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.
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 ⓘ |
Referenced by (1)
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