On the Diophantine equation π₯Β³Β±π¦Β³=π^{π}Β±π^{π}
Abstract
In this note we consider the title Diophantine equation from both a theoretical as well as experimental point of view. In particular, we prove that for k = 4 , 6 k=4, 6 and each choice of the signs our equation has infinitely many coprime positive integer solutions ( x , y , a , b ) (x, y, a, b) such that no partial sum in the expression x 3 Β± y 3 β ( a k Β± b k ) x^3 \pm y^3-(a^k \pm b^k) vanishes. The same is true for each k β’ 0 ( mod 4 ) k\not \equiv 0\pmod {4} and the equation x 3 Β± y 3 = a k β b k x^3\pm y^3=a^k-b^k . For k = 5 , 7 k=5, 7 and all choices of the signs we computed all coprime positive integer solutions ( x , y , a , b ) (x, y, a, b) of x 3 Β± y 3 = a k + b k x^3\pm y^3=a^k+b^k
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Analytic Number Theory Research
Advanced Mathematical Identities
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 3
- Pages
- 913β925
- Publication date
- 16 January 2026
- Language
- EN