Evidence for Fermat's Little Theorem

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

p=11:   Powers in  the ring 11

a a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a 11
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 17

a a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a 11 a 12 a 13 a 14 a 15 a 16 a 17
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


Converted by Mathematica      April 24, 2000