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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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If $r$ is a prime and $a \equiv b \pmod r,$ what can you tell me about the relationship of $(a|r)$ and $(b|r)?$ Here $r$ does not divide either. Furthermore the horizontal way of writing the Legendre/Jacobi symbol was introduced by Dickson, so it's good enough for me. –  Will Jagy Dec 6 '12 at 2:11
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It's probably also a good idea to figure out what $q$ is modulo 4. –  Andreas Blass Dec 6 '12 at 2:15
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Note that if $p\equiv 3\pmod{4}$ then $q\equiv 1\pmod{4}$, so always $(p/q)=(q/p)$. –  André Nicolas Dec 6 '12 at 2:43

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

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