0 solved out of 3 shown

OPEN

Let $A\subset\mathbb{N}$ be the set of $n$ such that for every prime $p\mid n$ there exists some $d\mid n$ such that $d\equiv 1\pmod{p}$. Is it true that there exists some constant $c>0$ such that for all large $N$
\[\frac{\lvert A\cap [1,N]\rvert}{N}=\exp(-(c+o(1))\sqrt{\log N}\log\log N).\]

Erdős could prove that there exists some constant $c>0$ such that for all large $N$
\[\exp(-c\sqrt{\log N}\log\log N)\leq \frac{\lvert A\cap [1,N]\rvert}{N}\]
and
\[\frac{\lvert A\cap [1,N]\rvert}{N}\leq \exp(-(1+o(1))\sqrt{\log N\log\log N}).\]
Erdős asked about this because $\lvert A\cap [1,N]\rvert$ provides an upper bound for the number of integers $n\leq N$ for which there is a non-cyclic simple group of order $n$.

OPEN

Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ - in particular, is it true that $c(n) \gg n^n$?

A problem first investigated by Hadwiger, who proved the lower bound
\[c(n) \geq 2^n+2^{n-1}.\]
It is easy to see that $c(2)=6$. Meier conjectured $c(3)=48$. Burgess and Erdős [Er74b] proved
\[c(n) \ll n^{n+1}.\]
Erdős wrote 'I am certain that if $n+1$ is a prime then $c(n)>n^n$.'.

OPEN

Let $h(n)$ be maximal such that $2^n-1,3^n-1,\ldots,h(n)^n-1$ are pairwise coprime.

Does, for every prime $p$, the density $\delta_p$ of integers with $h(n)=p$ exist? Does $\liminf h(n)=\infty$? Is it true that if $p$ is the greatest prime such that $p-1\mid n$ and $p>n^\epsilon$ then $h(n)=p$?

It is easy to see that $h(n)=n+1$ if and only if $n+1$ is prime, and that $h(n)$ is unbounded for odd $n$.

It is probably true that $h(n)=3$ for infinitely many $n$.