Fast computation of the number of solutions to $x_1^2+\cdots+x_k^2 \equiv \lambda \pmod{n}$

TL;DR

This paper presents explicit formulas for counting solutions to the quadratic congruence involving sums of squares modulo n, enabling fast computation of the solution count for various parameters.

Contribution

The paper introduces closed-form formulas for the solution count function _{k,}(p^s) with constant arithmetic complexity, improving computational efficiency.

Findings

Derived explicit formulas for _{k,}(p^s)

Achieved constant-time computation for solution counts

Enhanced understanding of quadratic congruences modulo prime powers

Abstract

In this paper we study the multiplicative function $\rho_{k,\lambda}(n)$ that counts the number of incongruent solutions of the equation $x_1^2+\cdots+x_k^2 \equiv \lambda\pmod{n}$. In particular we give closed explicit formulas for $\rho_{k,\lambda}(p^s)$ with a arithmetic complexity of constant order.