Darmon, Henri; Merel, Loïc Winding quotients and some variants of Fermat’s Last Theorem. (English) Zbl 0976.11017 J. Reine Angew. Math. 490, 81-100 (1997). 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. Reviewer: D.Poulakis (Thessaloniki) Cited in 7 ReviewsCited in 89 Documents MSC: 11D41 Higher degree equations; Fermat’s equation 11G05 Elliptic curves over global fields 14H52 Elliptic curves Keywords:Frey curves; Galois representations; modular forms; modular curves; modular elliptic curve; Fermat’s Last Theorem Citations:Zbl 0805.11028; Zbl 0877.11015 PDFBibTeX XMLCite \textit{H. Darmon} and \textit{L. Merel}, J. Reine Angew. Math. 490, 81--100 (1997; Zbl 0976.11017) Full Text: EuDML Online Encyclopedia of Integer Sequences: a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n cubes, or 0 if no such magic square exists.