Every integer is a quadratic residue for infinitely many primes
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).
- Artin's conjecture on primitive roots
- Every integer that is not a perfect square is a quadratic nonresidue for infinitely many primes
- Dirichlet's theorem on primes in arithmetic progressions
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 .