Question

Let $P$ be the set of prime numbers and $S=\{t:2^t-1$ is a prime $\}$. Then,

• A. $S\subset P$
• B. $P\subset S$
• C. $S=P$
• D. None of these

Answer 1

Answer: (A)

(a) Let $x\notin P$
$\to x$ is a composite number.
Let us now assume that $x\in S$
$\Rightarrow 2^x-1=m(\text{ where }m$ is a prime number)
$\Rightarrow 2^x=m+1$
which is not true for all composite numbers, say for $x=4$ because $2^4=16$ which cannot be equal to the sum of any prime number $m$ and 1 .
Thus, we arrive at a contradiction
$\Rightarrow x\notin S$
Thus, when $x\notin P$, we arrive at $x\notin S$.
So, $S\subset P$