User:Alangarrod/Sandbox

Composite Symmetry
The composites due to the first n primes can be seen to form an infinitely repeating symmetrical pattern whose period is Pn * Pn-1 * Pn-2 .. * 2.

All of the even composites are divisible by 2 and are therefore not of great interest. We can visualize the odd composites that are divisible by 3 in the table below. NOTE - The period of repetition - 6 (3 * 2), which is even, is also shown.

New Composites due to 3
Some important facts are apparent from the above table:-


 * The period of repetition is 3 * 2 i.e. Pn * 2


 * The table only shows 4 cycles of repetition, but this will evidently repeat to infinity

We can then say that any odd positive integer n is divisible by 3 if, and only if n (Mod 6) = 3 (see Modular Arithmetic)
 * One odd number in three is divisible by 3

We can now look at the effect of introducing the next prime number, 5. The table shows 2 cycles of repetition here but again this will repeat to infinity.

New Composites Due to 5
Here the table contains entries for all odd composites indicating the smallest prime that divides them. It again also shows the even composite that is the period of repetition, in this case 5*3*2 = 30. Those cells containing 5 alone are the new odd composites due to the introduction of 5.

What is also of interest is the fact that if 5*3*2 is treated as 0 working in the reverse direction then the pattern of composites is exactly the same as it is in the forward direction. This is logical since the primes that we have taken into account so far: 2,3 and 5 all meet at both zero and 5*3*2 as well as any multiple of this.

We can say that any positive integer K is divisible by 5 (and not 3) if K (Mod 30) = 5 or 25 - The positions of the two 5s in the repeated pattern. As we will see later in this text there is a neat way of determining what these newly introduced composites will be without the use of a table.

If we were to continue this by introducing the number 7 then we would have a table with a period of 7*5*3*2 {=210} cells.

If we were to then draw this table we could see quite easily which composites were divisible by 7 and not divisible by 3 or 5.