fundamental theorem of arithmetic

E451510

mathematical theorem theorem in number theory

The fundamental theorem of arithmetic states that every integer greater than 1 can be written uniquely (up to the order of factors) as a product of prime numbers.

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All labels observed (3)

Label	Occurrences
fundamental theorem of arithmetic canonical	1
prime factorization theorem	1
unique factorization theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T4552078 — 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: fundamental theorem of arithmetic
Context triple: [An Introduction to the Theory of Numbers, coversConcept, fundamental theorem of arithmetic]

A. prime number theorem
The prime number theorem is a fundamental result in number theory that describes how prime numbers become less frequent and provides an approximate formula for the number of primes less than a given large number.
B. 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.
C. 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.
D. Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
E. 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.
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: fundamental theorem of arithmetic
Target entity description: The fundamental theorem of arithmetic states that every integer greater than 1 can be written uniquely (up to the order of factors) as a product of prime numbers.

A. prime number theorem
The prime number theorem is a fundamental result in number theory that describes how prime numbers become less frequent and provides an approximate formula for the number of primes less than a given large number.
B. 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.
C. 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.
D. Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
E. 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.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in number theory ⓘ
alsoKnownAs	prime factorization theorem ⓘ linked to: fundamental theorem of arithmetic unique factorization theorem ⓘ linked to: fundamental theorem of arithmetic
appearsIn	Disquisitiones Arithmeticae ⓘ
appliesTo	integers greater than 1 ⓘ
clarification	Uniqueness means that if n = p1⋯pk = q1⋯ql with primes pi, qj, then k = l and the primes are the same up to permutation. ⓘ
conclusion	Every integer greater than 1 has at least one prime factorization. ⓘ Every integer greater than 1 has at most one prime factorization up to ordering of factors. ⓘ
dependsOn	Euclidean algorithm ⓘ linked to: Euclidean algorithm for polynomials basic properties of divisibility ⓘ
doesNotHoldIn	all integral domains ⓘ
domainOfQuantification	set of positive integers greater than 1 ⓘ
example	12 = 2^2 × 3 is the unique prime factorization of 12 up to order ⓘ 30 = 2 × 3 × 5 is the unique prime factorization of 30 up to order ⓘ
excludes	integer 0 ⓘ integer 1 ⓘ
field	number theory ⓘ
generalizationOf	unique factorization in principal ideal domains ⓘ
historicalAttribution	Carl Friedrich Gauss popularized its modern formulation ⓘ
holdsIn	ring of integers ⓘ
implies	existence of prime factorization for each integer greater than 1 ⓘ uniqueness of prime factorization for each integer greater than 1 ⓘ
importance	foundational result in elementary number theory ⓘ
logicalForm	existence and uniqueness theorem ⓘ
relatesToConcept	arithmetic of integers ⓘ composite number ⓘ divisibility ⓘ greatest common divisor ⓘ least common multiple ⓘ prime number ⓘ
role	basis for many proofs involving integers and primes ⓘ
specialCaseOf	unique factorization domain theory ⓘ
statement	Every integer greater than 1 can be written as a product of prime numbers. ⓘ This factorization into primes is unique up to the order of the factors. ⓘ
usedFor	defining greatest common divisors via prime exponents ⓘ proving properties of arithmetic functions ⓘ proving properties of divisibility ⓘ proving that there are infinitely many primes ⓘ proving the Euclidean algorithm properties ⓘ proving uniqueness of representation in base systems ⓘ
usedIn	algebra ⓘ coding theory ⓘ computational number theory ⓘ cryptography ⓘ elementary number theory ⓘ
yearFormalized	1801 ⓘ

How these facts were elicited

Referenced by (3)

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

An Introduction to the Theory of Numbers → coversConcept → fundamental theorem of arithmetic ⓘ

fundamental theorem of arithmetic → alsoKnownAs → unique factorization theorem ⓘ

linked to: fundamental theorem of arithmetic

fundamental theorem of arithmetic → alsoKnownAs → prime factorization theorem ⓘ