3 problems

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…

Squarefree-gap conjecture. For any $\epsilon>0$ and $x$ sufficiently large in terms of $\epsilon$, the interval $(x,x+x^\epsilon]$ always contains a squarefree number.

Random-matrix conjecture. The limiting exponent is