Some Diophantine equations and inequalities with primes

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.