## 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

### Citations:

Zbl 0805.11028; Zbl 0877.11015
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