= Fixed odd numerator Legendre-symbol periodicity
{c}
{title2=$p\equiv q\pmod{4n}\Longrightarrow(n/p)=(n/q)$}
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.
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