  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,
D\'enes conjectured that these are the only ones. Using methods of Darmon, we
prove this conjecture for $p\equiv1$ mod~4. We link the case $p\equiv3$ mod~4
to conjectures of Frey and Darmon about elliptic curves over~$\Q$ with
isomorphic mod~$p$ Galois representations.
