While working another question I asked here, I started looking at $a^{F(k)}$ where $F(k)$ is the $k$th Fibonacci number. I looked at $a^{F(k)}\mod k$ for no reason other than to start somewhere, which gave some interesting results early on for $a=3,5$ but appeared to have no real pattern when I computed better lists for $2\le a\le10, 1\le k\le35$. Here's the numbers, because why not:
2 : [' 0', ' 0', ' 1', ' 0', ' 2', ' 4', ' 2', ' 0', ' 7', ' 8', ' 6', ' 4', ' 6', ' 4', ' 4', ' 0', '15', '16', '13', '12', ' 4', ' 2', ' 4', '16', ' 7', ' 2', ' 4', ' 8', '19', '16', '16', ' 0', '25', '26']
3 : [' 0', ' 1', ' 0', ' 3', ' 3', ' 3', ' 3', ' 3', ' 0', ' 7', ' 4', ' 9', ' 9', ' 5', ' 9', '11', '12', ' 9', '15', ' 3', ' 9', ' 3', ' 9', ' 9', '18', ' 3', ' 0', '27', '21', '21', '12', '19', '27', '11']
4 : [' 0', ' 0', ' 1', ' 0', ' 4', ' 4', ' 4', ' 0', ' 4', ' 4', ' 3', ' 4', '10', ' 2', ' 1', ' 0', ' 4', ' 4', '17', ' 4', '16', ' 4', '16', '16', '24', ' 4', '16', ' 8', '13', '16', ' 8', ' 0', '31', '30']
5 : [' 0', ' 1', ' 1', ' 1', ' 0', ' 1', ' 5', ' 5', ' 4', ' 5', ' 9', ' 1', ' 5', ' 3', '10', '13', ' 3', '13', ' 9', ' 5', ' 4', ' 5', '21', ' 1', ' 0', ' 5', '25', '13', ' 4', '25', ' 5', '21', ' 4', '27']
6 : [' 0', ' 0', ' 0', ' 0', ' 1', ' 0', ' 6', ' 0', ' 0', ' 6', ' 2', ' 0', ' 2', ' 6', ' 6', ' 0', '10', ' 0', ' 5', '16', '15', ' 6', '13', ' 0', ' 1', ' 6', ' 0', '20', '22', ' 6', ' 6', ' 0', '15', '14']
7 : [' 0', ' 1', ' 1', ' 3', ' 2', ' 1', ' 0', ' 7', ' 7', ' 3', ' 8', ' 1', '11', ' 7', ' 4', ' 7', ' 6', ' 7', '11', ' 7', ' 7', ' 7', '20', ' 1', ' 7', ' 7', '22', ' 7', '20', ' 1', '14', ' 7', '31', '29']
8 : [' 0', ' 0', ' 1', ' 0', ' 3', ' 4', ' 1', ' 0', ' 1', ' 2', ' 7', ' 4', ' 8', ' 8', ' 4', ' 0', ' 9', '10', '12', ' 8', ' 1', ' 8', '18', '16', '18', ' 8', '10', ' 8', '15', '16', ' 4', ' 0', '16', '32']
9 : [' 0', ' 1', ' 0', ' 1', ' 4', ' 3', ' 2', ' 1', ' 0', ' 9', ' 5', ' 9', ' 3', '11', ' 6', ' 9', ' 8', ' 9', '16', ' 9', '18', ' 9', '12', ' 9', '24', ' 9', ' 0', ' 1', ' 6', '21', '20', ' 9', ' 3', '19']
10: [' 0', ' 0', ' 1', ' 0', ' 0', ' 4', ' 3', ' 0', ' 1', ' 0', '10', ' 4', ' 4', '12', '10', ' 0', '11', '10', ' 3', ' 0', '16', '10', '15', '16', ' 0', '10', '19', '20', '18', '10', '18', ' 0', ' 1', '22']
I explored the first and last digits of the numbers as well as a couple other modulus things with them, and nothing interesting arose. The numbers themselves don't seem to have much pattern to them of course, as they're just powers of a number. I can't seem to find anything interesting or of use to it, at least not with what I've tried. Any suggestions or know of anything which might apply here?
