We show that the equation x2+y5=z7 has no solution in nonzero coprime integers, assuming a classification theorem from Putz's thesis, which has not appeared in a refereed journal. Equivalently, none of the equations x2+y5=z7, x2+y7=z5, x5+y7=z2 has a solution in nonzero coprime integers. Putz attaches to a solution an octic algebra, the fibre of a Belyi map of degree 8, and proves that it is always isomorphic to one fixed octic field L8. We prove unconditionally that no solution has fibre isomorphic to L8, by a fifth-power descent over a field L24 of degree 24 that Putz introduced, completed by fifth-power residue symbols at five auxiliary primes. We also prove, without assuming the generalized Riemann hypothesis, that L24 has class number 1, using a criterion of Belabas, Diaz y Diaz and Friedman.
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We prove that the equations x3+y3=zn and x3+y3=3zn have no solution in coprime nonzero integers for any n≥3. For the first equation the open cases were prime exponents p≡1(mod3) above 109 outside a set of congruence classes treated by Chen and Siksek; in these cases the solution 13+23=32 blocks the modular method. We replace Kraus's Frey curve by a hypergeometric motive of rank three over K=Q(−3), whose parameter at a solution is 3-adically close enough to a point of maximally unipotent monodromy for inertia at −3 to act unipotently; the parameter of 13+23=32 is not. By a theorem of Calegari, Emerton and Gee, the mod p representation of the fiber, twisted by a cubic character, is the reduction of a member of a compatible system of minimal ramification, and we show that the member at a prime above 3 is irreducible and ordinary at −3. Ramification and unconditional discriminant bounds show that its reductions have trivial semisimplification, and as the Galois group of the maximal pro-3 extension of K unramified outside −3 is generated by one inertia group, the member stabilizes its ordinary line, a contradiction. For x3+y3=3zn the same argument also works at the primes p≡2(mod3), which are inert in K, and Frey curves and descents treat the remaining exponents.
Primarily AI-generated textHuman understanding: no partsFrey representationscompatible systems of Galois representationsgeneralized Fermat equationhypergeometric motivesmodular method