A congruence involving Mersenne numbers

Question

Can you provide a proof or counterexample to the following claim ?

Let $p$ be a prime number greater than two then :

$$2^{2^{p-1}-1} \equiv 1 \pmod{2^p-1}$$

Pari/GP implementation .

I have tested this claim up to $5 \cdot 10^4$ .

I was searching for a counterexample using the following Pari/GP code :

MersenneFermat(lb,ub)={
forprime(p=lb,ub,
if(!(lift(Mod(2,2^p-1)^(2^(p-1)-1))==1),print(p)))
}

Answer 1

From Fermat's little theorem we know that $p\mid 2^{p-1}-1$ which means that $2^{p-1}-1=k\cdot p$.
Then, $2^{2^{p-1}-1}-1=2^{kp}-1=(2^p)^k-1=(2^p-1)(2^{p(k-1)}+2^{p(k-1)}+\ldots+1)$ which is a multiple of $2^p-1$ which is equivalent to what you ask.