2.II.17B

Suppose $p$ is an odd prime and $a$ an integer coprime to $p$. Define the Legendre symbol $\left(\frac{a}{p}\right)$, and state (without proof) Euler's criterion for its calculation.

For $j$ any positive integer, we denote by $r_{j}$ the (unique) integer with $\left|r_{j}\right| \leq(p-1) / 2$ and $r_{j} \equiv a j \bmod p$. Let $l$ be the number of integers $1 \leq j \leq(p-1) / 2$ for which $r_{j}$ is negative. Prove that

$\left(\frac{a}{p}\right)=(-1)^{l} .$

Hence determine the odd primes for which 2 is a quadratic residue.

Suppose that $p_{1}, \ldots, p_{m}$ are primes congruent to 7 modulo 8 , and let

$N=8\left(p_{1} \cdots p_{m}\right)^{2}-1$

Show that 2 is a quadratic residue for any prime dividing $N$. Prove that $N$ is divisible by some prime $p \equiv 7 \bmod 8$. Hence deduce that there are infinitely many primes congruent to 7 modulo 8 .

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