16 problems

A Mersenne prime is a prime of the form $2^n-1$, and let $s_c$ denote the relevant supremal DGM-quality quantity as a function of $c$. Lenstra–Pomerance–Wagstaff conjecture. There…

Gall-Rahav conjecture. The polynomial $\sigma(M^{2h})$ is divisible by a non-Mersenne prime.

For a prime $p$, a Mersenne number is $M_p=2^p-1$; write $M_n$ for the Mersenne primes indexed by their prime exponents, and let $\gamma$ denote the Euler–Mascheroni constant. The…

Wright-Euler Mersenne Exponent Hypothesis. The values $C(n)$ generate candidate exponents

Let $A$ be the coefficient ring and let … be a Drinfeld module. For each prime polynomial $P$, let $g_{P,\phi}$ denote the auxiliary polynomial associated with $P$ and $\phi$. Prim…

Mersenne-prime infinitude conjecture. There are infinitely many Mersenne primes.

A Mersenne prime is a prime of the form $2^n-1$. Mersenne-prime conjecture. There are infinitely many Mersenne primes. The paper presents this as a standard unresolved special case…

Let $\rho$ be a prime and let $M_\rho=2^\rho-1$. The Mersenne-prime residue conjecture. If $M_\rho$ is a Mersenne prime, then exactly one of the following holds:…

Let $A$ be an even nonsplitting perfect polynomial over $\mathbb{F}_2$, and consider its odd divisors. A prime is called Mersenne or 2-Mersenne according to the terminology of the…

Let $S$ be a 2-Mersenne prime and let $h\geq 2$. The polynomial sum-of-divisors function is denoted by $\sigma$. 2-Mersenne divisor conjecture. The integer $\sigma(S^{2h})$ is divi…

Let $M$ be a Mersenne prime and let $h\geq 2$. The polynomial sum-of-divisors function is denoted by $\sigma$. Mersenne divisor conjecture. The integer $\sigma(M^{2h})$ is divisibl…

Kaiser's conjecture. The exponent $n$ of a Mersenne prime of the form $2^n-1$ has to be singular; that is, exactly one Collatz matrix has $n_C=n$.

Let $D(X)$ denote the Collatz–Kakutani path length of a positive integer $X$, and let $n$ be the index of a Mersenne number $M_n=2^n-1$. The source studies the ratio $D(M_n)/n$ and…

Let $M_n=2^n-1$ be a Mersenne number with index $n$, and let $D(M_n)$ be its Collatz–Kakutani path length. The source compares the sequence of Mersenne primes with general Mersenne…

Let $M_n=2^n-1$ be a Mersenne number, where $n$ is its index, and let $D(M_n)$ denote its path length from the root of the Collatz–Kakutani tree. Ohira–Watanabe's path-length conje…

Let $\mathcal{M}_n$ denote the $n$-th Mersenne prime. Wagstaff's conjecture. Their sizes grow doubly exponentially, in the sense that … where $\gamma=0.57721566\ldots$ is the Euler…