All the Solutions of the Diophantine Equation $p^3 + q^2 = z^3$

Abstract

In this paper, it is established that the title equation has exactly four solutions, all of which are exhibited.

The Infinitude of Solutions to the Diophantine Equation p3 + q = z3 when p, q are Primes

“…The literature contains a very large number of articles on non-linear such individual equations involving primes and powers of all kinds. Among them are for example [1,2,5,6]. The title equation stems from the equation p x + q y = z 2 .…”

The Class of Diophantine Equations p^4 + q^y = z^4 when y = 1, 2, 3 is Insolvable for all Primes p and q

“…The literature contains a very large number of articles on non-linear such individual equations involving primes and powers of all kinds. Among them are for example [1,3,5,7].…”

“…One may now ask the question whether or not the equation p n + q y = z n has solutions for all values y where 1 ≤ y ≤ n -1. When n = 3, the author [3] established that the equation p 3 + q 2 = z 3 (y = 2) has exactly four solutions in all of which p = 7. In one solution q is prime, whereas in the other three solutions q is composite.…”

A Note on the Diophantine Equation p3 + q2 = z4 when p is Prime

“…The literature contains a very large number of articles on non-linear such individual equations involving primes and powers of all kinds. Among them are for example [1,3,4,6,7,8]. The title equation stems from the equation p x + q y = z 2 .…”

“…Whereas in most articles, the values x, y are investigated for the solutions of the equation, in this paper these values are fixed positive integers. In the equation p 3 + q 2 = z 4 (1) we consider all primes p ≥ 2 and q > 1. We are mainly interested in how many solutions exist for any given prime p. This is established in Section 2.…”