Consecutive numbers with the same Legendre symbol

Abstract:

Let $p$ be an odd prime, and $R_{p}$ be a complete set of residues $(\operatorname {mod} p)$. The goal of the paper is to determine all the values of $n\ (n\in R_{p})$ such that $\big (\frac {n}{p}\big ) = \big (\frac {n+1}{p}\big )$ or $\big (\frac {n-1}{p}\big )= \big (\frac {n}{p}\big ) =\big (\frac {n+1}{p}\big )$, where $\big (\frac {\cdot }{p}\big )$ is the Legendre symbol.