Fixed odd numerator Legendre-symbol periodicity ($p\equiv q(\mathrm{mod}4n)⟹(n/p)=(n/q)$)

For a fixed positive odd integer $n$, the Legendre symbol $(n/p)$ for odd prime denominators depends only on $p$ modulo $4n$. For prime numerator $r$, quadratic reciprocity expresses it as $(p/r)$ times a sign determined by $p$ modulo four. Factorization of $n$ and multiplicativity prove the general statement. If a denominator divides $n$, the congruence forces the other prime denominator to be identical, so the zero cases agree too.