Is any Mersenne number $M_p$ divisible by $p+2$? More precisely, does there exist a natural number $p$ such that $(2^p-1)/(p+2)$ is also a natural number? It seems to me that this is a really simple problem (with the answer "no"), but I couldn't find anything on the web. There are some facts known about division by $p+1$, but nothing useful for $p+2$.

Another proof is as follows. First for $p=2$, it follows because $2^p-1=3<4=p+2$.
For odd primes $p$, there is a theorem (see Mersenne Prime for proof) that states that every factor of $2^p-1$ is of the form $2kp+1$ for some integer $k\geq0$. Thus we want a $p$ for which $p+2=2kp+1$. There is no such prime because for $p>2$, we have $1<p+2<2p+1$ thus requiring $0<k<1$, a contradiction to the theorem.
