Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N_0$ where $\mathbb N_0=\{0\}\cup\mathbb N$, where $a,b\in\mathbb N$. This topology is often referred to as the *Furstenberg topology* or the *profinite topology*. The space $\mathbb N_\tau:=(\mathbb N,\tau)$ is homeomorphic to the space $\mathbb Q$ of rational numbers (being a second-countable regular countable space without isolated points). So $\mathbb N_\tau$ has many non-trivial homeomorphisms. But I know no non-trivial homeomorphism of $\mathbb N_\tau$ which would be *multiplicative* is the sense that $f(x\cdot y)=f(x)\cdot f(y)$ for any $x,y\in\mathbb N$.

Problem 1.Is the identity function a unique multiplicative homeomorphism of $\mathbb N_\tau$?

An affirmative answer to this problem follows from an affirmative answer to

Problem 2.Are there prime numbers $a,b,c$ such that for any $d\in\mathbb N$ there exists $n\in\mathbb N$ such that $a^n\equiv 1\!\!\!\mod\! d\;\;$ but $\;\;b^n\not\equiv 1\!\!\!\mod\! c$?

positiveintegers... (+1 to the question, anyway!) $\endgroup$ – Qfwfq Sep 9 '18 at 10:50