Unlösbarkeit der Gleichung $alpha x^n + beta y^n = z^n$ und Gleichverteilung von $root n of{alpha x^n + beta y^n}$

摘要

We show that the unsolvability of the Diophantine equation $alpha x^n + beta y^n = z^n$ is equivalent to a good uniform distribution of the set ${ root n of{alpha x^n + beta y^n} }$ . The proof depends on the asymptotic evaluation of the Gauss sum $sum_{x, y} e (root n of{alpha x^n + beta y^n})$ .