 Infinitely many prime quadratic nonresidues of a nonsquare integer

For an odd positive nonsquare integer $a$, there are infinitely many primes $p$ with Legendre symbol $(a/p)=-1$. Choose a residue $n$ modulo $a$ with Jacobi symbol $(n/a)=-1$ by making it a quadratic nonresidue at one prime occurring to odd multiplicity and a square at every other prime. The Chinese remainder theorem also imposes $n\equiv 1(\mathrm{mod}4)$ and $n\equiv 1$ at any finite list of previously found primes. Then Jacobi reciprocity gives $(a/n)=-1$, so a prime factor of $n$ has $(a/p)=-1$ and is outside that list.