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

TL;DR

This work proves unconditional existence of very small prime quadratic residues and nonresidues within short arithmetic progressions n = a + qm in the regime x ≈ (log p)(log log p)^{4+ε}, for q ≪ log log p and coprime a,q, and extends the results to small prime k-th power residues/nonresidues with k | (p−1) and k ≪ log log p. Central to the approach are new finite-field characteristic functions for residues and nonresidues, together with exponential-sum bounds and a fiber-mapping analysis that isolates the main term and tightly controls the error. The paper derives explicit main-term counts with densities δ(k,q,a) = c(k,q,a)/(kφ(q)) and demonstrates these densities are determined by correction constants c(k,q,a) reflecting entanglement among primes, residue classes, and k-th power conditions. Numerical data for large primes corroborate the theoretical predictions, and the work outlines several open problems and conjectures related to correction constants and patterns of consecutive residues. Overall, the results significantly sharpen prior bounds on the size and frequency of small prime residues/nonresidues in arithmetic progressions and introduce robust techniques for higher-power generalizations.

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 $k\ll\log\log p$.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof : Proof
• Lemma 3.1
• proof : Proof
• Theorem 4.1
• Theorem 4.2