!}$ isdivisible by allprime numbers that are atmost $n {\displaystyle n}$ , and by no larger prime numbers.^[52] More precise information about its divisibility is given byLegendre's formula, which gives the exponent of each prime $p {\displaystyle p}$ in the prime factorization of $n! {\displaystyle n!}$ as^[53]^[54] $\sum _{i=1}^{\infty }\left\lfloor {\frac {n}{p^{i}}}\right\rfloor ={\frac {n-s_{p}(n)}{p-1}}.$ Here $s_{p}(n)$ denotes the sum of thebase- $p {\displaystyle p}$ digitsof $n {\displaystyle n}$ , and the exponent given by this formula can also be interpreted in advanced mathematics as thep-adic valuation of the factorial.^[54] Applying Legendre's formula to the product formula forbinomial coefficients producesKummer's theorem, a similar result on the exponent of each prime in the factorization of a binomial coefficient.^[55] Grouping the prime factors of the factorial intoprime powers in different ways produces themultiplicative partitions of factorials.^[56]

The special case of Legendre's formula for $p=5$ gives the number oftrailing zeros in the decimal representation of the factorials.^[57] According to this formula, the number of zeros can be obtained by subtracting the base-5 digits of $n {\displaystyle n}$ from $n {\displaystyle n}$ , and dividing the result by four.^[58] Legendre's formula implies that the exponent of the prime $p=2$ is always larger than the exponent for $p=5$ , so each factor of five can be paired with a factor of two to produce one of these trailing zeros.^[57] The leading digits of the factorials are distributed according toBenford's law.^[59] Every sequence of digits, in any base, is the sequence of initial digits of some factorial number in that base.^[60]

Another result on divisibility of factorials,Wilson's theorem, states that $(n-1)!+1$ is divisible by $n {\displaystyle n}$ if and only if $n {\displaystyle n}$ is aprime number.^[52] For any giveninteger $x {\displaystyle x}$ , theKempner function of $x {\displaystyle x}$ is given by the smallest $n {\displaystyle n}$ for which $x {\displaystyle x}$ divides $n! {\displaystyle n!}$ .^[61] For almost all numbers (all but a subset of exceptions withasymptotic density zero), it coincides with the largest prime factorof $x {\displaystyle x}$ .^[62]

The product of two factorials, $m!\cdot n!$ , always evenly divides $(m+n)!$ .^[63] There are infinitely many factorials that equal the product of other factorials: if $n {\displaystyle n}$ is itself any product of factorials, then $n! {\displaystyle n!}$ equals that same product multiplied by one more factorial, $(n-1)!$ . The only known examples of factorials that are products of other factorials but are not of this "trivial" form are $9!=7!\cdot 3!\cdot 3!\cdot 2!$ , $10!=7!\cdot 6!=7!\cdot 5!\cdot 3!$ , and $16!=14!\cdot 5!\cdot 2!$ .^[64] It would follow from theabc conjecture that there are only finitely many nontrivial examples.^[65]

Thegreatest common divisor of the values of aprimitive polynomial of degree $d {\displaystyle d}$ over the integers evenly divides $d! {\displaystyle d!}$ .^[63]

Continuous interpolation and non-integer generalization

[edit]

The gamma function (shifted one unit left to match the factorials) continuously interpolates the factorial to non-integer values

Absolute values of the complex gamma function, showing poles at non-positive integers

Main article:Gamma function

There are infinitely many ways to extend the factorials to acontinuous function.^[66] The most widely used of these^[67] uses thegamma function, which can be defined for positive real numbers as theintegral $\Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx.$ The resulting function is related to the factorial of a non-negative integer $n {\displaystyle n}$ by the equation $n!=\Gamma (n+1),$ which can be used as a definition of the factorial for non-integer arguments.At all values $x {\displaystyle x}$ for which both $\Gamma (x)$ and $\Gamma (x-1)$ are defined, the gamma function obeys thefunctional equation $\Gamma (n)=(n-1)\Gamma (n-1),$ generalizing therecurrence relation for the factorials.^[66]

The same integral converges more generally for anycomplex number $z {\displaystyle z}$ whose real part is positive. It can be extended to the non-integer points in the rest of thecomplex plane by solving for Euler'sreflection formula $\Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin \pi z}}.$ However, this formula cannot be used at integers because, for them, the $\sin \pi z$ term would produce adivision by zero. The result of this extension process is ananalytic function, theanalytic 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 hassimple poles. Correspondingly, this provides a definition for the factorial at all complex numbers other than the negative integers.^[67]One property of the gamma function, distinguishing it from other continuous interpolations of the factorials, is given by theBohr–Mollerup theorem, which states that the gamma function (offset by one) is the onlylog-convex function on the positive real numbers that interpolates the factorials and obeys the same functional equation. A related uniqueness theorem ofHelmut Wielandt states that the complex gamma function and its scalar multiples are the onlyholomorphic 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.^[68]

Other complex functions that interpolate the factorial values includeHadamard's gamma function, which is anentire function over all the complex numbers, including the non-positive integers.^[69]^[70] In thep-adic numbers, it is not possible to continuously interpolate the factorial function directly, because the factorials of large integers (a dense subset of thep-adics) converge to zero according to Legendre's formula, forcing any continuous function that is close to their values to be zero everywhere. Instead, thep-adic gamma function provides a continuous interpolation of a modified form of the factorial, omitting the factors in the factorial that are divisible byp.^[71]

Thedigamma function is thelogarithmic 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 theharmonic numbers, offset by theEuler–Mascheroni constant.^[72]

Computation

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TI SR-50A, a 1975 calculator with a factorial key (third row, center right)

The factorial function is a common feature inscientific calculators.^[73] It is also included in scientific programming libraries such as thePython mathematical functions module^[74] and theBoost C++ library.^[75] If efficiency is not a concern, computing factorials is trivial: just successively multiply a variable initializedto $1 {\displaystyle 1}$ by the integers upto $n {\displaystyle n}$ . The simplicity of this computation makes it a common example in the use of different computer programming styles and methods.^[76]

The computation of $n! {\displaystyle n!}$ can be expressed inpseudocode usingiteration^[77] as

define factorial(n):f := 1  fori := 1, 2, 3, ...,n:f :=f *i  returnf

or usingrecursion^[78] based on its recurrence relation as

define factorial(n):  if (n = 0) return 1  returnn * factorial(n − 1)

Other methods suitable for its computation includememoization,^[79]dynamic programming,^[80] andfunctional programming.^[81] Thecomputational complexity of these algorithms may be analyzed using the unit-costrandom-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 $n! {\displaystyle n!}$ in time $O(n)$ , and the iterative version uses space $O(1)$ . Unless optimized fortail recursion, the recursive version takes linear space to store itscall stack.^[82] However, this model of computation is only suitable when $n {\displaystyle n}$ is small enough to allow $n! {\displaystyle n!}$ to fit into amachine word.^[83] The values 12! and 20! are the largest factorials that can be stored in, respectively, the32-bit^[84] and64-bit integers.^[85]Floating point can represent larger factorials, but approximately rather than exactly, and will still overflow for factorials larger than $170!$ .^[84]

The exact computation of larger factorials involvesarbitrary-precision arithmetic, because offast growth andinteger overflow. Time of computation can be analyzed as a function of the number of digits or bits in the result.^[85] By Stirling's formula, $n! {\displaystyle n!}$ has $b=O(n\log n)$ bits.^[86] TheSchönhage–Strassen algorithm can produce a $b {\displaystyle b}$ -bit product in time $O(b\log b\log \log b)$ , and fastermultiplication algorithms taking time $O(b\log b)$ are known.^[87] However, computing the factorial involves repeated products, rather than a single multiplication, so these time bounds do not apply directly. In this setting, computing $n! {\displaystyle n!}$ by multiplying the numbers from 1to $n {\displaystyle n}$ in sequence is inefficient, because it involves $n {\displaystyle n}$ multiplications, a constant fraction of which take time $O(n\log ^{2}n)$ each, giving total time $O(n^{2}\log ^{2}n)$ . A better approach is to perform the multiplications as adivide-and-conquer algorithm that multiplies a sequence of $i {\displaystyle i}$ 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 $O(n\log ^{3}n)$ : 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.^[88]

