The Diophantine equation $2 x^2 + 1 = 3^n$

THE DIOPHANTINE EQUATION x2+2k =yn, II

New results regarding the full solution of the diophantine equationx2+2k=yn in positive integers are obtained. These support a previous conjecture, without providing a complete proof.

ON THE DIOPHANTINE EQUATION x2+p2k+1 = 4yn

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

” THE DIOPHANTINE EQUATION x n = Dy 2 + 1 ”