Every integer is a quadratic residue for infinitely many primes
From Number
Contents
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).
Related facts
Stronger facts
- 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
Facts used
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
.