Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / C (Number Theory)

The quadratic reciprocity states that for distinct odd primes $a,b$,For fixed odd prime $r$, first suppose that $p,q\ne r$. The congruence $p\equiv q(\mathrm{mod}4r)$ gives the same residues modulo $r$ and modulo four. Reciprocity yieldsIf $p=r$, the congruence forces $r\mid q$, so $q=r$ and both symbols are zero; the same handles $q=r$.

For $n={\prod }_j{r}_j^{a_j}$, where all $r_j$ are odd primes, $4r_j$ divides $4n$. Thus $p\equiv q(\mathrm{mod}4n)$ implies equality of each prime-numerator Legendre symbol. Multiplicativity givesThis also covers symbols that vanish; for $n=1$ both sides are one.