Let $m,n$ be two positive integers and $2^{2n+1}-1\, | \, 2^{2m+1}-1$. Suppose $P\_0$ be the largest prime number such that $P\_0 \, | \, 2^{2m+1}-1$. If $P\_0 \, | \, 2^{2n+1}-1$ then is the following equation true?
 $$2^{2n+1}-1 = 2^{2m+1}-1$$



equality of two numbers which are odd powers of 2 and satisfy a certain condition
No. $(2n+1,2m+1) = (17,51)$ is a counterexample because
$$
2^{51} - 1 = 7 \cdot 103 \cdot 2143 \cdot 11119 \cdot 131071
$$
and $131071 = 2^{17}-1$ is prime. The only other counterexample with $m \leq 100$ is $(2n+1,2m+1) = (37,111)$ with $P\_0 = (2^{37}-1)/223 = 616318177$.


P.S. **gp** code:



```
forstep(k=3,200,2,if(!isprime(k),f=factor(2^k-1)[,1]; r = znorder(Mod(2,f[#f])); if(r<k, print([r,k]))))

```

P.P.S. The condition $2^{2n+1} - 1 \mid 2^{2m+1} - 1$ is equivalent to
$2n+1 \mid 2m+1$.


