In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), the factorial satisfies
exactly when n is a prime number. In other words, any integer n > 1 is a prime number if, and only if, (n − 1)! + 1 is divisible by n.The Universal Book of Mathematics. David Darling, p. 350.
History The theorem was first stated by Ibn al-Haytham . Edward Waring announced the theorem in 1770 without proving it, crediting his student John Wilson for the discovery.Edward Waring, Meditationes Algebraicae (Cambridge, England: 1770), page 218 (in Latin). In the third (1782) edition of Waring's Meditationes Algebraicae, Wilson's theorem appears as problem 5 on page 380. On that page, Waring states: "Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger." (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.) Lagrange gave the first proof in 1771.Joseph Louis Lagrange, "Demonstration d'un théorème nouveau concernant les nombres premiers" (Proof of a new theorem concerning prime numbers), Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres (Berlin), vol. 2, pages 125–137 (1771). There is evidence that Leibniz was also aware of the result a century earlier, but never published it.Giovanni Vacca (1899) "Sui manoscritti inediti di Leibniz" (On unpublished manuscripts of Leibniz), Bollettino di bibliografia e storia delle scienze matematiche ... (Bulletin of the bibliography and history of mathematics), vol. 2, pages 113–116; see page 114 (in Italian). Vacca quotes from Leibniz's mathematical manuscripts kept at the Royal Public Library in Hanover (Germany), vol. 3 B, bundle 11, page 10: Original : Inoltre egli intravide anche il teorema di Wilson, come risulta dall'enunciato seguente:"Productus continuorum usque ad numerum qui antepraecedit datum divisus per datum relinquit 1 (vel complementum ad unum?) si datus sit primitivus. Si datus sit derivativus relinquet numerum qui cum dato habeat communem mensuram unitate majorem."Egli non giunse pero a dimostrarlo.Translation : In addition, he [Leibniz] also glimpsed Wilson's theorem, as shown in the following statement:"The product of all integers preceding the given integer, when divided by the given integer, leaves 1 (or the complement of 1?) if the given integer be prime. If the given integer be composite, it leaves a number which has a common factor with the given integer [which is] greater than one."However, he didn't succeed in proving it. See also: Giuseppe Peano, ed., Formulaire de mathématiques, vol. 2, no. 3, page 85 (1897).
Example For each of the values of n from 2 to 30, the following table shows the number (n − 1)! and the remainder when (n − 1)! is divided by n. (In the notation of modular arithmetic, the remainder when m is divided by n is written m mod n.) As expected, when n is prime. The background color is blue for prime values of n, gold for composite values. + Table of factorial and its remainder modulo n 2 1 1 3 2 2 4 6 2 5 24 4 6 120 0 7 720 6 8 5040 0 9 40320 0 10 362880 0 11 3628800 10 12 39916800 0 13 479001600 12 14 6227020800 0 15 87178291200 0 16 1307674368000 0 17 20922789888000 16 18 355687428096000 0 19 6402373705728000 18 20 121645100408832000 0 21 2432902008176640000 0 22 51090942171709440000 0 23 1124000727777607680000 22 24 25852016738884976640000 0 25 620448401733239439360000 0 26 15511210043330985984000000 0 27 403291461126605635584000000 0 28 10888869450418352160768000000 0 29 304888344611713860501504000000 28 30 8841761993739701954543616000000 0
Proofs As a biconditional (if and only if) statement, the proof has two halves: to show that equality does not hold when is composite, and to show that it does hold when is prime.
Composite modulus Suppose that is composite. Therefore, it is divisible by some prime number where . Because divides , there is an integer such that . Suppose for the sake of contradiction that were congruent to modulo . Then would also be congruent to modulo : indeed, if then for some integer , and consequently is one less than a multiple of . On the other hand, since , one of the factors in the expanded product is . Therefore . This is a contradiction; therefore it is not possible that when is composite.
In fact, more is true. With the sole exception of the case , where , if is composite then is congruent to 0 modulo . The proof can be divided into two cases: First, if can be factored as the product of two unequal numbers, , where , then both and will appear as factors in the product and so is divisible by . If has no such factorization, then it must be the square of some prime larger than 2. But then , so both and will be factors of , and so divides in this case, as well.
Prime modulus
The first two proofs below use the fact that the residue classes modulo a prime number form a finite field (specifically, a prime field).
Elementary proof
The result is trivial when , so assume is an odd prime, . Since the residue classes modulo form a field, every non-zero residue has a unique multiplicative inverse . Euclid's lemma implies that the only values of for which are . Therefore, with the exception of , the factors in the expanded form of can be arranged in disjoint pairs such that product of each pair is congruent to 1 modulo . This proves Wilson's theorem.
For example, for , one has
Proof using Fermat's little theorem
Again, the result is trivial for p = 2, so suppose p is an odd prime, . Consider the polynomial
g has degree , leading term , and constant term . Its roots are 1, 2, ..., .
Now consider
h also has degree and leading term . Modulo p, Fermat's little theorem says it also has the same roots, 1, 2, ..., .
Finally, consider
f has degree at most p − 2 (since the leading terms cancel), and modulo p also has the roots 1, 2, ..., . But Lagrange's theorem says it cannot have more than p − 2 roots. Therefore, f must be identically zero (mod p), so its constant term is . This is Wilson's theorem.
Proof using elementary Group Theory
Let p be an odd prime. Let denote the multiplicative group of integers modulo p, such that . Since is cyclic, there exists a group isomorphism , where is the additive group of integers modulo p - 1. Therefore:
Since p was an odd prime, has order 2 inside . Since isomorphisms preserve the orders of elements, it follows that also has order exactly 2. The only such element in is .
Proof using the Sylow theorems
It is possible to deduce Wilson's theorem from a particular application of the Sylow theorems. Let p be a prime. It is immediate to deduce that the symmetric group has exactly elements of order p, namely the p-cycles . On the other hand, each Sylow p-subgroup in is a copy of . Hence it follows that the number of Sylow p-subgroups is . The third Sylow theorem implies
Multiplying both sides by gives
that is, the result.
Applications
Primality tests In practice, Wilson's theorem is useless as a primality test because computing (n − 1)! modulo n for large n is computationally complex.Lagrange, p. 132: "cette méthode devient extrémement laborieuse, & presque impracticable"
Quadratic residues Using Wilson's Theorem, for any odd prime , we can rearrange the left hand side of
to obtain the equality
This becomes
or
We can use this fact to prove part of a famous result: for any prime p such that p ≡ 1 (mod 4), the number (−1) is a square (quadratic residue) mod p. For this, suppose p = 4k + 1 for some integer k. Then we can take m = 2k above, and we conclude that (m!)2 is congruent to (−1) (mod p).
Formulas for primes Wilson's theorem has been used to construct formulas for primes, but they are too slow to have practical value.
p-adic gamma function Wilson's theorem allows one to define the p-adic gamma function.
Gauss's generalization Gauss provedGauss, DA, art. 78 that
where p represents an odd prime and a positive integer. That is, the product of the positive integers less than and relatively prime to is one less than a multiple of when is equal to 4, or a power of an odd prime, or twice a power of an odd prime; otherwise, the product is one more than a multiple of . The values of m for which the product is −1 are precisely the ones where there is a primitive root modulo m.
See also Wilson prime Table of congruences Agoh–Giuga conjecture
Notes
References
The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. English translation: German translation: .
External links
Mizar system proof: http://mizar.org/version/current/html/nat_5.html#T22