Suppose that $r$ is a homogeneous linear recurrence sequence of order $>1$ with nonnegative integer coefficients, not all $0$, and nonnegative initial values, not all $0$. Suppose that $S$ and $T$ are finite sets of numbers in $r$. Let $S'$ be the product of numbers in $S$, and let $T'$ be the product of numbers in $T$. If $S' = T'$, must $S = T$? 


This question generalizes [Distinctness of products of Fibonacci numbers](https://mathoverflow.net/q/238505), in which $S$ and $T$ are sets of Fibonacci numbers, for which the answer is "yes", as a corollary of Carmichael's theorem. 



Distinct products of terms from a linear recurrence sequence
Consider the recurrence $a\_{n+2} = a\_{n+1} + 2 a\_n$, $a\_0 = 1$, $a\_1 = 2$, whose solution is $a\_n = 2^n$. Then $\prod\_{j \in J} a\_j = 2^{\sum J}$, so there are infinitely many counterexamples to your conjecture.


EDIT: Similarly for $a\_{n+2} = c a\_{n+1} + d a\_n$, $a\_0 = 1$, $a\_1 = t$, where $t^2 = c t + d$.


