Small Prime $k$th Power Residues and Nonresidues in Arithmetic Progressions

N. A. Carella

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arXiv · 2405.13159

Small Prime $k$th Power Residues and Nonresidues in Arithmetic Progressions

N. A. Carella

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Abstract

Let $p$ be a large odd prime, let $x=\log p)(\log\log p)^{3+\varepsilon}$ and let $q\ll\log\log p$ be an integer, where $\varepsilon>0$ is a small number. This note proves the existence of small prime quadratic residues and small prime quadratic nonresidues in the arithmetic progression $a+qm\ll x$, with relatively prime $1\leq a<q$, unconditionally. The same results are generalized to small prime $k$th power residues and nonresidues, where $k\mid p-1$ and $k\ll\log\log p$.

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N. A. Carella. 2024-05-21. Small Prime $k$th Power Residues and Nonresidues in Arithmetic Progressions. https://arxiv.org/abs/2405.13159

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