THE NUMBER OF SOLUTIONS OF CONGRUENCE OF HOMOGENEOUS QUADRATIC POLYNOMIALS WITH PRIME MODULUS

Abstract

This research focuses on the proofs of the formula to calculate the number of solutions of the congruence , where  is a homogeneous quadratic polynomial with integer coefficients and  is a prime (referred to as congruence of a homogeneous quadratic polynomial with prime modulus). The study studies the problem naturally through relatively elementary results, including those from number theory and quadratic forms, to construct the formula to calculate the number of solutions of the aforementioned congruence. Unlike other proofs using advanced knowledge, the research results not only provide the formula to calculate the number of solutions but also demonstrate that all solutions of a congruence of a homogeneous quadratic polynomial with prime modulus are entirely determined by applying algebraic transformations to quadratic forms.