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 $\{p\}=\bigcap\_{i=1}^k\Pi(2^{n\_i}-1)$?
> 
> 
> 




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**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 Boolean algebra generated by sets of prime divisors of the numbers $2^n-1$
$2^{11}-1=2047=23\cdot 89$ and neither prime divides a smaller such number. So $23 \mid 2^n-1$ exactly when $n=11m.$ Similarly $89 \mid 2^n-1$ exactly when $n=11m.$ So $\{23,89\}$ is in the Boolean Algebra but neither of its singleton subsets is. 


Here is a list for $n \leq 33$ of the set of primes which divide $2^n-1$ but not any $2^m-1$ for $1 \lt m \lt n$. These sets (excluding $\{\}$) are the atoms of the Boolean Algebra:


$[2, \left\{ 3 \right\} ],[3, \left\{ 7 \right\} ],[4, \left\{ 5
 \right\} ],[5, \left\{ 31 \right\} ],[6, \left\{ \right\} ],[7,
 \left\{ 127 \right\} ],[8, \left\{ 17 \right\} ],[9, \left\{ 73
 \right\} ],[10, \left\{ 11 \right\} ],$


$[11, \left\{ 23,89 \right\} ],[
12, \left\{ 13 \right\} ],[13, \left\{ 8191 \right\} ],[14, \left\{ 43
 \right\} ],[15, \left\{ 151 \right\} ],[16, \left\{ 257 \right\} ],[
17, \left\{ 131071 \right\} ],$$[18, \left\{ 19 \right\} ],[19, \left\{ 
524287 \right\} ],[20, \left\{ 41 \right\} ],[21, \left\{ 337
 \right\} ],[22, \left\{ 683 \right\} ],[23, \left\{ 47,178481
 \right\} ],$


$[24, \left\{ 241 \right\} ],[25, \left\{ 601,1801
 \right\} ],[26, \left\{ 2731 \right\} ],[27, \left\{ 262657 \right\} 
],[28, \left\{ 29,113 \right\} ],$


$[29, \left\{ 233,1103,2089 \right\} ]
,[30, \left\{ 331 \right\} ],[31, \left\{ 2147483647 \right\} ],[32,
 \left\{ 65537 \right\} ],[33, \left\{ 599479 \right\} ]$


