Mathematical Reflections 2013, Issue 2 - Problem J260

Problem:
Solve in integers the equation $$x^4-y^3=111.$$

Proposed by Jos e Hern andez Santiago.

Solution:
We claim that the equation has no integer solution. As a matter of fact, suppose that the given equation has an integer solution. Let us consider the equation modulo $13$. Since $x^2 \equiv 0,\pm 1, \pm 3, \pm 4 \pmod{13}$, then $x^4 \equiv 0,1,3,-4 \pmod{13}$. Moreover $y^3 \equiv 0, \pm 1, \pm 5 \pmod{13}$. Hence, $$x^4-y^3 \equiv 0, \pm 1, \pm 2, \pm 3,\pm 4, \pm5, 6 \pmod{13},$$ but $111 \equiv -6 \pmod{13}$, a contradiction.