fundamental theorem of arithmetic
E451510
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| fundamental theorem of arithmetic canonical | 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]
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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.
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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.
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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
NERFINISHED
ⓘ
unique factorization theorem NERFINISHED ⓘ |
| appearsIn | Disquisitiones Arithmeticae NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: fundamental theorem of arithmetic Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.