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arXiv · 2607.11068

Exact Frequencies of Consecutive Quadratic Residue and Nonresidue Patterns Modulo a Prime

Brahim Mittou
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Abstract

This paper determines the exact frequency of all consecutive sign patterns of lengths two and three formed by the quadratic character modulo an odd prime $p$. Using basic properties of character sums over finite fields, we derive exact counting formulas for pairs $(n, n+1)$ and triples $(n-1, n, n+1)$ exhibiting specific sequence patterns in \(\{\pm 1\}\). The frequencies of pairs are classified entirely by \(p \pmod 4\), whereas triples exhibit a more complex dependency on \(p \pmod 8\) and the Jacobsthal sum $T_p=\sum_{x\in\mathbb F_p}\chi(x^3-x)$.

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Brahim Mittou. 2026-07-13. Exact Frequencies of Consecutive Quadratic Residue and Nonresidue Patterns Modulo a Prime. https://arxiv.org/abs/2607.11068

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