The Diophantine Equation x 3 = dy 3 +1

On the Diophantine Equation x^6+ky^3=z^6+kw^3

Given the positive integers m,n, solving the well known symmetric Diophantine equation xm+kyn=zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x6+ky3=z6+kw3 has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is n...

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...

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

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...

THUE ’ S THEOREM AND THE DIOPHANTINE EQUATION x 2 − Dy

A constructive version of a theorem of Thue is used to provide representations of certain integers as x2 −Dy2, where D = 2, 3, 5, 6, 7.