Fermat.nb

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

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