On the Diophantine Equation 7^x+n^y=z^3
Abstract
By using some properties of congruence, we prove that the non-negative integer solutions of the Diophantine equation 7^x+n^y=z^3, where n is a positive integer with n\equiv \pm2,\pm3 \pmod{7}, are (x,y,z)=(0,1,\sqrt[3]{n+1}) and (x,y,z)=(\log_{7} (3n^{2k}+3n^k+1),3k,n^k+1), where k is a non-negative integer.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Analytic Number Theory Research
Advanced Differential Equations and Dynamical Systems
Publication details
- Journal / source
- International journal of mathematics and computer science
- Pages
- 71–73
- Publication date
- 1 January 2026
- Language
- EN