Every

natural number greater than 1 has a

unique prime factorisation, up to

permutation of the

factors.

Every nonzero non-unit (not 1 or -1) integer has a unique prime factorisation, up to permutation of the factors and replacement of some factors by their associates.

An integral domain for which the above holds (substituting `irreducible' for `prime'; I think it matters) is called a `unique factorisation domain'. In all UFDs, an element is prime iff it is irreducible.

Every principal ideal domain (and hence every Euclidean domain) is a unique factorization domain.