The fundamental theorem of arithmetic

The formal name of the result we have already met is telling: the ancient Greeks regarded it as the theorem, the brick on which the whole of arithmetic rests.

Theorem — Fundamental theorem of arithmetic (FTA)

Every natural number $n>1$ can be written in a unique way (up to the order of the factors) as a product of prime numbers: $n = p_1^{a_1}\cdot p_2^{a_2}\cdots p_k^{a_k}, \quad p_1<p_2<\cdots<p_k, \quad a_i\ge 1.$

The writing with prime bases in increasing order and exponents $\ge 1$ is the canonical form of the factorisation: it is the one we shall use to compare numbers, compute GCD and LCM, and recognise perfect squares.

Links

Topics: Numbers and operations
Concepts: Prime factorisation · Prime number · Fundamental theorem of arithmetic
Methods: FTA factorisation
Skills: Factorise