6 problems

The converse conjecture. If $S(q)\neq\emptyset$, then there exists a prime $p\equiv1\pmod q$ such that $z(p)\mid\pi(q)$.

A Cunningham chain is a finite sequence of primes of the form $\{p,2p+1,4p+3,\ldots,2^{n-1}(p+1)-1\}$. The arbitrarily long Cunningham chains conjecture. There are arbitrarily larg…

The a-Sophie Germain prime probability conjecture. The probability that a given prime $p$ is $a$-SGP converges, as $p\to\infty$, to the common asymptotic value represented by

The 2-Sophie Germain prime probability conjecture. The probability that a given prime $p$ is $2$-SGP converges to $0.76$ as $p\to\infty$.

A Brazilian number is an integer whose base-$b$ representation has all the same digits for some $1<b<n-1$. A Brazilian prime is a Brazilian number that is prime. A Sophie Germain p…

Let $x\geq 1$ be large, and let $\Lambda$ be the von Mangoldt function. Germain prime conjecture. … where … This predicts a quantitative distribution for Sophie Germain primes, nam…