A Note on Generalized Sophie Germain Primes: In the Direction of Legendre's Extended Sophie Germain Primes

[摘要] This article considers primes $p$ such that $2kp+1$ is also prime for some natural number $k$. And, we prove that if $2$ is a primitive root of $2kp+1$, then $p\equiv k\pmod{4}$. We also present a few more observations concerning these primes and the least primitive root of $2kp+1$.

[学科分类] 社会科学、人文和艺术（综合）

[关键词] Primitive roots;primality;Sophie Germain primes;Legendre primes.