Prove that every odd number can be written as $(2^n-1)/A$ or $(2^n+1)/A$, $n$ & $A$ are some integers.

Question

I found an interesting fact that every odd number can be written as

$(2^n-1)/A$ or $(2^n+1)/A$, where $n$ & $A$ are some integers.

If the odd number is $N$, then $n ≤ (N-1)/2$.

I have checked from $3$ to $101$ and it is true for all these odd numbers.

ex. $101=(2^{50}+1)/11147523830125$

Is there a general proof for this odd number expression form?

Or a proof that this statement is wrong?

Answer 1

For all odd numbers $N$, $2^{\phi(N)} \equiv 1 \pmod{N}$.

$\phi(N)$ is even. Hence $2^ { \frac{\phi(N)}{2}} \equiv \pm 1 \pmod{N}$

Now show that $ \frac{ \phi(N)}{2} \leq \frac{ N-1}{2}$. Equality holds when $N$ is a prime.