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Winding quotients and some variants of Fermat’s Last Theorem. (English) Zbl 0976.11017

The paper under review deals with the equations \[ (1) \quad x^n+ y^n= 2z^n,\;n\geq 3, \qquad (2) \quad x^n+ y^n= z^2,\;n\geq 4, \qquad (3) \quad x^n+ y^n= z^3,\;n\geq 3. \] By [H. Darmon, Int. Math. Res. Not. 10, 263-274 (1993; Zbl 0805.11028)] and [K. Ribet, Acta Arith. 79, 7-16 (1997; Zbl 0877.11015)], if \(n\) is a prime, then the following results are valid:
(a) Equation (1) has no nontrivial solutions when \(n\equiv 1\pmod 4\),
(b) Equations (2) has no nontrivial primitive solutions when \(n\equiv 1\pmod 4\),
(c) Suppose that every elliptic curve over \(\mathbb{Q}\) is modular. Then equation (3) has no nontrivial primitive solution when \(n\equiv 1\pmod 3\).
In this paper the author extends these results to the case of general \(n\). The proof follows the same method as in the proof of Fermat’s Last Theorem.

MSC:

11D41 Higher degree equations; Fermat’s equation
11G05 Elliptic curves over global fields
14H52 Elliptic curves
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