Quadratic nonresidue

A nonzero congruence class modulo an odd prime is a quadratic nonresidue when it is not a square. In the convention of quadratic residue, zero belongs to neither class. Thus $3,5,6$ are the quadratic nonresidues modulo seven, while $2,6,7,8,10$ are the quadratic nonresidues modulo eleven. No integer $x$ satisfies $x^2\equiv 3(\mathrm{mod}7)$, as the complete square map under quadratic residue shows.

A quadratic nonresidue is not an arithmetic dead end: multiplying by any fixed quadratic nonresidue $n$ exchanges the quadratic residues and the quadratic nonresidues. If $nq$ were a square for a square $q$, division by $q$ would make $n$ a square. Hence $n{Q}_p$ consists entirely of quadratic nonresidues; it has $(p-1)/2$ elements, because multiplication by nonzero $n$ is a bijection. It therefore equals the entire other coset of the quadratic residue subgroup. Modulo seven, $3\{1,2,4\}=\{3,6,5\}$.

The product of two quadratic nonresidues is consequently a quadratic residue: write them as $n{q}_1,n{q}_2$ in that same coset, and their product is $n^2{q}_1{q}_2$, a square. For example, $3\cdot 5\equiv 1(\mathrm{mod}7)$, whereas $2\cdot 3\equiv 6(\mathrm{mod}7)$ remains a quadratic nonresidue. The multiplicativity of the Legendre symbol packages these rules as ordinary sign multiplication.

For an odd prime $p$, the least quadratic nonresidue $n(p)$ is the smallest positive integer whose nonzero residue class is a quadratic nonresidue modulo $p$.

On a dyadic interval $Q<p\le 2Q$, every such exceptional prime makes all $Q^{0.01}$-smooth integers quadratic residues. By positive density of power-smooth integers, there are $\gg Q^2$ of these integers up to $Q^2$. Apply the variance form of the large sieve or its forbidden-residue consequence with interval length $Q^2$ and prime cutoff $2Q$. Each exceptional prime excludes $(p-1)/2$ residue classes and contributes at least a fixed positive amount to the sieve denominator. Hence each dyadic interval contains $O(1)$ exceptional primes. Summing the intervals gives the logarithmic bound.

// Target: analytic-number-theory.bigb

If, for every $\eta >0$, the smooth numbers up to $T$ with all prime factors at most $T^{\eta }$ have density at least $\kappa (\eta )>0$, then for fixed $\epsilon >0$ there are only $O_{\epsilon }(1)$ primes $p\le N$ with $n(p)>N^{\epsilon }$. Apply the variance form of the large sieve to the indicator function of the $N^{\epsilon }$-smooth numbers up to $N^2$. For every exceptional prime, all these numbers are nonzero quadratic residues, so half its nonzero residue classes are empty. Their variance contribution is at least one third of the square of the total count. The large sieve bounds the sum of contributions by a constant times $N^2$ times that count. The density assumption with $\eta =\epsilon /2$ finishes the argument.