3 research outputs found
Dynamics of the Fibonacci Order of Appearance Map
The \textit{order of appearance} of a positive integer in the
Fibonacci sequence is defined as the smallest positive integer such that
divides the -th Fibonacci number. A \textit{fixed point} arises
when, for a positive integer , we have that the
Fibonacci number is the smallest Fibonacci that divides. In other words,
.
In 2012, Marques proved that fixed points occur only when is of the
form or for all non-negative integers . It
immediately follows that there are infinitely many fixed points in the
Fibonacci sequence. We prove that there are infinitely many integers that
iterate to a fixed point in exactly steps. In addition, we construct
infinite families of integers that go to each fixed point of the form . We conclude by providing an alternate proof that all positive integers
reach a fixed point after a finite number of iterations.Comment: 10 pages, 2 figure