New observations on primitive roots modulo primes

TL;DR

This paper presents new theoretical observations and conjectures about primitive roots modulo primes, including sums involving primitive roots, and explores related properties of quadratic nonresidues and primitive prime divisors, supported by numerical evidence.

Contribution

It introduces new theorems on sums over primitive roots, formulates 35 conjectures on primitive roots and related properties, and connects these to computational algorithms.

Findings

Established a theorem on sums of Legendre symbols over primitive roots

Formulated 35 conjectures on primitive roots and related properties

Proposed a polynomial time algorithm for finding square roots modulo primes

Abstract

We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$, and $(\frac{\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $g<p$ modulo $p$ with $g-1$ a square, and that for any prime $p>3$ there is a prime $q<p$ with the Bernoulli number $B_{q-1}$ a primitive root modulo $p$. We also make related observations on quadratic nonresidues modulo primes and primitive prime divisors of some combinatorial sequences. For example, based on heuristic arguments we conjecture that for any prime $p>3$ there exists a Fibonacci number $F_k<p/2$ which is a quadratic nonresidue modulo $p$; this implies that there is a deterministic polynomial time algorithm to find square roots of quadratic residues modulo a prime $p>3$.