The Quadratic Diophantine Equations x^2− P(t)y^2− 2P′(t)x + 4P(t)y + (P′(t))^2− 4P(t) − 1 = 0

  •  Amara Chandoul    
  •  Diego Marques    
  •  Samira Shaban Albrbar    


Let P := P(t) be a non square polynomial. In this paper, we consider the number of integer solutions of Diophantine equation

E : x2− P(t)y2− 2P′(t)x + 4P(t)y + (P′(t))2− 4P(t) − 1 = 0.

We derive some recurrence relations on the integer solutions (xn,yn) of E. In the last section, we consider the same problem over finite fields Fpfor primes p ≥ 5. Our main results are generaliations of previous results given by Ozcok and Tekcan (Ozkoc and Tekcan, 2010).

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