If $p$ is a prime other then $2$, express the general prime factor of $2^p-1$ in terms of p and some other integer.
Let prime $q\mid (2^p-1)\implies ord_q2\mid p$
If $ord_q2=1,2^1\equiv 1\pmod q\implies q\mid (2-1)$ which is impossible.
SO, $ord_q2=p$ and $ord_q2\mid\phi(q)\implies p\mid\phi(q)\implies p\mid(q-1)$
For prime $q>2,q-1$ is even as $q$ must be odd.
So, $2\mid (q-1)\implies lcm(2,p)\mid (q-1)$
But $lcm(2,p)=2p$ as $p$ is odd.
So, $2pk=q-1\implies q=2pk+1$ for some natural number $k$