# 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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topKenneth 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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