Let $\Pi$ be the set of odd prime numbers and let $\mathcal P(\Pi)$ be the Boolean algebra of subsets of $\Pi$.

For a number $x$ denote by $\Pi(x)$ the set of odd prime divisors of $x$.

Problem. Does each singleton $\{p\}\subset \Pi$ belong to the Boolean algebra generated by the family $\{\Pi(2^n-1):n\ge 2\}$ in $\mathcal P(\Pi)$?

We can also ask a stronger

Question. Is it true that for every odd prime number $p$ there exit numbers $n_1,\dots,n_k$ such that $\{p\}=\bigcap_{i=1}^k\Pi(2^{n_i}-1)$?

Let $\Pi$ be the set of odd prime numbers and let $\mathcal P(\Pi)$ be the Boolean algebra of subsets of $\Pi$.

For a number $x$ denote by $\Pi(x)$ the set of odd prime divisors of $x$.

Problem. Does each singleton $\{p\}\subset \Pi$ belong to the Boolean algebra generated by the family $\{\Pi(2^n-1):n\ge 2\}$ in $\mathcal P(\Pi)$?

We can also ask a stronger

Question. Is it true that for every odd prime number $p$ there exit numbers $n_1,\dots,n_k$ such that $\{p\}=\bigcap_{i=1}^k\Pi(2^{n_i}-1)$?

Added in Edit. The number $p=5$ mentioned in the answer of Ofir Gorodetsky gives a counterexamle to Question but not to Problem as $\{5\}=\Pi(2^4-1)\setminus\Pi(2^2-1)=\{3,5\}\setminus\{3\}$.

The Small Fermat Theorem implies that any odd prime number $p$ divides $2^{p-1}-1$. But $2^{p-1}-1$ can have other prime divisors. For example, $2^{7-1}-1=63=7\cdot 9$.

Question. Is it true that for any odd prime number $p$ there are numbers $n_1,\dots,n_k$ such that $p$ is the unique common prime divisor of the numbers $2^{n_1}-1,\dots,2^{n_k}-1$?

Let $\Pi$ be the set of odd prime numbers and let $\mathcal P(\Pi)$ be the Boolean algebra of subsets of $\Pi$.

For a number $x$ denote by $\Pi(x)$ the set of odd prime divisors of $x$.

Problem. Does each singleton $\{p\}\subset \Pi$ belong to the Boolean algebra generated by the family $\{\Pi(2^n-1):n\ge 2\}$ in $\mathcal P(\Pi)$?

We can also ask a stronger

Question. Is it true that for every odd prime number $p$ there exit numbers $n_1,\dots,n_k$ such that $\{p\}=\bigcap_{i=1}^k\Pi(2^{n_i}-1)$?