Artin's conjecture on primitive roots

An elliptic curve ${\displaystyle E}$ given by ${\displaystyle y^{2}=x^{3}+ax+b}$, Lang and Trotter gave a conjecture for rational points on ${\displaystyle E/(\mathbb {F_{p}} )}$ analogous to Artin's primitive root conjecture.[4]

Specifically, they said there exists a constant ${\displaystyle C_{E}}$ for a given point of infinite order ${\displaystyle P}$ in the set of rational points ${\displaystyle E(\mathbb {Q} )}$ such that the number ${\displaystyle N(P)}$ of primes (${\displaystyle p\leq x}$) for which the reduction of the point ${\displaystyle P{\pmod {p}}}$ denoted by ${\displaystyle {\bar {P}}}$ generates the whole set of points in ${\displaystyle \mathbb {F_{p}} }$ in ${\displaystyle E}$, denoted by ${\displaystyle {\bar {E}}(\mathbb {F_{p}} )}$. Here we exclude the primes which divide the denominators of the coordinates of ${\displaystyle P}$.

Moreover, Lang and Trotter conjectured that ${\displaystyle N(P)\sim C_{E}\left({\frac {x}{\log x}}\right)}$.[5] Gupta and Murty proved the Lang and Trotter conjecture for ${\displaystyle E/\mathbb {Q} }$ with complex multiplication under the Generalized Riemann Hypothesis.[6] Moreover, Gupta and Murty proved also that it is possible to add 3 natural even numbers and obtain an odd natural number.[7]

Krishnamurty proposed the question how often the period of the decimal expansion ${\displaystyle 1/p}$ of a prime ${\displaystyle p}$ is even.

The claim is that the period of the decimal expansion of a prime in base ${\displaystyle g}$ is even if and only if ${\displaystyle g^{\left({\frac {p-1}{2^{j}}}\right)}\not \equiv 1{\bmod {p}}}$ where ${\displaystyle j\geq 1}$ and ${\displaystyle j}$ is unique and p is such that ${\displaystyle p\equiv 1+2^{j}\mod {2^{j}}}$.

The result was proven by Hasse in 1966.[4][8]