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.

  • $\begingroup$ 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 Aug 8 '16 at 22:03

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 $t^2 = c t + d$.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.