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.