Solving some parametric quadratic Diophantine equation over ? and F_p

摘要

Let t ≥ 2 be an integer. In this work, we consider the integer solutions to the Diophantine equation D: x~2 + (t - t~2)y~2 + (4 - 8t)x + (8t~2 - 8t)y + 3 = 0 over Z and over finite fields ~(Fp) for primes p ≥ 2, respectively. We also derive some algebraic identities related to the integer solutions of D including recurrence relations and continued fractions.