On the equation $a^p+2^{a}lpha{b}^p+c^p=0$

Abstract

We discuss the equation $a^p + 2^\a b^p + c^p =0$ in which $a$, $b$, and $c$ are non-zero relatively prime integers, $p$ is an odd prime number, and $\a$ is a positive integer. The technique used to prove Fermat's Last Theorem shows that the equation has no solutions with $\a>1$ or $b$ even. When $\a=1$ and $b$ is odd, there are the two trivial solutions $(\pm 1,\mp 1,\pm 1)$. In 1952, Dénes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for $p\equiv 1$ mod~4. We link the case $p\equiv 3$ mod~4 to conjectures of Frey and Darmon about elliptic curves over~$\Q$ with isomorphic mod~$p$ Galois representations.