72-406: Kramp is a surname. Notable people with the surname include: Christian Kramp (1760–1826), French mathematician who worked primarily with factorials Daryl Kramp (1947–2024), Canadian politician Paul L. Kramp (1887–1975), Danish zoologist who worked extensively on jellyfish v t e Surnames associated with
144-717: A double exponential function . Its growth rate is similar to n n {\displaystyle 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: ln n ! = ∑ x = 1 n ln x ≈ ∫ 1 n ln x d x = n ln n − n + 1. {\displaystyle \ln n!=\sum _{x=1}^{n}\ln x\approx \int _{1}^{n}\ln x\,dx=n\ln n-n+1.} Exponentiating
216-408: A 1685 treatise by John Wallis , a study of their approximate values for large values of n {\displaystyle 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
288-467: A common example in the use of different computer programming styles and methods. The computation of n ! {\displaystyle n!} can be expressed in pseudocode using iteration as or using recursion based on its recurrence relation as Other methods suitable for its computation include memoization , dynamic programming , and functional programming . The computational complexity of these algorithms may be analyzed using
360-545: A definition for the factorial at all complex numbers other than the negative integers. 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
432-421: A factor of two to produce one of these trailing zeros. The leading digits of the factorials are distributed according to Benford's law . Every sequence of digits, in any base, is the sequence of initial digits of some factorial number in that base. Another result on divisibility of factorials, Wilson's theorem , states that ( n − 1 ) ! + 1 {\displaystyle (n-1)!+1}
504-697: A function of the number of digits or bits in the result. By Stirling's formula, n ! {\displaystyle n!} has b = O ( n log n ) {\displaystyle b=O(n\log n)} bits. The Schönhage–Strassen algorithm can produce a b {\displaystyle b} -bit product in time O ( b log b log log b ) {\displaystyle O(b\log b\log \log b)} , and faster multiplication algorithms taking time O ( b log b ) {\displaystyle O(b\log b)} are known. However, computing
576-487: A modified form of the factorial, omitting the factors in the factorial that are divisible by p . 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 . The factorial function
648-404: A proof of Euclid's theorem that the number of primes is infinite. When n ! ± 1 {\displaystyle n!\pm 1} is itself prime it is called a factorial prime ; relatedly, Brocard's problem , also posed by Srinivasa Ramanujan , concerns the existence of square numbers of the form n ! + 1 {\displaystyle n!+1} . In contrast,
720-422: A sequence. Factorials appear more broadly in many formulas in combinatorics , to account for different orderings of objects. For instance the binomial coefficients ( n k ) {\displaystyle {\tbinom {n}{k}}} count the k {\displaystyle k} -element combinations (subsets of k {\displaystyle k} elements) from
792-399: A set with n {\displaystyle n} elements, and can be computed from factorials using the formula ( n k ) = n ! k ! ( n − k ) ! . {\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.} The Stirling numbers of the first kind sum to the factorials, and count
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#1732787232258864-468: A specific person led you to this page, you may wish to change that link by adding the person's given name (s) to the link. Retrieved from " https://en.wikipedia.org/w/index.php?title=Kramp&oldid=1205514306 " Category : Surnames Hidden categories: Articles with short description Short description is different from Wikidata All set index articles Christian Kramp Christian Kramp (8 July 1760 – 13 May 1826)
936-418: A subset of exceptions with asymptotic density zero), it coincides with the largest prime factor of x {\displaystyle x} . The product of two factorials, m ! ⋅ n ! {\displaystyle m!\cdot n!} , always evenly divides ( m + n ) ! {\displaystyle (m+n)!} . There are infinitely many factorials that equal
