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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, in which $S$ and $T$ are sets of Fibonacci numbers, for which the answer is "yes", as a corollary of Carmichael's theorem.

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  • $\begingroup$ Let $r_i = 2 F_i$, which is characterized by the same recurrence relation as the Fibonacci numbers. Then if $S = \{0, 1\}, T = \{2\}$, we have $S' = 2 \times 2 = 4, T' = 4$. I think that is a counterexample, right? You may need to say that there are finitely many exceptions. $\endgroup$
    – user44191
    Commented Aug 8, 2016 at 22:03

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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 $t^2 = c t + d$.

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