On the Equation x2 1 + x2 2 + x2 3 + x2 4 = N with Variables such that x1x2x3x4 + 1 is an Almost-Prime

On the largest prime factor of x2-1

In this paper, we find all integers x such that x − 1 has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer n we can find the largest positive integer x such that all prime factors of each of x, x + 1, . . . , x+ n are less than 100.

ON THE EQUATION x2+2a ·3b =yn

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.

A Simple Criterion for Solvability of Both X2-DY2= c and x2-Dy2=-c

This article provides a simple criterion for the simultaneous solvability of the Diophantine equations X2 −DY 2 = c and x2 −Dy2 = −c when c ∈ Z, and D ∈ N is not a perfect square.