The Diophantine equation aX 4 – bY 2 = 1

The Diophantine Equation B 2 X 4 ? Dy 2 = 1

If b and d are given positive integers with b > 1, then we show that the equation of the title possesses at most one solution in positive integers X; Y. Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic elds. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with pr...

On the Exponential Diophantine Equation ( 4 m 2 + 1

Let m be a positive integer. Then we show that the exponential Diophantine equation (4m2 + 1)x + (5m2 − 1)y = (3m)z has only the positive integer solution (x, y, z) = (1, 1, 2) under some conditions. The proof is based on elementary methods and Baker’s method. Mathematics Subject Classification: 11D61

Integer Solutions of the Equation y 2 = Ax 4 + B

Let A ∈ {k2(k2l2 + 1), 4k2(k2(2l − 1)2 + 1)}, where k and l are positive integers, and let B be a non-zero square-free integer such that |B| < √ A. In this paper we determine all the possible integer solutions of the equation y2 = Ax4 + B by using terms of Lucas sequences of the form mx2.

The Diophantine Equation x4 + 2 y4 = z4 + 4 w4

The Diophantine Equation b²X^4 - dY² = 1

If b and d are given positive integers with b > 1, then we show that the equation of the title possesses at most one solution in positive integers X, Y . Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with...