Use our version of quadratic reciprocity to prove Euler's version, which is that (for p, q odd primes) (-1)^((p-1)/2)p is a quadratic residue modulo q if and only if q is a quadratic residue modulo p.
Use our version of quadratic reciprocity to prove Euler's version, which is that (for p, q odd primes) (-1)^((p-1)/2)p is a quadratic residue modulo q if and only if q is a quadratic residue modulo p.