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An example of a PrimeSwing computation:
As this example shows an efficient computation of the factorial function reduces to an efficient computation of the swinging factorial n≀. Some information about these numbers can be found here and here. The prime factorization of the swing numbers is crucial for the implementation of the PrimeSwing algorithm.
A concise description of this algorithm is given in this write-up (pdf) and in the SageMath link below (Algo 5).
Link | Content | |
Algorithms | A very short description of 21 algorithms for computing the factorial function n!. | |
X | Julia factorial | *NEW* The factorial function based on the swinging factorial which in turn is computed via prime factorization implemented in Julia. |
Mini Library | The factorial function, the binomial function, the double factorial, the swing numbers and an efficient prime number sieve implemented in Scala and GO. | |
Browse Code | Various algorithms implemented in Java, C# and C++. |
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SageMath | Implementations in SageMath. | |
LISP | Implementations in Lisp. | |
Benchmarks | Benchmark 2013: With MPIR 2.6 you can calculate 100.000.000! in less than a minute provided you use one of the fast algorithms described here. | |
Conclusions | Which algorithm should we choose? | |
Download | Download a test application and benchmark yourself. | |
X | Approximations | A unique collection! Approximation formulas. |
Gamma quot | Bounds for Gamma(x+1)/Gamma(x+1/2) | |
Gamma shift | Why is Gamma(n)=(n-1)! and not Gamma(n)=n! ? | |
X |
Hadamard |
Hadamard's Gamma function and a new factorial function [MathJax version] |
History | Not even Wikipedia knows this! The early history of the factorial function. |
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Notation | On the notation n! | |
Binary Split | For coders only. Go to the page of the day. | |
Sage / Python | Implementation of the swing algorithm. | |
‼ | Double Factorial | The fast double factorial function. |
Prime Factorial | Primfakultaet ('The Primorial', in German.) | |
Bibliography | Bibliography on Inequalities for the Gamma function. | |
X | Bernoulli & Euler |
Exotic Applications: Inclusions for the Bernoulli and Euler numbers. |
Binomial | Fast Binomial Function (Binomial Coefficients). | |
Variations | A combinatorial generalization of the factorial. | |
X | Stieltjes' CF | On Stieltjes' Continued Fraction for the Gamma Function. |
al-Haytham / Lagrange |
The ignorance of some western mathematicians. A deterministic factorial primality test. |
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Factorial Digits | Number of decimal digits of 10n! | |
Calculator | Calculate n! for n up to 9.999.999.999 . | |
RPN-Factorial | The retro-factorial page! | |
Permutations | Awesome! Permutation trees, the combinatorics of n!. | |
Perm. trees | Download a pdf-poster with 120 permutation trees! | |
Gamma LogGamma |
Plots of the factorial (gamma) function. | |
External links | Some bookmarks. |
Fast-Factorial-Functions: The Homepage of Factorial Algorithms. (C) Peter Luschny, 2000-2017. All information and all source code in this directory is free under the Creative Commons Attribution-ShareAlike 3.0 Unported License. This page is listed on the famous "Dictionary of Algorithms and Data Structures" at the National Institute of Standards and Technology's web site (NIST). Apr. 2003 / Apr. 2017 : 800,000 visitors! Thank you!