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June 2021 Some Diophantine equations and inequalities with primes
Roger Baker
Funct. Approx. Comment. Math. 64(2): 203-250 (June 2021). DOI: 10.7169/facm/1912

Abstract

We consider the solutions to the inequality \[|p_1^c + \cdots + p_s^c - R| < R^{-\eta}\] (where $c > 1$, $c \not\in \mb N$ and $\eta$ is a small positive number; $R$ is large). We obtain new ranges of $c$ for which this has many solutions in primes $p_1, \ldots, p_s$, for $s = 2$ (and `almost all' $R$), $s=3$, 4 and~5. We also consider the solutions to the equation in integer parts \[[p_1^c] + \cdots + [p_s^c] = r\] where $r$ is large. Again $c> 1$, $c\not\in \mb N$. We obtain new ranges of $c$ for which this has many solutions in primes, for $s=3$ and 5.

Citation

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Roger Baker. "Some Diophantine equations and inequalities with primes." Funct. Approx. Comment. Math. 64 (2) 203 - 250, June 2021. https://doi.org/10.7169/facm/1912

Information

Published: June 2021
First available in Project Euclid: 13 November 2020

Digital Object Identifier: 10.7169/facm/1912

Subjects:
Primary: 11N36
Secondary: 11L20 , 11P55

Keywords: exponential sums , the alternative sieve , the Davenport-Heilbronn method , the Hardy-Littlewood method

Rights: Copyright © 2021 Adam Mickiewicz University

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Vol.64 • No. 2 • June 2021
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