Prime Number Assignation

The most essential feature of using a random number to find a potential prime number is the ordinal assignation to prime numbers. Since 2 and 3 gave us the first seven prime numbers, the spreadsheet can be expanded all the way to 23 squared, or 529. On the spreadsheet, 529 is located in the 6x+1 column when x= 88. If we assign 23 as the seventh prime, it will facilitate the test for primes. The spreadsheet indicates that before 529, only 174 numbers are necessary to look for prime numbers. All 174 numbers are located on the 6x±1 columns. Essentially, we can choose any number at random using the following equations located on the spreadsheet:

x = N (N is a natural number, will yield 2 results)

6x-1 =  P (left side potential prime) *

6x+1 = P (right side potential prime) *

*Note: when selecting P = 6x±1, ensure P is a natural odd number greater than 4. The equation must not yield a decimal result. 

The next step is to divide by the assigned primes, depending on the location of the random number. If the number is P = 89, then divide P by 5, 7, and 11. If 89 is not divisible by 5, 7, and 11 then the number is prime. This eliminates the need to check 89 by dozens of divisors. Location is everything. 89 is located between 49 and 121 or the square of prime numbers 7 and 11. The squares are the prime limits. Those limits, limit the number of divisors to check from primality. If you look at the spreadsheet, the square primes are in bold in the 6x+1 column. 5 is the first prime. The second prime 7. The third prime 11 and so on. Every time a prime number is found, then it is another prime limit by squaring it. When squaring a prime, solve for x using the equation P = 6x+1. x will yield the row its located in. 

The Number 7

Now that we are aware of the prime number table, one can see the importance of the first few prime numbers provided by 2 and 3. The table is simpler if 5 is assigned as the first prime number in the table (P1) to facilitate the process. This means, 7 can be the second prime in the table or P₂ to further increase the table and the possibility to find primes at random. The random selection can only be used if prime limits are accepted and used. The first prime limit is introduced is 25. 25 is the square of the first prime number and the introduction of 5 as the deciding divisor for determining primes. When reaching 49, or 7 squared, a new limit is set. Basically, the same primes are used to test numbers and locate more primes. After 49, 5 and 7 are the determining divisors for primality. The spreadsheet will illustrate the primes and composites between 53 and 120, determined by dividing by 5 or 7. We can ignore any number that ends if 5, but it's important to justify the role 5 has on the spreadsheet. 

If 6x±1 = P, and P is between 53 and 120, then divide by 5 and 7, therefore if P ≠ 6x+1/ P_1, P₂ then the number is prime.

Prime Number Spreadsheet

I have created a spreadsheet using Google sheets to illustrate the location of prime numbers. The numbers with a yellow background are prime numbers. Two yellow numbers 1n the same row indicates twin primes. The non-primes will remain with the same white background, but the column labeled "composite test" will illustrate the reason why the number is not a prime using the prime number test, as explained on the post, "Prime Number Test." I will update the table as much as possible. You can view the table using the link provided:

Prime Number Spreadsheet

The square primes are in bold because it introduces a new number to test for primality. It will also show the squaring of primes will always be located on the "6x+1" column. 

Prime Number Birthdays

 Happy Prime Birthday!

Awesome people who have a prime birthday will have a prime number as month, day, and year.

If your birthday lands on the following criteria, then you have a prime birthday:

Month: February (2), March (3), May (5), July (7), and November (11)

Day: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

Year: 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, and some day, 2027.

Jascha Heifetz, a Russian-American violinist, was born on February 2, 1901. He was regarded as one of the greatest violinists of all time.

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.

The First Primes

"Just because we can't find a solution, it doesn't mean there isn't one" -Andrew Wiles

Prime numbers are very interesting and mysterious. The irregular nature of their location creates a barrier in which a formula is virtually impossible to find. I know for a fact all prime numbers greater than 5 are located near multiples of 6 or more precisely, at 6x±1, where x is a natural number. Solving for x is a simple way to test if a number is almost a prime. A few more steps are necessary if it’s a legitimate prime number. First, we must understand one thing: 1, 2 and 3, the first three prime numbers (1 still qualifies as a prime), are the key to unlock primes and are the basis of all primes. 

1, 2, and 3 are the factors of 6, the first perfect number. 1 is the deciding factor whether a number is prime or not and where the number is possibly located by adding or subtracting 1 from 6 and its multiples (6-1, 6+1). 2 and 3 are the only consecutive prime numbers and the only ones necessary to find prime numbers between 4 and 24. 

1 and the first prime numbers, have given us a starting point to find the first few primes. If we divide [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] by [2 and 3]; that leaves out 5, 7, 11, 13, 17, 19, and 23. All prime numbers. Now we have something to work with.