Darmon, Henri The equation \(x^ 4-y^ 4=z^ p\). (English) Zbl 0794.11014 C. R. Math. Acad. Sci., Soc. R. Can. 15, No. 6, 286-290 (1993). Let \(p\) be a prime \(\geq 11\). Consider the equation \(x^ 4- y^ 4= z^ p\), in rational integers, \(x\), \(y\), \(z\), with \(\text{gcd}(x,y)=1\). In this note, under the hypothesis that the Shimura-Taniyama conjecture is true, the author shows the above equation has no non-trivial integer solution \((x,y,z)\) in the following two cases: (i) \(p\equiv 1\pmod 4\), (ii) \(z\) is even. Reviewer: D.Poulakis (Thessaloniki) Cited in 1 ReviewCited in 11 Documents MSC: 11D41 Higher degree equations; Fermat’s equation 11G05 Elliptic curves over global fields Keywords:integer solution; elliptic curve; conductor; Galois representation PDFBibTeX XMLCite \textit{H. Darmon}, C. R. Math. Acad. Sci., Soc. R. Can. 15, No. 6, 286--290 (1993; Zbl 0794.11014)