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.
-
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
- 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 ⓘ |
| relatedTo |
Fermat's little theorem
ⓘ
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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.
linked to: Wilson's theorem