Alain Togbé

ON THE DIOPHANTINE EQUATION x2 + 2a · 5b = yn

In this note, we find all the solutions of the Diophantine equation x2 + 2a · 5b = yn in positive integers x, y, a, b, n with x and y coprime and n ≥ 3.

On the solutions of a family of quartic Thue equations

In this paper, we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields. In fact, we prove that for n < 5 × 10 6 and n ≥ N = 1.191 x 10 19 , with n, n + 2, n 2 + 4 square-free, the Thue equation Φ n (x,y) = x 4 - n 2 x 3 y - (n 3 + 2n 2 + 4n + 2)x 2 y 2 - n 2 xy 3 + y 4 = 1 has no integral solution except the trivial ones: (1, 0), (-1,0), (0,1), (0, -1).

ON THE DIOPHANTINE EQUATION x + 5 13 = y

In this note, we find all the solutions of the Diophantine equation x 2 + 5 a 13 b = y n in positive integers x, y, a, b, n ≥ 3 with x and y coprime.

On a family of diophantine triples {K,A2K + 2A, (A + 1)2K + 2(A + 1)} with two parameters II

AbstractLet A and k be positive integers. In this paper, we study the Diophantine quadruples

THE EXPONENTIAL DIOPHANTINE EQUATION n x + ( n + 1) y = ( n + 2) z REVISITED

\{ k,A^2 k + 2A,(A + 1)^2 k + 2(A + 1)d\} .

A GENERALIZATION OF A THEOREM OF BUMBY ON QUARTIC DIOPHANTINE EQUATIONS

If d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then

On the diophantine equation x 2 + 2 α 5 β 13 γ = y n

\begin{gathered} d = (4A^4 + 8A^3 + 4A^2 )k^3 + (16A^3 + 24A^2 + 8A)k^2 + \hfill \\ + (20A^2 + 20A + 4)k + (8A + 4) \hfill \\ \end{gathered}