1008-419: Is 1 {\displaystyle 1} , or in symbols, 0 ! = 1 {\displaystyle 0!=1} . There are several motivations for this definition: The earliest uses of the factorial function involve counting permutations : there are n ! {\displaystyle n!} different ways of arranging n {\displaystyle n} distinct objects into
1080-534: Is 1, according to the convention for an empty product . Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature , and by Jewish mystics in the Talmudic book Sefer Yetzirah . The factorial operation is encountered in many areas of mathematics, notably in combinatorics , where its most basic use counts the possible distinct sequences –
1152-574: Is a common feature in scientific calculators . It is also included in scientific programming libraries such as the Python mathematical functions module and the Boost C++ library . If efficiency is not a concern, computing factorials is trivial: just successively multiply a variable initialized to 1 {\displaystyle 1} by the integers up to n {\displaystyle n} . The simplicity of this computation makes it
1224-550: Is a single multiplication of a number with O ( n log n ) {\displaystyle 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 series to O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} . Consequentially,
1296-663: 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 by n {\displaystyle n} : n ! = n ⋅ ( n − 1 ) ! . {\displaystyle n!=n\cdot (n-1)!.} For example, 5 ! = 5 ⋅ 4 ! = 5 ⋅ 24 = 120 {\displaystyle 5!=5\cdot 4!=5\cdot 24=120} . The factorial of 0 {\displaystyle 0}
1368-563: Is divisible by n {\displaystyle n} if and only if n {\displaystyle n} is a prime number . For any given integer x {\displaystyle x} , the Kempner function of x {\displaystyle x} is given by the smallest n {\displaystyle n} for which x {\displaystyle x} divides n ! {\displaystyle n!} . For almost all numbers (all but
1440-440: 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 Γ ( z ) Γ ( 1 − z ) = π sin π z . {\displaystyle \Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin \pi z}}.} However, this formula cannot be used at integers because, for them,
1512-536: Is to perform the multiplications as a divide-and-conquer algorithm that multiplies a sequence of i {\displaystyle i} numbers by splitting it into two subsequences of i / 2 {\displaystyle 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 ) {\displaystyle O(n\log ^{3}n)} : one logarithm comes from
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#17327872322581584-967: The ∼ {\displaystyle \sim } symbol means that, as n {\displaystyle 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: n ! ∼ 2 π n ( n e ) n ( 1 + 1 12 n + 1 288 n 2 − 139 51840 n 3 − 571 2488320 n 4 + ⋯ ) . {\displaystyle 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
1656-416: The sin π z {\displaystyle \sin \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
1728-419: The abc conjecture that there are only finitely many nontrivial examples. The greatest common divisor of the values of a primitive polynomial of degree d {\displaystyle d} over the integers evenly divides d ! {\displaystyle d!} . There are infinitely many ways to extend the factorials to a continuous function . The most widely used of these uses
1800-532: 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. As a function of n {\displaystyle n} , the factorial has faster than exponential growth , but grows more slowly than
1872-471: The Wallis product , which expresses π {\displaystyle \pi } as a limiting ratio of factorials and powers of two. The result of these corrections is Stirling's approximation : n ! ∼ 2 π n ( n e ) n . {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\,.} Here,
1944-465: The exponential generating function , which for a combinatorial class with n i {\displaystyle n_{i}} elements of size i {\displaystyle i} is defined as the power series ∑ i = 0 ∞ x i n i i ! . {\displaystyle \sum _{i=0}^{\infty }{\frac {x^{i}n_{i}}{i!}}.} In number theory ,
2016-414: The factorial of a non-negative integer n {\displaystyle n} , denoted by n ! {\displaystyle n!} , is the product of all positive integers less than or equal to n {\displaystyle n} . The factorial of n {\displaystyle n} also equals the product of n {\displaystyle n} with
