The conjecture that there are only finitely many triples of relatively prime integer powers , , for which
(1)
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with
(2)
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Darmon and Merel (1997) have shown that there are no relatively prime solutions with . Ten solutions are known,
(3)
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for , and
(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(Mauldin 1997).
The following table summarizes known solutions (Poonen et al. 2005). Any remaining solutions would satisfy the Tijdeman-Zagier conjecture, also known popularly as Beal's conjecture (Elkies 2007).
exponents | reference |
(2, 3, 7) | Poonen et al. (2005) |
Wiles | |
(2, 3, 8), (2, 3, 9), (2, 4, 5), | Bruin (2004) |
(2, 4, 6), (3, 3, 4), (3, 3, 5) | |
(2, 4, 7) | Ghioca |
, | Darmon-Merel |
Bennett | |
Bennett-Skinner |
It is not known if the analogous conjecture for , , and Gaussian integers holds. Known solutions include
(13)
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(14)
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(E. Pegg Jr., pers. comm., March 30, 2002).