Journal of Integer Sequences, Vol. 17 (2014), Article 14.11.6

Counting Solutions of Quadratic Congruences in Several Variables Revisited

László Tóth
Department of Mathematics
University of Pécs
Ifjúság útja 6
7624 Pécs
Institute of Mathematics
Department of Integrative Biology
Universität für Bodenkultur
Gregor Mendel-Straße 33
1180 Vienna


Let Nk(n,r,a) denote the number of incongruent solutions of the quadratic congruence a1x12 + ··· + akxk2n (mod r), where a = (a1, ... ,ak) ∈ Zk, nZ, rN. We give short direct proofs for certain less known compact formulas on Nk(n,r,a), valid for r odd, which go back to the work of Minkowski, Bachmann and Cohen. We also deduce some other related identities and asymptotic formulas which do not seem to appear in the literature.

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(Concerned with sequences A000089 A000188 A060594 A060968 A062570 A062775 A062803 A086932 A086933 A087561 A087687 A087784 A088964 A088965 A089002 A089003 A091143 A096018 A096020 A208895 A227553 A229179 A240547.)

Received July 1 2014; revised versions received September 21 2014; November 8 2014. Published in Journal of Integer Sequences, November 9 2014.

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