Fermat's little theorem

E146189

result in number theory 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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All labels observed (2)

Label	Occurrences
Fermat's little theorem canonical	4
Fermat’s little theorem	2

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]

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
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 ⓘ linked to: 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 ⓘ linked to: 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 encryption ⓘ linked to: ElGamal Fermat primality test ⓘ RSA cryptosystem ⓘ linked to: RSA cryptography ⓘ modular exponentiation algorithms ⓘ primality testing ⓘ

How these facts were elicited

Referenced by (6)

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

Pierre de Fermat → notableWork → Fermat's little theorem ⓘ

Euler’s theorem → generalizes → Fermat’s little theorem ⓘ

linked to: Fermat's little theorem

Fermat primality test → basedOn → Fermat's little theorem ⓘ

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

Carmichael number → relatedTo → Fermat’s little theorem ⓘ

linked to: Fermat's little theorem

Euler criterion → involvesConcept → Fermat's little theorem ⓘ

All labels observed (2)

How this entity was disambiguated Show

Statements (46)

How these facts were elicited Show

Referenced by (6)

How this entity was disambiguated

How these facts were elicited