2088-459: The gamma function , which can be defined for positive real numbers as the integral Γ ( z ) = ∫ 0 ∞ x z − 1 e − x d x . {\displaystyle \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
2160-583: The gamma function . 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 . The notation n ! {\displaystyle n!} for factorials was introduced by the French mathematician Christian Kramp in 1808. Many other notations have also been used. Another later notation | n _ {\displaystyle \vert \!{\underline {\,n}}} , in which
2232-408: The permutations – of n {\displaystyle n} distinct objects: there are n ! {\displaystyle n!} . In mathematical analysis , factorials are used in power series for the exponential function and other functions, and they also have applications in algebra , number theory , probability theory , and computer science . Much of the mathematics of
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2304-406: The prime number theorem , so the time for the first step is O ( n log 2 n ) {\displaystyle 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
2376-418: The analysis of brute-force searches over permutations, factorials arise in the lower bound of log 2 n ! = n log 2 n − O ( n ) {\displaystyle \log _{2}n!=n\log _{2}n-O(n)} on the number of comparisons needed to comparison sort a set of n {\displaystyle n} items, and in
2448-403: The analysis of chained hash tables , where the distribution of keys per cell can be accurately approximated by a Poisson distribution. 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
2520-547: 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. The word "factorial" (originally French: factorielle ) was first used in 1800 by Louis François Antoine Arbogast , in the first work on Faà di Bruno's formula , but referring to a more general concept of products of arithmetic progressions . The "factors" that this name refers to are
2592-432: The coefficients of other Taylor series (in particular those of the trigonometric and hyperbolic functions ), where they cancel factors of n ! {\displaystyle n!} coming from the n {\displaystyle n} th derivative of x n {\displaystyle x^{n}} . This usage of factorials in power series connects back to analytic combinatorics through
2664-400: 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. Other complex functions that interpolate the factorial values include Hadamard's gamma function , which is an entire function over all the complex numbers, including
2736-721: The correction terms: n ! ∼ 2 π n ( n e ) n exp ( 1 12 n − 1 360 n 3 + 1 1260 n 5 − 1 1680 n 7 + ⋯ ) . {\displaystyle 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
2808-515: 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 to O ( n log 2 n ) {\displaystyle 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 ) {\displaystyle O(n\log ^{2}n)} , because each
2880-496: The denominators of power series , most notably in the series for the exponential function , e x = 1 + x 1 + x 2 2 + x 3 6 + ⋯ = ∑ i = 0 ∞ x i i ! , {\displaystyle 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
2952-471: The equation n ! = Γ ( n + 1 ) , {\displaystyle 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 Γ ( x ) {\displaystyle \Gamma (x)} and Γ ( x − 1 ) {\displaystyle \Gamma (x-1)} are defined,
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3024-420: The factorial function are commonly used as an example of different computer programming styles, and are included in scientific 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 fast multiplication algorithms for numbers with
3096-407: 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 than exponential growth . Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of
3168-907: The factorial implies that n ! {\displaystyle n!} is divisible by all prime numbers that are at most n {\displaystyle n} , and by no larger prime numbers. More precise information about its divisibility is given by Legendre's formula , which gives the exponent of each prime p {\displaystyle p} in the prime factorization of n ! {\displaystyle n!} as ∑ i = 1 ∞ ⌊ n p i ⌋ = n − s p ( n ) p − 1 . {\displaystyle \sum _{i=1}^{\infty }\left\lfloor {\frac {n}{p^{i}}}\right\rfloor ={\frac {n-s_{p}(n)}{p-1}}.} Here s p ( n ) {\displaystyle s_{p}(n)} denotes
