Quadratic Reciprocity Law

theoremquadratic-reciprocity

Statement of Main Law

For distinct odd primes $p$ and $q$, if $p\equiv 1\mathrm{mod}4$ or $q\equiv 1\mathrm{mod}4$ then $p\equiv \square \mathrm{mod}q$ if and only if $q\equiv \square \mathrm{mod}p$, alternatively, if $p\equiv 3\mathrm{mod}4$ and $q\equiv 3\mathrm{mod}4$, then $p\equiv \square \mathrm{mod}q$ if and only if $-q\equiv \square \mathrm{mod}p$.

Statement of Main Law with Legendre Symbols

For distinct odd primes $p$ and $q$,

1. $p$ or $q$ is $1\mathrm{mod}4⟹\left(\frac{q}{p}\right)=\left(\frac{p}{q}\right)$
2. $p$ and $q$ are $3\mathrm{mod}4⟹\left(\frac{q}{p}\right)=-\left(\frac{p}{q}\right)$

Statement of Supplementary Laws

See

• Note - Quadratic Reciprocity Law and the Legendre Symbol
• Proof of Some Square Patterns