Counting solutions of quadratic congruences in several variables revisited

الأصل · EN

Let Nₖ(n,r,a) denote the number of incongruent solutions of the quadratic congruence a₁x₁²++aₖxₖ²≡ n (mod r), where a=(a₁,,aₖ)∈ Zᵏ, n∈ Z, r∈ N. 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.