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Quaestiones Mathematicae

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

Wang Tingting, Jiang Yingzhao

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




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