The Number 5

 After 2 and 3, the next prime number found is 5. A very special number because it opens the door for the location of the next set of prime numbers and provides a particular pattern for primes. 5 is a result of the equation 5 = 6x-1. When solving for x, x = 1. This solution helps designate primes at a unique location on a table.

x    6x-1     6x     6x+1

1        5        6        7

The inclusion of 6x helps understand the importance of 6 as a locator of primes since all primes greater than 3 are located near multiples of 6. It it's possible to pick any number at random for x and use the equations above to find a potential prime. Further math will be necessary to determine primality. Still, a simple method is there to locate a prime. For example, if we choose x = 7, then 6x-1 = 41, 6x = 42, and 6x+1 = 43. Therefore:

x    6x-1     6x     6x+1

1        5        6        7

7      41       42      43

We now know numbers 41 and 43 are located on the seventh row of the table. This process helps facilitate the location of primes. 41 and 43 are prime numbers and twin primes. By continuing the table all the way to 25, it is evident that when squaring primes greater than or equal to 5, they will always be located on the 6x+1 column and thus, proving he relationship between primes and 6. 

x    6x-1     6x     6x+1

1        5        6        7

2        11     12      13

3        17     18      19

4        23     24      25

With the first part of the table complete, we only need to divide each number by 2 or 3 to determine primality. All the numbers under the 6x±1 columns will be prime except 25 because it is not only divisible by 5, but also the square of the first prime on the table. It is very important to assign 5 as P₁ because it will facilitate the discovery of primes. After 25, dividing by 2 and 3 is no longer necessary because the 25, or the square prime, has introduced another testing factor until the next square prime, or 49. 

 x    6x-1     6x     6x+1

5        29       30        31

6        35        36       37

7        41        42      43

8        47        48       49     All the numbers on the table on the 6x±1 columns only need to be divided by 5 to determine

primality except 49. All the numbers on the table at 6x±1 when x = 5, 6, 7, and 8, in bold are prime because they are not divisible by five. We knew in advance that 7 was the second prime on the table, therefore it was not necessary to test the number due to its composite nature. Nevertheless, the introduction of 49 will introduce a new testing number: 7.