ON THE DIOPHANTINE EQUATION x2 + d2l + 1 = yn

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 EQUATION x2+2a ·3b =yn

ON THE DIOPHANTINE EQUATION Ax2+22m =yn

Let h denote the class number of the quadratic field Q( √−A) for a square free odd integer A> 1, and suppose that n> 2 is an odd integer with (n,h)= 1 and m> 1. In this paper, it is proved that the equation of the title has no solution in positive integers x and y if n has any prime factor congruent to 1 modulo 4. If n has no such factor it is proved that there exists at most one solution with ...

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

ON THE DIOPHANTINE EQUATION x m − 1 x − 1 = yn − 1 y − 1

There is no restriction in assuming that y > x in (1) and thus we have m > n. This equation asks for integers having all their digits equal to one with respect to two distinct bases and we still do not know whether or not it has finitely many solutions. Even if we fix one of the four variables, it remains an open question to prove that (1) has finitely many solutions. However, when either the b...