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- W1750343326 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, D'enes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for $pequiv1$ mod~4. We link the case $pequiv3$ mod~4 to conjectures of Frey and Darmon about elliptic curves over~$Q$ with isomorphic mod~$p$ Galois representations." @default.
- W1750343326 created "2016-06-24" @default.
- W1750343326 creator A5029473852 @default.
- W1750343326 date "1995-08-01" @default.
- W1750343326 modified "2023-09-27" @default.
- W1750343326 title "On the equation $a^p + 2^alpha b^p + c^p =0$" @default.
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