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Prove that for any $m,n\in\mathbb{Z}^+$, the number $2^n-1$ is divisible by $(2^m-1)^2$ iff the number $n$ is divisible by $m(2^m-1)$.

My work so far:

It's an if and only if statement, so the common approach is to split the proof into two "if then" proofs ($A\rightarrow B \text{ and } B\rightarrow A$).

Starting with $A\rightarrow B$, for a direct proof, I would assume $A$ and prove $B$. I can use the definition of divisibility:

Assume $n=m(2^m-1)k$ for some integer $k$

Then I need to prove that $2^n-1=(2^m-1)^2h$ for some integer $h$

So I'm thinking I need to start with the expression $2^n-1$ and transform it to $(2^m-1)^2h$, or vice versa, and I need to use the assumption $n=m(2^m-1)k$ somehow.

I notice that the LS of the equality contains the variable $n$ while the RS contains $m$, and the assumption relates $n$ to $m$, so that could be a way forward. For example, simple substitution gives $2^{m(2^m-1)k}-1$, but I'm not sure what I could do from here. Perhaps there's a divisibility theorem that could help?

Other things I notice: $2^n-1$ and $(2^m-1)^2$ are always odd. Also, the divisors in both cases are multiples of $2^m-1$.

Any help/hints appreciated.

yroc
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1 Answers1

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Hints. $$\gcd(2^n-1,2^m-1) = 2^{\gcd(m,n)}-1,\tag{1} $$ hence in order that $(2^m-1)\mid (2^n-1)$, we must have $m\mid n$, and this is a sufficient condition, too, since: $$ y^a-1 = (y-1)\cdot(y^{a-1}+\ldots+y+1).\tag{2} $$ Suppose now that $n=mh$. We have: $$ 2^n-1 = (2^m-1)\cdot\left(2^{m(h-1)}+\ldots+2^{m}+1\right)\tag{3} $$ and: $$ 2^{mu}=\left((2^m-1)+1\right)^{u}\equiv 1\pmod{(2^m-1)}\tag{4} $$ hence in order that $(2^m-1)^2\mid (2^n-1)$ we must have $(2^m-1)\mid h$, and it is also a sufficient condition. Putting all together: $$ (2^m-1)^2\mid (2^n-1)\quad\Longleftrightarrow\quad m(2^m-1)\mid n. \tag{5} $$

Jack D'Aurizio
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