Near-primitive roots

Details Near-primitive roots Near-primitive roots

2011-12-21

arxiv.org/abs/1112.5090v1

Mathematics

Number Theory

12 pages, 2 tables

Scientific paper

Given an integer $t\ge 1$, a rational number $g$ and a prime $p\equiv 1({\rm mod} t)$ we say that $g$ is a near-primitive root of index $t$ if $\nu_p(g)=0$, and $g$ is of order $(p-1)/t$ modulo $p$. In the case $g$ is not minus a square we compute the density, under the Generalized Riemann Hypothesis (GRH), of such primes explicitly in the form $\rho(g)A$, with $\rho(g)$ a rational number and $A$ the Artin constant. We follow in this the approach of Wagstaff, who had dealt earlier with the case where $g$ is not minus a square. The outcome is in complete agreement with the recent determination of the density using a very different, much more algebraic, approach due to Hendrik Lenstra, the author and Peter Stevenhagen.

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