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bob bundy
2013-07-11 22:21:01

Welcome to the forum.

This topic is not my area of expertise (if indeed I have one at all ) but I can get you started I think.

The underlying mathematics for this is group theory.  There's a bit on this page

http://en.wikipedia.org/wiki/Generating_set_of_a_group

If the operation of a group is multiplication, then 1 is the identity element because

x.1 = x for all x in the group.

A member of the group is called a generator if you can make all the other elements of the group from it.

Let's say the group is G = {1,2,3,4,5,6} and the operation is multiplication, mod 7

3 is a generator because

3^1 = 3 = 3mod7
3^2 = 9 = 2mod7;
3^3 =27 = 6mod7;
3^4 = 81 = 4mod7;
3^5 = 243 = 5mod7;
3^6 = 729 = 1mod7

Once you get to the identity like this, higher powers of 3 will just generate the same elements again (eg 3^7 = 3mod7)

Note: You can save some effort by using the previous answer to get the next, taking advantage of the mod

eg 3^3 = 3^2 x 3 = 2 x3mod 7 = 6mod7

On the other hand, 2 isn't a generator because

2^1 = 2;
2^2 = 4;
2^3 = 8 = 1mod7

I've reached the identity before generating all the group elements.  Higher powers will just keep producing 2,4 and 1 over and over.

Bob