13 problems

Dynamical Mersenne-prime conjecture. For every $n_0>0$, there exists a prime integer $n>n_0$ such that, for every prime $q$ with $1<q\leq \left\lfloor\sqrt{M(n)}\right\rfloor$, the…

Bateman–Selfridge–Wagstaff conjecture. If two of these three statements are true, then the third is also true.

Infinitude conjecture. There are infinitely many integers $n$ such that $M(n)$ is prime.

Let $N$ be an index of a highly-composite Mersenne number if $\tau(2^N-1)>\tau(2^m-1)$ for every integer $m$ with $1\leqslant m<N$. Highly-composite Mersenne index conjecture. If…

Let $p$ be an odd prime, and define the least occurrence exponent $k$ to be the least positive integer such that $p$ divides the Mersenne number $2^k-1$, when such an integer exist…

Distinct-prime-divisor conjecture. If $\ell\geq 3$, then $2^n-1$ is divisible by at least $2^\ell$ distinct primes.

Let $p>p_0$ be prime, assume the distinct-factor bound conjecture and the uniform-distribution conjecture, and write … where $q_1<\cdots<q_k$ are the prime factors and…

Let $p\ge 3$ be prime and set $N_p=2^p-1$. The squarefreeness conjecture. There exists $p_0$ such that, if $p>p_0$, then $N_p$ is squarefree. Squarefreeness would rule out repeated…

Let $p\ge 3$ be prime and set $N_p=2^p-1$. Write $\omega(N_p)$ for the number of distinct prime factors of $N_p$. The distinct-factor bound conjecture. There exists $p_0$ such that…

Let $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime, and let $P(d)$ denote the Benford probability of leading digit $d$. The waiting time between successive occurrences of a leadi…

Let $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime, and consider the leading digits of consecutive terms $M_n,M_{n+1},\ldots,M_{n+k-1}$. A digit is Benford-distributed when its p…

Let $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime, let $D(M_n)$ denote the leading digit of $M_n$, and define the Benford error by … Here $P(d)$ is the Benford probability of th…

Let $M_p=2^p-1$ be the Mersenne number associated with a prime number $p$. For a positive integer $q$, let $G_q/R_q$ denote the cycle decomposition of the multiplication-by-$2$ map…