On the equation a^p + 2^alpha b^p + c^p =0, by Kenneth A. Ribet

This paper has appeared in Acta Arith. 79 (1997), no. 1, 7-16, and so the dvi version has been removed. We discuss the equation a^p + 2^n b^p + c^p =0 in which a, b, and c are non-zero relatively prime integers, p is an odd prime number, and n is a positive integer. The technique used to prove Fermat's Last Theorem shows that the equation has no solutions with n or b even. When n and b is odd, there are the two trivial solutions (1,-1,1) and (-1,1,-1). In 1952, Dénes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for p congruent to 1 mod 4. We link the case p congruent to 3 mod 4 to conjectures of Frey and Darmon about elliptic curves over Q with isomorphic mod p Galois representations.

Kenneth A. Ribet <ribet@math.berkeley.edu>