Euler’s theorem

E300757

result in modular arithmetic theorem in number theory

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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All labels observed (2)

Label	Occurrences
Euler's theorem	2
Euler’s theorem canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T2815395 — 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: Euler’s theorem
Context triple: [Euler’s totient function φ(n), roleIn, Euler’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. de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
C. Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
D. Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
E. Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
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: Euler’s theorem
Target entity description: 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.

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. de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
C. Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
D. Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
E. Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	result in modular arithmetic ⓘ theorem in number theory ⓘ
appearsIn	introductory number theory textbooks ⓘ
appliesTo	any integer a with gcd(a,n)=1 ⓘ any positive integer n ≥ 1 ⓘ
assumption	gcd(a,n)=1 ⓘ
category	elementary number theory ⓘ multiplicative number theory ⓘ
conclusion	a^{φ(n)} ≡ 1 (mod n) ⓘ
contrastsWith	Wilson's theorem ⓘ surface form: Wilson’s theorem
dependsOn	finiteness of (Z/nZ)^× ⓘ group exponent properties ⓘ
domain	integers coprime to n ⓘ
equivalentForm	For a in (Z/nZ)^×, a^{\| (Z/nZ)^× \|} = 1 ⓘ
field	modular arithmetic ⓘ number theory ⓘ
generalizes	Fermat's little theorem ⓘ surface form: Fermat’s little theorem
hasComponent	Euler’s totient function φ(n) ⓘ condition of coprimality ⓘ modulo n congruence relation ⓘ
holdsIn	multiplicative group of units modulo n ⓘ
implies	a^{k·φ(n)} ≡ 1 (mod n) for any integer k ⓘ a^{φ(n)+m} ≡ a^{m} (mod n) when gcd(a,n)=1 ⓘ
isStrongerThan	Fermat’s little theorem for composite moduli ⓘ
language	mathematical notation ⓘ
namedAfter	Leonhard Euler ⓘ
proofMethod	combinatorial counting argument ⓘ group-theoretic argument ⓘ
relatedTo	Chinese remainder theorem ⓘ group theory ⓘ multiplicative group modulo n ⓘ
requires	φ(n) to be well-defined ⓘ
specialCase	Fermat’s little theorem when n is prime ⓘ
statedAs	If gcd(a,n)=1 then a^{φ(n)} ≡ 1 (mod n) ⓘ
usedFor	computing modular inverses ⓘ reducing large exponents modulo n ⓘ simplifying congruences ⓘ
usedIn	RSA ⓘ surface form: RSA cryptosystem modular exponentiation algorithms ⓘ primality testing methods ⓘ public-key cryptography ⓘ
usesConcept	Euler’s totient function φ(n) ⓘ surface form: Euler’s totient function greatest common divisor ⓘ modular congruence ⓘ
validFor	composite moduli ⓘ prime moduli ⓘ
yearIntroducedApprox	18th century ⓘ

How these facts were elicited

Referenced by (3)

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

Euler’s totient function φ(n) → roleIn → Euler’s theorem ⓘ

Fermat's little theorem → relatedTo → Euler’s theorem ⓘ

this entity surface form: Euler's theorem

Fermat's little theorem → generalizedBy → Euler’s theorem ⓘ