New observations on primitive roots modulo primes

zbMATH Open1463.11004arXiv1405.0290MaRDI QIDQ3386033

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Publication date: 14 January 2021

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 $su{m}_g(fracg+cp)$, where $g$ runs over all the primitive roots modulo $p$ among $1,ldots,p-1$, and $(fraccdotp)$ 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 $p>3$.

Full work available at URL: https://arxiv.org/abs/1405.0290

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