On the simultaneous Pell equations x2 - (4m2 - 1)y2 = y2 - pz2 = 1

Abstract

Let m be a positive integer, and let p be an odd prime. By using certain properties of Pell and quartic diophantine equations with some elementary number theory methods, we prove that the system of equations x² - (4m² - 1)y² = 1 and y² - pz² = 1 has positive integer solutions (x, y, z) if and only if p ≡ 7(mod 8) and m = ¹/₄ (f²-1), where (f, g) is a positive integer solution of the equation f²-pg² = 2. Further, if the above condition is satisfied, then the system of equations has only the positive integer solution (x, y, z) = ( ¹/₂ (f⁴ - 2f² - 1); f² - 1; fg).

Keywords: Simultaneous Pell equations, solvability condition