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

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 $T_p=\sum_{x\in\mathbb F_p}χ(x^3-x)$.