A simple solution of the diophantine equation $x^3 + y^3 = z^2 + t^2$

The diophantine equation x3 3 + y3 + z 3 − 2xyz = 0

We will be presenting two theorems in this paper. The first theorem, which is a new result, is about the non-existence of integer solutions of the cubic diophantine equation. In the proof of this theorem we have used some known results from theory of binary cubic forms and the method of infinite descent, which are well understood in the purview of Elementary Number Theory(ENT). In the second th...

On searching for solutions of the Diophantine equation x3 + y3 +2z3 = n

We propose an efficient search algorithm to solve the equation x3 + y3 + 2z3 = n for a fixed value of n > 0. By parametrizing |z|, this algorithm obtains |x| and |y| (if they exist) by solving a quadratic equation derived from divisors of 2|z|3±n. Thanks to the use of several efficient numbertheoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. ...

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)...

Practical Solution of the Diophantine Equation

Let p be an odd prime and a, b positive integers. In this note we prove that the problem of the determination of the integer solutions to the equation y2 = x(x + 2apb)(x − 2apb) can be easily reduced to the resolution of the unit equation u+ √ 2v = 1 over Q( √ 2, √ p). The solutions of the latter equation are given by Wildanger’s algorithm.

A SIMPLE SOLUTION OF THE DIOPHANTINE EQUATION x+y = z+t