ON THE EQUATION a^{p} + 2^{a}lphab^{p} + c^{p} = 0

Kenneth Ribet

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

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