# Smallest quadratic nonresidue

From Number

## Definition

Let be a prime number. The **smallest quadratic nonresidue** modulo is the smallest positive integer such that the congruence class of modulo is not a square; in other words, is the smallest quadratic nonresidue modulo .

The smallest quadratic nonresidue modulo a prime is always a prime.

## Facts and conjectures

- Extended Riemann hypothesis: This states that the smallest quadratic nonresidue modulo is less than . This is a conjecture, and has not been proved.

### Facts

- Smallest quadratic nonresidue is less than squareroot plus one: This states that the smallest quadratic nonresidue modulo is less than . This has been proved.
- Every prime is the smallest quadratic nonresidue for infinitely many primes: This follows from the fact that every integer that is not a perfect square is a quadratic nonresidue for infinitely many primes.