On the subject of the set S := { x , x = 6n ± 1 with n being a natural number }

For every x = 6n ± 1 (n being natural), let i be called the primary index number for x, such that when x = 6n - 1, i = 2n - 1, and when x = 6n + 1, i = 2n. Call n the secondary index number.

Claim 1: Every element of S that is not prime may be written as the product of two other elements of S.

Example 1: 25 = 6*4+1 so 25 is in S. 25 = 5*5 so it is not prime. 5 = 6*1 - 1, so 5 is in S. So 25 which is in S and not prime can be written as the product of two numbers also in S.

Claim 2: If m is an element of S is not prime, then i, the index number for m, may be written as i = 2kx + j OR i = 2kx - (j+1), where x is an element of set S, j is the primary index number for x, and k is a natural number.

Claim 2.1: In this form x is a factor of m and k is the secondary index number for another factor of m, call this factor y (y in S), where y = 6k - 1 when i = 2kx - (j+1), and y = 6k + 1 when i = 2kx + j.

Example 2: For m = 25, i = 8. We can write 8 = 2*1*5 - (1+1), where 5 is in S (shown in example 1), and 1, so the other factor of 25 is 6*1 - 1 = 5 (because we wrote 8 = 2*1*5 - (1+1)). So the factors of of 25 are 5 and 5.