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If $p$ and $q = 10p+3$ are odd primes, show that the Legendre symbols $(\frac{p}{q})$ and $(\frac{3}{p})$ are equal.

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    Note that if $p\equiv 3\pmod{4}$ then $q\equiv 1\pmod{4}$, so always $(p/q)=(q/p)$.2012-12-06

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The law of quadratic reciprocity states $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}.$

we also know that when $a \equiv a' \pmod q$ we have $\left(\frac{a}{q}\right)=\left(\frac{a'}{q}\right).$

Therefore $\left(\frac{p}{q}\right)\left(\frac{10p+3}{p}\right)=\left(\frac{p}{q}\right)\left(\frac{3}{p}\right)=(-1)^{(p-1)(5p+1)}=1.$