# On the equation ${a}^{p}+{2}^{\alpha }{b}^{p}+{c}^{p}=0$

Acta Arithmetica (1997)

• Volume: 79, Issue: 1, page 7-16
• ISSN: 0065-1036

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## Abstract

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We discuss the equation ${a}^{p}+{2}^{\alpha }{b}^{p}+{c}^{p}=0$ 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.

## How to cite

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Kenneth A. Ribet. "On the equation $a^p + 2^α b^p + c^p = 0$." Acta Arithmetica 79.1 (1997): 7-16. <http://eudml.org/doc/206967>.

@article{KennethA1997,
abstract = {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.},
author = {Kenneth A. Ribet},
journal = {Acta Arithmetica},
keywords = {Fermat's last theorem; -th powers in arithmetic progression; exponential diophantine equations; higher degree diophantine equations},
language = {eng},
number = {1},
pages = {7-16},
title = {On the equation $a^p + 2^α b^p + c^p = 0$},
url = {http://eudml.org/doc/206967},
volume = {79},
year = {1997},
}

TY - JOUR
AU - Kenneth A. Ribet
TI - On the equation $a^p + 2^α b^p + c^p = 0$
JO - Acta Arithmetica
PY - 1997
VL - 79
IS - 1
SP - 7
EP - 16
AB - 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.
LA - eng
KW - Fermat's last theorem; -th powers in arithmetic progression; exponential diophantine equations; higher degree diophantine equations
UR - http://eudml.org/doc/206967
ER -

## References

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