Some remarks on the diophantine equation $(x^2 - 1)(y^2 - 1) = (z^2 - 1)^2$

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

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.

The Diophantine Equation 8x + py = z2

Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p ≡ ± 3(mod  8), then the equation 8 (x) + p (y) = z (2) has no positive integer solutions (x, y, z); (ii) if p ≡ 7(mod  8), then the equation has only the solutions (p, x, y, z) = (2 (q) - 1, (1/3)(q + 2), 2, 2 (q) + 1), where q is an odd prime with q ≡ 1(mod  3); (iii) if p ≡ 1(mod  8)...

Some Remarks on Diophantine Approximations

BY P. ERDOS, University of Syracuse . [Received 27 July, 1948 .] 1 . The present note contains some disconnected remarks on diophatine approximations . First we collect a few well-known results about continued fractions, which we shall use later' . Let a be an irrational number, q, < q2 < . . . be the sequence of the denominators of its convergents . For almost all a we have fork > k,,(a), qk+,...

Some remarks on diophantine equations and diophantine approximation

We give many equivalent statements of Mahler’s generalization of the fundamental theorem of Thue. In particular, we show that the theorem of Thue–Mahler for degree 3 implies the theorem of Thue for arbitrary degree ≥ 3, and we relate it with a theorem of Siegel on the rational integral points on the projective line P(K) minus 3 points. Classification MSC 2010: 11D59; 11J87; 11D25