# Every integer is a quadratic residue for infinitely many primes

## Statement

Let  be a nonzero integer. Then, there are infinitely many primes  such that  is a quadratic residue modulo . (The statement also holds for , if we declare  to be a quadratic residue).

## Proof

By fact (1), there are infinitely many primes  such that  divides  for some natural number . We can exclude the primes that divide  itself (if we follow the convention that  is not considered a quadratic residue) and we still obtain that there are infinitely many primes  for which  is a quadratic residue.

Note that this proof generalizes to -adic residues for any .