A friend asked me the following question. What -values satisfy the equation
It occurred to me to ask a more general question. Let and . Let be the composition of with itself times. Similarly, let be the composition of with itself times.
What are the solutions to ?
Is it possible answer this question? Always? Sometimes? Never?
I’ll post my thoughts on this in a follow-up post.
[Update: I forgot to add that when I was looking up this problem online I found that it was a problem in the 1994 Russian Math Olympiad.]