 Euclid's proof shows that every finite list of primes is incomplete.The key idea is to multiply together the primes in any given list and add formula_42 If the list consists of the primes formula_43 this gives the number
By the fundamental theorem, formula_45 has a prime factorization
with one or more prime factors.formula_45 is evenly divisible by each of these factors, but formula_45 has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of formula_45 can be in the given list.