Fermat's little theorem
E146189
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Fermat's little theorem canonical | 1 |
| Fermat’s little theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1281482 — 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: Fermat's little theorem Context triple: [Pierre de Fermat, notableWork, Fermat's little theorem]
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A.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
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B.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
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C.
Pierre de Fermat
Pierre de Fermat was a 17th-century French mathematician renowned for his work in number theory, probability, and analytic geometry, and especially for Fermat's Last Theorem.
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D.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
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E.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
- 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: Fermat's little theorem Target entity description: 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.
-
A.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
B.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
-
C.
Pierre de Fermat
Pierre de Fermat was a 17th-century French mathematician renowned for his work in number theory, probability, and analytic geometry, and especially for Fermat's Last Theorem.
-
D.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
E.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in number theory
ⓘ
theorem ⓘ |
| appliesTo |
finite fields
ⓘ
modular arithmetic ⓘ |
| assumption | p is greater than 1 ⓘ |
| category | elementary number theory ⓘ |
| condition |
a is an integer
ⓘ
gcd(a,p) = 1 for the form a^(p−1) ≡ 1 (mod p) ⓘ p is a prime number ⓘ |
| failsFor | composite moduli in general ⓘ |
| field | number theory ⓘ |
| generalizedBy |
Euler’s theorem
ⓘ
surface form:
Euler's theorem
|
| hasConsequence |
existence of multiplicative inverses modulo a prime
ⓘ
structure of the multiplicative group modulo a prime is cyclic of order p−1 ⓘ |
| hasVariant |
exponent p form
ⓘ
exponent p−1 form ⓘ |
| historicalPeriod | 17th century mathematics ⓘ |
| implies |
a^(p−1) − 1 is divisible by p when gcd(a,p)=1
ⓘ
p divides a^p − a for any integer a ⓘ |
| importance |
basic result taught in undergraduate number theory
ⓘ
fundamental tool in modern public-key cryptography ⓘ |
| isPartOf | classical results on primes ⓘ |
| languageOfOriginalFormulation | Latin ⓘ |
| namedAfter | Pierre de Fermat ⓘ |
| proofTechnique |
combinatorial arguments
ⓘ
group theory ⓘ induction on exponents ⓘ |
| relatedConcept |
Carmichael number
ⓘ
Fermat pseudoprime ⓘ |
| relatedTo |
Chinese remainder theorem
ⓘ
Euler’s theorem ⓘ
surface form:
Euler's theorem
Wilson's theorem ⓘ |
| statement |
For any integer a and prime p, a^p ≡ a (mod p).
ⓘ
If p is a prime and a is an integer not divisible by p, then a^(p−1) ≡ 1 (mod p). ⓘ |
| topic |
congruences
ⓘ
multiplicative group of integers modulo p ⓘ properties of prime numbers ⓘ |
| usedFor |
computing modular inverses when modulus is prime
ⓘ
simplifying large exponent computations modulo a prime ⓘ |
| usedIn |
Diffie–Hellman key exchange
ⓘ
ElGamal ⓘ
surface form:
ElGamal encryption
Fermat primality test ⓘ RSA ⓘ
surface form:
RSA cryptosystem
cryptography ⓘ modular exponentiation algorithms ⓘ primality testing ⓘ |
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: Fermat's little theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Fermat’s little theorem