3240-722: 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 1 to 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 ) {\displaystyle O(n\log ^{2}n)} each, giving total time O ( n 2 log 2 n ) {\displaystyle O(n^{2}\log ^{2}n)} . A better approach
3312-620: The factorial, used to analyze comparison sorting , can be very accurately estimated using Stirling's approximation. In the formula below, the O ( 1 ) {\displaystyle O(1)} term invokes big O notation . log 2 n ! = n log 2 n − ( log 2 e ) n + 1 2 log 2 n + O ( 1 ) . {\displaystyle \log _{2}n!=n\log _{2}n-(\log _{2}e)n+{\frac {1}{2}}\log _{2}n+O(1).} The product formula for
3384-665: The factorials arise through the binomial theorem , which uses binomial coefficients to expand powers of sums. 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 . Their use in counting permutations can also be restated algebraically: the factorials are the orders of finite symmetric groups . In calculus , factorials occur in Faà di Bruno's formula for chaining higher derivatives. In mathematical analysis , factorials frequently appear in
3456-431: The factorials. Daniel Bernoulli and Leonhard Euler interpolated the factorial function to a continuous function of complex numbers , except at the negative integers, the (offset) gamma function . Many other notable functions and number sequences are closely related to the factorials, including the binomial coefficients , double factorials , falling factorials , primorials , and subfactorials . Implementations of
3528-401: The factorization of a binomial coefficient. Grouping the prime factors of the factorial into prime powers in different ways produces the multiplicative partitions of factorials . The special case of Legendre's formula for p = 5 {\displaystyle p=5} gives the number of trailing zeros in the decimal representation of the factorials. According to this formula,
3600-471: The first results of Paul Erdős , was based on the divisibility properties of factorials. The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials. Factorials are used extensively in probability theory , for instance in the Poisson distribution and in the probabilities of random permutations . In computer science , beyond appearing in
3672-427: The gamma function obeys the functional equation Γ ( n ) = ( n − 1 ) Γ ( n − 1 ) , {\displaystyle \Gamma (n)=(n-1)\Gamma (n-1),} generalizing the recurrence relation for the factorials. The same integral converges more generally for any complex number z {\displaystyle z} whose real part
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#17327872322583744-624: The generalised factorial function which applied to non- integers . His work on factorials is independent of that of James Stirling and Vandermonde . He was the first to use the notation n ! ( Elements d'arithmétique universelle , 1808). In fact, the more general concept of factorial was found at the same time by Arbogast . I use the very simple notation n ! to designate the product of numbers decreasing from n to unity, i.e. n ( n − 1)( n − 2) ... 3 . 2 . 1. The constant use in combinatorial analysis, in most of my proofs, that I make of this idea, has made this notation necessary. ... I have given it
3816-449: The most salient property of factorials is the divisibility of n ! {\displaystyle n!} by all positive integers up to n {\displaystyle 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 ! ± 1 {\displaystyle n!\pm 1} , leading to
3888-499: The name 'faculty'. Arbogast has substituted the name 'factorial' which is clearer and more French. In adopting his idea I congratulate myself on paying homage to the memory of my friend. Kramp's function , a scaled complex error function , is today better known as the Faddeeva function . This article about a French mathematician is a stub . You can help Misplaced Pages by expanding it . Factorial In mathematics ,
3960-798: The next smaller factorial: n ! = n × ( n − 1 ) × ( n − 2 ) × ( n − 3 ) × ⋯ × 3 × 2 × 1 = n × ( n − 1 ) ! {\displaystyle {\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 × 4 ! = 5 × 4 × 3 × 2 × 1 = 120. {\displaystyle 5!=5\times 4!=5\times 4\times 3\times 2\times 1=120.} The value of 0!
