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The equation \(x^ 4-y^ 4=z^ p\). (English) Zbl 0794.11014

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.

MSC:

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