Abstract.
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 \(\sum_{x, y} e (\root n \of{\alpha x^n + \beta y^n})\).
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Langmann, K. Unlösbarkeit der Gleichung \(\alpha x^n + \beta y^n = z^n\) und Gleichverteilung von \(\root n \of{\alpha x^n + \beta y^n}\). Monatsh. Math. 143, 205–227 (2004). https://doi.org/10.1007/s00605-003-0197-0
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DOI: https://doi.org/10.1007/s00605-003-0197-0