3-method grouping of natural numbers

[Indroneil Ghosh]

Just felt like playing with numbers a few days ago and stumbled accross this...

 

Divide all natural numbers into three groups as... (:confused: Be patient and have a good look at them:) )

 

Group1:

1,4,7,10,13,16,19... and so on. General term: 3x+1

 

Group2:

2,5,8,11,14,17,20... and so on General term: 3y+2

 

Group3:

3,6,9,12,15,18,21... and so on General term: 3z

 

Let G1, G2 and G3 respectively represent any number from group 1, 2 and 3.

 

Try these out.

 

Additive laws:-

G1 + G2 = G3 (By this I mean add one number of group1 to one from group2. The result will be a member of group 3.)

G1 + G1 = G2

 

G2 + G2 = G1

 

G3 + G3 = G3

 

G1 + G3 = G1

 

G2 + G3 = G2

 

Multiplicative laws:-

(G1)(G1) = G1 (Again, this means multiply a number of group1 to another number from group1. The answer will also be in group1)

(G2)(G2) = G1

 

(G3)(G3) = G3

 

(G1)(G2) = G2

 

(G2)(G3) = G3

 

(G3)(G1) = G3

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I admit that I cant think of any use of all this:hihi: ... can anybody out there think of a good way of using this? And has anybody got anything to add to all this?

 

If anybody is interested in any kind of proof, just do this:

Consider elements of group1 as 3x + 1

consider elements of group2 as 3y + 2

consider elements of group3 as 3z

 

Then cooly perform the operations and attempt to get answers as 3a + 1 for group1, 3a + 2 for group2 and 3a for group3.

('a' may be anything. It is (x + z) for the proof of G1+G3=G3)

[ronthepon]

I dont understand a thing make this more reader friendly

[Qfwfq]

I admit that I cant think of any use of all this:hihi: ... can anybody out there think of a good way of using this? And has anybody got anything to add to all this?

What you posted is the usual way of constructing Z3 by congruence modulo 3 which is an equivalence relation (a relation which is reflexive, symmetric and transitive). Your three groups are usually called equivalence classes.

 

It can be done for any number besides 3 and Zn, with addition, is a cyclic additive group, with multiplication it's also a ring. If n is prime, as 3 is, Zn is also a field. It's a topic of algebra and number theory and certainly has uses, including in cryptography, you can find a lot of stuff about it.