Wilson's theorem

E530311

theorem in number theory

Wilson's theorem is a result in number theory stating that a positive integer n > 1 is prime if and only if the factorial of (n − 1) is congruent to −1 modulo n.

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Observed surface forms (1)

Surface form	Occurrences
Wilson’s theorem	1

Statements (43)

Predicate	Object
instanceOf	theorem in number theory ⓘ
appearsIn	elementary number theory textbooks ⓘ
appliesTo	positive integers n > 1 ⓘ
assumes	n is an integer greater than 1 ⓘ
canBeProvedUsing	pairing of inverses modulo p ⓘ properties of multiplicative group modulo p ⓘ
category	results about primes ⓘ results in modular arithmetic ⓘ
characterizes	prime numbers ⓘ
equivalenceType	if and only if condition ⓘ
equivalentFormulation	In (Z/pZ)*, the product of all elements equals −1. ⓘ The product of all nonzero residues modulo a prime p is congruent to −1 modulo p. ⓘ
example	For n = 4, (4 − 1)! = 6 ≡ 2 (mod 4), so 4 is not prime. ⓘ For n = 6, (6 − 1)! = 120 ≡ 0 (mod 6), so 6 is not prime. ⓘ For p = 5, (5 − 1)! = 24 ≡ −1 (mod 5). ⓘ For p = 7, (7 − 1)! = 720 ≡ −1 (mod 7). ⓘ
failsFor	composite numbers n > 4 ⓘ
field	number theory ⓘ
generalization	Wilson primes ⓘ
givesConditionFor	n to be prime ⓘ
hasConsequence	If (n − 1)! is not congruent to −1 modulo n, then n is composite. ⓘ If p is prime, then (p − 1)! + 1 is a multiple of p. ⓘ
historicalAttribution	first published proof by Joseph-Louis Lagrange ⓘ known to Ibn al-Haytham (Alhazen) before Wilson ⓘ
holdsFor	every prime number p ⓘ
implies	If (n − 1)! ≡ −1 (mod n), then n is prime. ⓘ If n is prime, then (n − 1)! + 1 is divisible by n. ⓘ
involvesOperation	multiplication of all nonzero residues modulo n ⓘ
isCriterionFor	primality ⓘ
isNot	efficient primality test for large n ⓘ
namedAfter	John Wilson NERFINISHED ⓘ
relatedTo	Fermat's little theorem NERFINISHED ⓘ group of units modulo p ⓘ primality test ⓘ
statement	A positive integer n > 1 is prime if and only if (n − 1)! ≡ −1 (mod n). ⓘ For a prime p, (p − 1)! ≡ −1 (mod p). ⓘ If n is composite and n > 4, then (n − 1)! ≡ 0 (mod n). ⓘ
usedAs	example of a necessary and sufficient condition in number theory ⓘ
usedIn	theoretical characterization of primes ⓘ
usesConcept	congruence modulo n ⓘ factorial ⓘ modular arithmetic ⓘ
yearProved	1771 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Fermat's little theorem → relatedTo → Wilson's theorem ⓘ

Euler’s theorem → contrastsWith → Wilson's theorem ⓘ

this entity surface form: Wilson’s theorem