Quadratic Mathematics | Part IB, 2001

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

For jj any positive integer, we denote by rjr_{j} the (unique) integer with rj(p1)/2\left|r_{j}\right| \leq(p-1) / 2 and rjajmodpr_{j} \equiv a j \bmod p. Let ll be the number of integers 1j(p1)/21 \leq j \leq(p-1) / 2 for which rjr_{j} is negative. Prove that

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

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

Suppose that p1,,pmp_{1}, \ldots, p_{m} are primes congruent to 7 modulo 8 , and let

N=8(p1pm)21N=8\left(p_{1} \cdots p_{m}\right)^{2}-1

Show that 2 is a quadratic residue for any prime dividing NN. Prove that NN is divisible by some prime p7mod8p \equiv 7 \bmod 8. Hence deduce that there are infinitely many primes congruent to 7 modulo 8 .

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