Evidence for Fermat's Little Theorem

For a given prime p, consider the positive powers of a, 0 ≤ a < p.  Each value of is congruent to a number from 0 to p-1.  Look for patterns.

p=11:   Powers in  the ring

a
0	0	0	0	0	0	0	0	0	0	0
1	1	1	1	1	1	1	1	1	1	1
2	4	8	5	10	9	7	3	6	1	2
3	9	5	4	1	3	9	5	4	1	3
4	5	9	3	1	4	5	9	3	1	4
5	3	4	9	1	5	3	4	9	1	5
6	3	7	9	10	5	8	4	2	1	6
7	5	2	3	10	4	6	9	8	1	7
8	9	6	4	10	3	2	5	7	1	8
9	4	3	5	1	9	4	3	5	1	9
10	1	10	1	10	1	10	1	10	1	10

p=17   Powers in  the ring

a
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2	4	8	16	15	13	9	1	2	4	8	16	15	13	9	1	2
3	9	10	13	5	15	11	16	14	8	7	4	12	2	6	1	3
4	16	13	1	4	16	13	1	4	16	13	1	4	16	13	1	4
5	8	6	13	14	2	10	16	12	9	11	4	3	15	7	1	5
6	2	12	4	7	8	14	16	11	15	5	13	10	9	3	1	6
7	15	3	4	11	9	12	16	10	2	14	13	6	8	5	1	7
8	13	2	16	9	4	15	1	8	13	2	16	9	4	15	1	8
9	13	15	16	8	4	2	1	9	13	15	16	8	4	2	1	9
10	15	14	4	6	9	5	16	7	2	3	13	11	8	12	1	10
11	2	5	4	10	8	3	16	6	15	12	13	7	9	14	1	11
12	8	11	13	3	2	7	16	5	9	6	4	14	15	10	1	12
13	16	4	1	13	16	4	1	13	16	4	1	13	16	4	1	13
14	9	7	13	12	15	6	16	3	8	10	4	5	2	11	1	14
15	4	9	16	2	13	8	1	15	4	9	16	2	13	8	1	15
16	1	16	1	16	1	16	1	16	1	16	1	16	1	16	1	16

One of the patterns that you might have noticed is indeed a theorem