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Generalized cyclotomic numbers of order 2 and the quadratic reciprocity law. (English) Zbl 1253.11005
Let $$p$$ be an odd prime number; for positive integers $$n$$, the author considers the number of solutions of congruences of the form $$a_1x_1^2 + \ldots + a_nx_n^2 \equiv a \bmod p$$. Using Dirichlet’s theorem on primes in arithmetic progression he then derives the quadratic reciprocity law from his results. Much simpler proofs along these lines have been obtained recently e.g. by W. Castryck [Am. Math. Mon. 115, No. 6, 550–551 (2008; Zbl 1228.11006)].
##### MSC:
 11A15 Power residues, reciprocity 11D79 Congruences in many variables