Geodesic
On the equation $a^p + 2^α b^p + c^p = 0$
Acta Arithmetica, Tome 79 (1997) no. 1, pp. 7-16.

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We discuss the equation $a^p + 2^α b^p + c^p = 0$ in which a, b, and c are non-zero 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.
DOI : 10.4064/aa-79-1-7-16

Kenneth Ribet 1

1 
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Kenneth Ribet. On the equation $a^p + 2^α b^p + c^p = 0$. Acta Arithmetica, Tome 79 (1997) no. 1, pp. 7-16. doi : 10.4064/aa-79-1-7-16. http://geodesic.mathdoc.fr/articles/10.4064/aa-79-1-7-16/

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