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**
Counting Solutions of Quadratic Congruences in Several Variables Revisited
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László Tóth

Department of Mathematics

University of Pécs

Ifjúság útja 6

7624 Pécs

Hungary

and

Institute of Mathematics

Department of Integrative Biology

Universität für Bodenkultur

Gregor Mendel-Straße 33

1180 Vienna

Austria

**Abstract:**

Let *N*_{k}(*n*,*r*,**a**)
denote the number of incongruent
solutions of the quadratic congruence
*a*_{1}*x*_{1}^{2}
+ ··· + *a*_{k}*x*_{k}^{2}
≡ *n* (mod *r*), where **a** =
(*a*_{1}, ... ,*a*_{k}) ∈ **Z**_{k},
*n* ∈ **Z**, *r* ∈ **N**. We give short
direct proofs for certain less known compact formulas on
*N*_{k}(*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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