Suppose that $S$ and $T$ are sets of Fibonacci numbers greater than $1$. Let $S^\*$ be the product of numbers in $S$, and likewise for $T^\*$. If $S^\*=T^\*$, must $S=T$? 



Distinctness of products of Fibonacci numbers
(**Finite** sets, of course, if you don't want to allow $S^\* = T^\* = \infty$)


Yes. We may assume wlog $S$ and $T$ are disjoint, and $\max S < \max T$. By [Carmichael's theorem](https://en.wikipedia.org/wiki/Carmichael's_theorem), every Fibonacci number except $1$, $8$ and $144$ has a prime factor that does not divide any earlier Fibonacci number. So $\max T$ can only be $8$ or $144$, and neither of these work. 


