The Diophantine Equation b²X^4 - dY² = 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...

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

Diophantine Equations Related with Linear Binary Recurrences

In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...

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...

Solving Fermat-type equations x + dy = z via modular Q-curves over polyquadratic fields

We solve the diophantine equations x + dy = zp for d = 2 and d = 3 and any prime p > 557. The method consists in generalizing the ideas applied by Frey, Ribet and Wiles in the solution of Fermat’s Last Theorem, and by Ellenberg in the solution of the equation x + y = zp, and we use Q-curves, modular forms and inner twists. In principle our method can be applied to solve this type of equations f...