Research on Equation \(\varphi\) (x)+2 = \(\varphi\) (x+2)

ON THE EQUATION x2+2a ·3b =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 DIOPHANTINE EQUATION x2+p2k+1 = 4yn

Points on y = x2 at rational distance

Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola y = x2 such that each of the six distances between pairs of points is rational? We demonstrate that there is a correspondence between all rational points satisfying this condition and orbits under a particular group action of rational points on a fiber product of (three copies of) an elliptic surf...

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.