Integer Solutions of a Special Diophantine Equation

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Let t≠1 be an integer. In this work, we determine the integer solutions of Diophantine equation D:x2+(2−t2)y2+(−2t2−2t+2)x+(2t5−6t3+4t)y−t8+4t6−4t4+2t3+t2−2t = 0 over Z and also over finite fields Fp for primes p≥2. Also we derive some recurrence relations on the integer solutions (xn,yn) of D and formulate the the n—th solution (xn,yn) by using the simple continued fraction expansion of xnyn.