Every integer that is not a perfect square is a quadratic nonresidue for infinitely many primes
From Number
Contents
Statement
Suppose is an integer that is not a perfect square. Then, there are infinitely many primes
such that
is a quadratic nonresidue modulo
.
Related facts and conjectures
Corollaries
Similar facts
- Artin's conjecture on primitive roots
- Every integer is a quadratic residue for infinitely many primes
- Every prime is the smallest quadratic nonresidue for infinitely many primes
Facts used
- Dirichlet's theorem on primes in arithmetic progressions
- Quadratic reciprocity
- Congruence condition for minus one to be a quadratic residue
- Congruence condition for two to be a quadratic residue
Proof
Proof idea
By a combination of facts (2), (3), and (4), we can show that whether or not is a quadratic residue modulo
depends on whether
lies in certain congruence classes modulo
. Further, if
is not a perfect square, there is at least one congruence class modulo
such that
is a quadratic nonresidue modulo any prime in that congruence class. Fact (1) now yields the infinitude.