Wilson's theorem
E530311
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Wilson's theorem canonical | 1 |
| Wilson’s theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570567 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Wilson's theorem Context triple: [Fermat's little theorem, relatedTo, Wilson's theorem]
-
A.
Fermat's little theorem
Fermat's little theorem is a fundamental result in number theory that characterizes how prime numbers interact with integer powers modulo that prime, forming the basis for many modern cryptographic algorithms.
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B.
Euler’s theorem
Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
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C.
Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
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D.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
E.
Fermat number
A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Wilson's theorem Target entity description: 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.
-
A.
Fermat's little theorem
Fermat's little theorem is a fundamental result in number theory that characterizes how prime numbers interact with integer powers modulo that prime, forming the basis for many modern cryptographic algorithms.
-
B.
Euler’s theorem
Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
-
C.
Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
-
D.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
E.
Fermat number
A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Wilson's theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Wilson’s theorem