Solving some parametric quadratic Diophantine equation over ZZ and FpFp

Let t ⩾ 2 be an integer. In this work, we consider the integer solutions to the Diophantine equation D: x2 + (t − t2)y2 + (4 − 8t)x + (8t2 − 8t)y + 3 = 0 over ZZ and over finite fields FpFp for primes p ⩾ 2, respectively. We also derive some algebraic identities related to the integer solutions of D including recurrence relations and continued fractions.