Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New observations on primitive roots modulo primes   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 $p>3$.

14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio