MULTI-FREY Q-CURVES AND THE DIOPHANTINE EQUATION a + b = c
نویسندگان
چکیده
We show that the equation a2 +b6 = cn has no nontrivial positive integer solutions with (a, b) = 1 via a combination of techniques based upon the modularity of Galois representations attached to certain Q -curves, corresponding surjectivity results of Ellenberg for these representations, and extensions of multi-Frey curve arguments of Siksek.
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We show that the Diophantine equation of the title has, for n > 1, no solution in coprime nonzero integers x, y and z. Our proof relies upon Frey curves and related results on the modularity of Galois representations.
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