4032-423: The non-positive integers. 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 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
4104-407: The number of bits in the factorial, a second comes from the multiplication algorithm, and a third comes from the divide and conquer. Even better efficiency is obtained by computing n ! from its prime factorization, based on the principle that exponentiation by squaring is faster than expanding an exponent into a product. An algorithm for this by Arnold Schönhage begins by finding the list of
4176-460: 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. Legendre's formula implies that the exponent of the prime p = 2 {\displaystyle p=2} is always larger than the exponent for p = 5 {\displaystyle p=5} , so each factor of five can be paired with
4248-423: The numbers n ! + 2 , n ! + 3 , … n ! + n {\displaystyle n!+2,n!+3,\dots n!+n} must all be composite, proving the existence of arbitrarily large prime gaps . An elementary proof of Bertrand's postulate on the existence of a prime in any interval of the form [ n , 2 n ] {\displaystyle [n,2n]} , one of
4320-464: The occupation of locksmith Germanic Schlosser , Kramp , Krampe Romance Serrurier Fabbro , Cerrajero Chaveiro, Lăcătuș Slavic Slesar Slesarev Zámečník Ślusarz Other Lakatos Lukkoseppä lukksepp Atslēdznieks Šaltkalvis [REDACTED] Surname list This page lists people with the surname Kramp . If an internal link intending to refer to
4392-454: The permutations of n {\displaystyle n} grouped into subsets with the same numbers of cycles. Another combinatorial application is in counting derangements , permutations that do not leave any element in its original position; the number of derangements of n {\displaystyle n} items is the nearest integer to n ! / e {\displaystyle n!/e} . In algebra ,
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#17327872322584464-458: The primes up to n {\displaystyle 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: The product of all primes up to n {\displaystyle n} is an O ( n ) {\displaystyle O(n)} -bit number, by
4536-949: 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 ) ! {\displaystyle (n-1)!} . The only known examples of factorials that are products of other factorials but are not of this "trivial" form are 9 ! = 7 ! ⋅ 3 ! ⋅ 3 ! ⋅ 2 ! {\displaystyle 9!=7!\cdot 3!\cdot 3!\cdot 2!} , 10 ! = 7 ! ⋅ 6 ! = 7 ! ⋅ 5 ! ⋅ 3 ! {\displaystyle 10!=7!\cdot 6!=7!\cdot 5!\cdot 3!} , and 16 ! = 14 ! ⋅ 5 ! ⋅ 2 ! {\displaystyle 16!=14!\cdot 5!\cdot 2!} . It would follow from
4608-782: The recursive version takes linear space to store its call stack . However, this model of computation is only suitable when n {\displaystyle n} is small enough to allow n ! {\displaystyle n!} to fit into a machine word . The values 12! and 20! are the largest factorials that can be stored in, respectively, the 32-bit and 64-bit integers . Floating point can represent larger factorials, but approximately rather than exactly, and will still overflow for factorials larger than 170 ! {\displaystyle 170!} . The exact computation of larger factorials involves arbitrary-precision arithmetic , because of fast growth and integer overflow . Time of computation can be analyzed as
4680-560: The result (and ignoring the negligible + 1 {\displaystyle +1} term) approximates n ! {\displaystyle n!} as ( n / e ) n {\displaystyle (n/e)^{n}} . 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 to n {\displaystyle {\sqrt {n}}} . The constant of proportionality for this correction can be found from
4752-417: The same number of digits. The concept of factorials has arisen independently in many cultures: 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. Christopher Clavius discussed factorials in a 1603 commentary on
4824-420: The sum of the base - p {\displaystyle p} digits of n {\displaystyle 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
4896-736: The terms of the product formula for the factorial. 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} n ! = 1 ⋅ 2 ⋅ 3 ⋯ ( n − 2 ) ⋅ ( n − 1 ) ⋅ n . {\displaystyle n!=1\cdot 2\cdot 3\cdots (n-2)\cdot (n-1)\cdot n.} This may be written more concisely in product notation as n ! = ∏ i = 1 n i . {\displaystyle n!=\prod _{i=1}^{n}i.} If this product formula
4968-480: 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 ! {\displaystyle n!} in time O ( n ) {\displaystyle O(n)} , and the iterative version uses space O ( 1 ) {\displaystyle O(1)} . Unless optimized for tail recursion ,
5040-547: 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. 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 . Other important works of early European mathematics on factorials include extensive coverage in
5112-518: Was a French mathematician, who worked primarily with factorials . Christian Kramp's father was his teacher at grammar school in Strasbourg . Kramp studied medicine and graduated; however, his interests certainly ranged outside medicine, for in addition to a number of medical publications he published a work on crystallography in 1793. In 1795, France annexed the Rhineland area in which Kramp
5184-568: Was carrying out his work and after this he became a teacher at Cologne (this city was French from 1794 to 1815), teaching mathematics, chemistry , and physics . Kramp could read and write in German and French. Kramp was appointed professor of mathematics at Strasbourg , the town of his birth, in 1809. He was elected to the geometry section of the French Academy of Sciences in 1817. As Bessel , Legendre and Gauss did, Kramp worked on
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