![]() |
![]() |
---|---|
0 | 1 |
1 | 1 |
2 | 2 |
3 | 6 |
4 | 24 |
5 | 120 |
6 | 720 |
7 | 5,040 |
8 | 40,320 |
9 | 362,880 |
10 | 3,628,800 |
11 | 39,916,800 |
12 | 479,001,600 |
13 | 6,227,020,800 |
14 | 87,178,291,200 |
15 | 1,307,674,368,000 |
20 | 2,432,902,008,176,640,000 |
25 | 15,511,210,043,330,985,984,000,000 |
50 | 3.04140932... × 1064 |
70 | 1.19785717... × 10100 |
450 | 1.73336873... × 101,000 |
3,249 | 6.41233768... × 1010,000 |
25,206 | 1.205703438... × 10100,000 |
47,176 | 8.4485731495... × 10200,001 |
100,000 | 2.8242294079... × 10456,573 |
1,000,000 | 8.2639316883... × 105,565,708 |
9.99... × 10304 | 1 × 103.045657055180967... × 10307 |
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,
The notation n! was introduced by Christian Kramp in 1808.
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The factorial function is formally defined by
The above definition incorporates the instance
as an instance of the fact that the product of no numbers at all is 1. This fact for factorials is useful, because
Factorials have many applications in number theory. In particular, n! is necessarily divisible by all prime numbers up to and including n. As a consequence, n > 5 is a composite number if and only if
A stronger result is Wilson's theorem, which states that
if and only if p is prime.
Adrien-Marie Legendre found that the multiplicity of the prime p occurring in the prime factorization of n! can be expressed exactly as
This fact is based on counting the number of factors of the integers from
to
. The number of multiples of
in the numbers
to
are given by
; however, this formula counts those numbers with two factors of
only once. Hence another
factors of
must be counted too. Similarly for three, four, five factors, to infinity. The sum is finite since
is less than or equal to
for only finitely many values of
, and the floor function results in 0 when applied to
The only factorial that is also a prime number is 2, but there are many primes of the form , called factorial primes.
All factorials greater than 0! and 1! are even, as they are all multiples of 2.
As n grows, the factorial n! becomes larger than all polynomials and exponential functions (but slower than double exponential functions) in n.
When n is large, n! can be estimated quite accurately using Stirling's approximation:
A weak version that can easily be proved with mathematical induction is
The logarithm of the factorial can be used to calculate the number of digits in a given base the factorial of a given number will take. It satisfies the identity:
Note that this function, if graphed, looks approximately linear; but the factor , and thereby the slope of the graph, does grow arbitrarily large, although quite slowly. The graph of
for
between 0 and 20,000 is shown in the figure on the right.
A simple approximation for log n! based on Stirling's approximation is
A much better approximation for log n! was given by Srinivasa Ramanujan:
One can see from this that log n! is Ο(n log n). This result plays a key role in the analysis of the computational complexity of sorting algorithms (see comparison sort).
The value of can be calculated by repeated multiplication if
is not too large. The largest factorial that most calculators can handle is 69!, because 70! > 10100 (except for most HP calculators which can handle 253! as their exponent can be up to 499). The calculator seen in Mac OS X, Microsoft Excel and Google Calculator can handle factorials up to 170!, which is the largest factorial less than 21024 (10400 in hexadecimal) and corresponds to a 1024 bit integer. The values 12! and 20! are the largest factorials that can be stored in, respectively, the 32 bit and 64 bit integers commonly used in personal computers. In practice, most software applications will compute these small factorials by direct multiplication or table lookup. Larger values are often approximated in terms of floating-point estimates of the Gamma function, usually with Stirling's formula.
For number theoretic and combinatorial computations, very large exact factorials are often needed. Bignum factorials can be computed by direct multiplication, but multiplying the sequence 1 × 2 × ... × n from the bottom up (or top-down) is inefficient; it is better to recursively split the sequence so that the size of each subproduct is minimized.
The asymptotically-best efficiency is obtained by computing n! from its prime factorization. As documented by Peter Borwein, prime factorization allows to be computed in time O(n(log n log log n)2), provided that a fast multiplication algorithm is used (for example, the Schönhage-Strassen algorithm).[1] Peter Luschny presents source code and benchmarks for several efficient factorial algorithms, with or without the use of a prime sieve.[2]
The factorial function can also be defined for non-integer values, but this requires more advanced tools from mathematical analysis. The function that "fills in" the values of the factorial between the integers is called the Gamma function, denoted for integers z no less than 1, defined by
Euler's original formula for the Gamma function was
The Gamma function is related to factorials in that it satisfies a similar recursive relationship:
Together with this yields the equation for any nonnegative integer
:
Based on the Gamma function's value for 1/2, the specific example of half-integer factorials is resolved to
For example
The Gamma function is in fact defined for all complex numbers except for the nonpositive integers
. It is often thought of as a generalization of the factorial function to the complex domain, which is justified for the following reasons:
Euler also developed a convergent product approximation for the non-integer factorials, which can be seen to be equivalent to the formula for the Gamma function above:
It can also be written as below:
The product converges quickly for small values of n.
Representation through the Gamma-function allows evaluation of factorial of complex argument. Equilines of amplitude and phase of factorial are shown in figure. Let . Several levels of constant modulus (amplitude)
and constant phase
are shown. The grid covers range
,
with unit step. The scratched line shows the level
.
Thin lines show internediate levels of constant modulus and constant phase. At poles , phase and amplitude are not defined. Equilines are dense in vicinity of singularities along negative integer values of the argument.
There are several other integer sequences similar to the factorial that are used in mathematics:
The primorial (sequence A002110 in OEIS) is similar to the factorial, but with the product taken only over the prime numbers.
denotes the double factorial of
and is defined recursively by
For example, 8!! = 2 · 4 · 6 · 8 = 384 and 9!! = 1 · 3 · 5 · 7 · 9 = 945. The sequence of double factorials (sequence A006882 in OEIS) for starts as
The above definition can be used to define double factorials of negative numbers:
The sequence of double factorials for starts as
while the double factorial of negative even integers is undefined.
Some identities involving double factorials are:
where is the Gamma function. The last equation above can be used to define the double factorial as a function of any complex number
, just as the Gamma function generalizes the factorial function. One should be careful not to interpret
as the factorial of
, which would be written
and is a much larger number (for
).
A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (), three (
), or more. The double factorial is the most commonly used variant, but one can similarly define the triple factorial (
) and so on. In general, the kth factorial, denoted by
, is defined recursively as
Some mathematicians have suggested an alternative notation of for the double factorial and similarly
for other multifactorials, but this has not come into general use.
In the same way that is not defined for integers, and
is not defined for even integers,
is not defined for
.
Also,
The so-called quadruple factorial, however, is not a multifactorial; it is a much larger number given by .
Neil Sloane and Simon Plouffe defined the superfactorial in 1995 as the product of the first factorials. So the superfactorial of 4 is
In general
The sequence of superfactorials starts (from ) as
This idea was extended in 2000 by Henry Bottomley to the superduperfactorial as the product of the first superfactorials, starting (from
) as
and thus recursively to any multiple-level factorial where the mth-level factorial of is the product of the first
th-level factorials, i.e.
where for
and
.
Clifford Pickover in his 1995 book Keys to Infinity defined the superfactorial of as
or as,
where the (4) notation denotes the hyper4 operator, or using Knuth's up-arrow notation,
This sequence of superfactorials starts:
Here, as is usual for compound exponentiation, the grouping is understood to be from right to left:
Occasionally the hyperfactorial of is considered. It is written as
and defined by
For n = 1, 2, 3, 4, ... the values H(n) are 1, 4, 108, 27648,... (sequence A002109 in OEIS).
The hyperfactorial function is similar to the factorial, but produces larger numbers. The rate of growth of this function, however, is not much larger than a regular factorial. However, H(14) = 1.85...×1099 is already almost equal to a googol, and H(15) = 8.09...×10116 is almost of the same magnitude as the Shannon number, the theoretical number of possible chess games.
The hyperfactorial function can be generalized to complex numbers in a similar way as the factorial function. The resulting function is called the K-function.