The fundamental theorem of arithmetic states that every integer greater than 1 can be factored uniquely (up to the ordering of the factors) into a product of powers of one or more primes. So denoting the primes 2, 3, 5 … as , , , …, each integer can be factored as , where and are the index and power of the prime in the factorization. For example, , or .
After that, each index and exponent can likewise be factored: . Because the recursive process ends with all indexes and exponents as 1, the ones can be suppressed without loss of meaning, so that .
This Demonstration shows this representation of any integer factorization.