18 problems

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

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…

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 $f(n)=\sum_{1\leqslant k\leqslant n}\tau(2^k-1)$, where $\tau(m)$ denotes the number of positive divisors of $m$. Ratio conjecture. The ratio $f(2n)/f(n)$ should stabilize and…

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.

For each positive integer $n$, let $P(2^n-1)$ denote the largest prime factor of the Mersenne number $2^n-1$. Erdős's conjecture. As $n$ tends to infinity, … The conjecture concern…

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, write $2^p-1=q_1\cdots q_k$ with distinct prime factors, and let $\mathcal A(p)$ be the selected set of odd integers $a\in[5,p-2]$ for which the associated p…

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 $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime, and let the leading digit of each $M_n$ be its first decimal digit. Independent Benford digits conjecture. In contrast to smoo…

Erdős's conjecture. The displayed limit is infinite. Stewart recently resolved this conjecture, so the claim is no longer open.

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…