Fundamental theorem of arithmetic
Every integer greater than 1 can be expressed as a product of primes in exactly one way, apart from the order of the factors.
Prime number: A natural number greater than 1 whose only positive divisors are 1 and itself. The theorem identifies primes as the indivisible factors from which every integer above 1 is built.
Integer: A whole number that is zero, positive, or negative. The theorem concerns integers greater than 1, whose prime factors are positive integers.
Least common multiple: The smallest positive integer divisible by each of a given set of integers. Prime exponents in factorizations give a direct way to compute least common multiples.
Unique factorization domain: An integral domain in which every nonzero nonunit factors uniquely into irreducible elements, up to units and order. The theorem is the integer case of this broader algebraic property.
Euclid's lemma: If a prime divides a product of integers, it divides at least one factor. This divisibility rule forces any prime in one factorization to appear in every other.
Divisibility: A relation in which one integer is a multiple of another integer. Prime factors are characterized through divisibility, the relation used to compare factorizations.
Divisor function: An arithmetic function that counts or sums the positive divisors of an integer. A number's prime exponents determine its number and sum of positive divisors.
Ring of integers of Q(√−5): The ring of integers consisting of numbers a + b√−5, with integers a and b. In this ring, 6 has distinct factorizations into irreducibles, unlike in the ordinary integers.
Mathematical induction: A proof method that establishes a base case and then derives each successive case from its predecessor. Induction proves that repeated prime extraction eventually factors every integer greater than 1.
Greatest common divisor: The largest positive integer that divides each of two or more integers. Coprimality, expressed through greatest common divisors, helps explain why prime factors cannot be redistributed.