# Conway's subprime Fibonacci sequences

I want to be certain I have the latest information on Conway's subprime Fibonacci sequences, arXiv-posted a year ago; I am referencing the status in a review. To wit, starting with $(0,1)$:1 $$0, 1, 1, 2, 3, 5, 4, 3, 7, 5, 6, 11, 17, 14, 31, 15, 23, 19, \ldots$$

Richard K. Guy, Tanya Khovanova, Julian Salazar:
"however, it seems more likely here than in the $3x + 1$ problem that sequences do not increase indefinitely. Here is an informal argument that supports such a conjecture,"

$\color{Red}{Q}$: Has it been established that some sequence increases indefinitely? Or no sequence, regardless of starting conditions, does not?

1"Start with the Fibonacci sequence 0, 1, 1, 2, 3, 5, . . . , but before you write down a composite term, divide it by its least prime factor so that this next term is not 8, but rather 8/2 = 4."

• Also published as Conway’s Subprime Fibonacci Sequences, Richard K. Guy, Tanya Khovanova and Julian Salazar, Mathematics Magazine Vol. 87, No. 5 (December 2014), pp. 323-337. Aug 7 '15 at 23:06