Even better efficiency is obtained by computingn! from its prime factorization, based on the principle thatexponentiation by squaring is faster than expanding an exponent into a product.^[86]^[89] An algorithm for this byArnold Schönhage begins by finding the list of the primes upto $n {\displaystyle n}$ , for instance using thesieve 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 $n {\displaystyle n}$ is an $O(n)$ -bit number, by theprime number theorem, so the time for the first step is $O(n\log ^{2}n)$ , 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 ageometric seriesto $O(n\log ^{2}n)$ . The time for the squaring in the second step and the multiplication in the third step are again $O(n\log ^{2}n)$ , because each is a single multiplication of a number with $O(n\log n)$ 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 seriesto $O(n\log ^{2}n)$ . Consequentially, the whole algorithm takestime $O(n\log ^{2}n)$ , proportional to a single multiplication with the same number of bits in its result.^[89]

Related sequences and functions

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Main article:List of factorial and binomial topics

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

Alternating factorial: Thealternating factorial is the absolute value of thealternating sum of the first $n {\displaystyle n}$ factorials, ${\textstyle \sum _{i=1}^{n}(-1)^{n-i}i!}$ . 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.^[90]
Bhargava factorial: TheBhargava factorials are a family of integer sequences defined byManjul Bhargava with similar number-theoretic properties to the factorials, including the factorials themselves as a special case.^[63]
Double factorial: The product of all the odd integers up to some odd positiveinteger $n {\displaystyle n}$ is called thedouble factorialof $n {\displaystyle n}$ , and denoted by $n!! {\displaystyle n!!}$ .^[91] That is, $(2k-1)!!=\prod _{i=1}^{k}(2i-1)={\frac {(2k)!}{2^{k}k!}}.$ For example,9!! = 1 × 3 × 5 × 7 × 9 = 945. Double factorials are used intrigonometric integrals,^[92] in expressions for thegamma function athalf-integers and thevolumes of hyperspheres,^[93] and in countingbinary trees andperfect matchings.^[91]^[94]
Exponential factorial: Just astriangular numbers sum the numbers from $1 {\displaystyle 1}$ to $n {\displaystyle n}$ , and factorials take their product, theexponential factorial exponentiates. The exponential factorial is defined recursivelyas $a_{0}=1,\ a_{n}=n^{a_{n-1}}$ . For example, the exponential factorial of 4 is $4^{3^{2^{1}}}=262144.$ These numbers grow much more quickly than regular factorials.^[95]
Falling factorial: The notations $(x)_{n}$ or $x^{\underline {n}}$ are sometimes used to represent the product of the greatest $n {\displaystyle n}$ integers counting up to andincluding $x {\displaystyle x}$ , equal to $x!/(x-n)!$ . This is also known as afalling factorial or backward factorial, and the $(x)_{n}$ notation is a Pochhammer symbol.^[96] Falling factorials count the number of different sequences of $n {\displaystyle n}$ distinct items that can be drawn from a universe of $x {\displaystyle x}$ items.^[97] They occur as coefficients in thehigher derivatives of polynomials,^[98] and in thefactorial moments ofrandom variables.^[99]
Hyperfactorials: Thehyperfactorial of $n {\displaystyle n}$ is the product $1^{1}\cdot 2^{2}\cdots n^{n}$ . These numbers form thediscriminants ofHermite polynomials.^[100] They can be continuously interpolated by theK-function,^[101] and obey analogues to Stirling's formula^[102] and Wilson's theorem.^[103]
Jordan–Pólya numbers: TheJordan–Pólya numbers are the products of factorials, allowing repetitions. Everytree has asymmetry group whose number of symmetries is a Jordan–Pólya number, and every Jordan–Pólya number counts the symmetries of some tree.^[104]
Primorial: Theprimorial $n\#$ is the product ofprime numbers less than or equalto $n {\displaystyle n}$ ; this construction gives them some similar divisibility properties to factorials,^[36] but unlike factorials they aresquarefree.^[105] As with thefactorial primes $n!\pm 1$ , researchers have studiedprimorial primes $n\#\pm 1$ .^[36]
Subfactorial: Thesubfactorial yields the number ofderangements of a set of $n {\displaystyle n}$ objects. It is sometimes denoted $! n {\displaystyle !n}$ , and equals the closest integerto $n!/e$ .^[29]
Superfactorial: Thesuperfactorial of $n {\displaystyle n}$ is the product of the first $n {\displaystyle n}$ factorials. The superfactorials are continuously interpolated by theBarnes G-function.^[106]

References

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Basic	Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10
Advanced(list)	Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array

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Series	Alternating Convergent Divergent Telescoping
Convergence	Absolute Conditional Uniform

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Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2^s + 1/3^s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)