$a^n-1$ and $b^n-1$ have the same set of prime factors for each $n\in\Bbb{Z}^+$, show that $a=b$. https://ift.tt/eA8V8J

Let $a,b$ be two positive integers, if $a^n-1$ and $b^n-1$ have the same set of prime factors for each $n\in\Bbb{Z}^+$, then $a=b$.

It seems to be not hard, but I have no idea. Any hint is welcome.

Edit: As the comment points out, it is probably not easy. I thought it is easy because it is kind of intuitively correct. I tried to study the Zsigomondy primes as $n$ increases, but it is not successful.

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