THE ENNISA FORMULA

The Secret of Prime and Twin Prime Number Distribution

#### The Laws of Prime Exclusion

PART I and Part II

Originally Published 1995 – Revised 11/11/2004

By Sollog Immanuel Adonai-Adoni www.Sollog.com

In this work I shall put forth a simple Euclidian Proof that shows the infinite field of primes revolves around a Mod 90 Algorithm that repeats infinitely throughout the infinite field of prime numbers.

I show also give the first 6 rules to enable anyone to quickly exclude most numbers from being prime. There are 3 more Prime Exclusion rules that I will give in Part II of this work.

The exclusion rules I am about to reveal narrows down the potential field of prime candidates greatly, this should help in developing prime testing programs that will run much more efficiently since the rules I am putting forth exclude most numbers from being possible prime candidates.

##### Primes are not Random the Prime Mod 90 Algorithm

The first item I put forth in this work is the FACT that prime numbers are not distributed randomly. There is absolutely a Mod 90 algorithm to prime numbers. The PROOF is available by using simple Euclidian number observation via Base 9 Number Reduction incorrectly called by some Theosophical Addition.

Base 9 Number Reduction is a simple way to compress numbers to their lowest integer from 1 to 9. You simply add the integers in any number and keep repeating the task until you are left with 1 integer from 1 to 9. Ex: 129 = 1+2+9=12 and 12=1+2=3: so 129=3 via Base 9 Number Reduction.

The following chart clearly shows the Mod 90 algorithm to the distribution of primes. This algorithm appears in the hidden sequence of numbers that Base 9 Number Reduction reveals. The total algorithm of the reduced number sequence does not appear in only a Mod 30 sequence, the hidden algorithm can only be fully seen if you use a Mod 90 chart.  While a Mod 30 chart (which I use in Part II) can reveal a simple pattern to primes where 8 prime columns are revealed, it does not show the actual hidden algorithm that Base 9 Number Reduction reveals. That number can only be shown via my Prime Mod 90 Chart. Here is my Prime Mod 90 Chart.

 Prime Compressed Twin Prime Compressed Twin Prime Compressed Twin 11 2 T 101 2 T 191 2 T 13 4 103 4 193 4 17 8 T 107 8 T 197 8 T 19 1 109 1 199 1 23 5 113 5 203 5 (NP) 29 2 T 119 2 (NP) 209 2 (NP) 31 4 121 4 (NP) 211 4 37 1 127 1 217 1 (NP) 41 5 131 5 221 5 (NP) 43 7 133 7 (NP) 223 7 47 2 137 2 T 227 2 T 49 4 (NP) 139 4 229 4 53 8 143 8 (NP) 233 8 59 5 T 149 5 T 239 5 T 61 7 151 7 241 7 67 4 157 4 247 4 (NP) 71 8 T 161 8 (NP) 251 8 73 1 163 1 253 1 (NP) 77 5 (NP) 167 5 257 5 79 7 169 7 (NP) 259 7 (NP) 83 2 173 2 263 2 89 8 179 8 T 269 8 T 91 1 (NP) 181 1 271 1 97 7 187 7 (NP) 277 7

COMPRESSED VALUE = Base 9 Reduction

T = TWIN

NP = NOT PRIME

The Prime Mod 90 Chart shows how only 24 numbers are prime candidates in a 90 number field. The reason I must use 90 numbers to produce the 24 number prime sequence is the hidden Base 9 Number Reduction Algorithm does not reveal itself if I reduce this chart to say a Mod 30 Chart with an 8 number sequence. This hidden 24 number sequence is similar to the Fibonacci Sequence. If you chart the Fibonacci Sequence you will find a similar hidden Base 9 Number Reduction algorithm that also repeats after each 24th Fibonacci Sequence.

If you analyze the Prime Mod 90 chart you can also see how all primes greater than 7  fit into a Mod 30 Algorithm as well. However, the Base 9 Number Reduction algorithm does not reveal itself unless you view a Mod 90 chart.

The FACT that a pattern of numbers (an algorithm) repeats over and over in the prime number sequence as my Prime Mod 90 chart shows is a simple Euclidian Proof that Prime Number Distribution IS NOT RANDOM. A Euclidian Proof simply means an observable pattern to numbers. It would be fairly easy to express this in an Algebraic expression as well, which is something most modern mathematicians would be used to doing. However, this work is intended for the masses and not just academics.

The Prime Mod 90 Chart also shows all the locations where the infinite Twin Prime distribution field exists Once again the Prime Mod 90 Chart is a simple Euclidian Proof to two famous math questions, those being; are Twin Primes Random and are they infinite?

The Prime Mod 90 Chart clearly shows 75% of Prime Candidates are also Twin Prime Candidates. So the clear answer is Twin Prime distribution is a repeating algorithm within the Prime algorithm in that it is a set field or constant within the field of prime candidates.

The Prime Field Formula

The following formula is true about where all prime candidates must exist, no prime can exist outside of this formula.

Every prime number above 7 is located within the limited field of whole integers expressed as

90(x) + n where x is a whole integer > 0 and n is equal to one of the following numbers:

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97

This can be reduced to

Every prime number above 7 is located within the limited field of whole integers expressed as

30(x) + n where x is a whole integer > 0 and n is equal to one of the following numbers:

11, 13, 17, 19, 23, 29, 31, 37

These formulas do no PROVE that a number is prime, it merely shows the very limited field where all prime candidates must exist. No prime number can exist outside these very limited prime candidate fields.

Now that I have shown exactly where all prime numbers have to exist, I will give the first 6 of 9 Prime Exclusion laws that can be used to quickly exclude primes.

##### The first 6 Laws of Prime Exclusion

The laws are

1.    Even number rule - All even numbers above 2 are not prime.

2.    Divisible by 5 rule - All numbers above 5 that end with a 5 are not prime.

3.    Repeating odd integer rule -  A prime cannot have repeating odd integers ex: 777 etc.

4.    Number Reduction 3 remainder – An odd integer above 9 when reduced cannot equal a remainder of 3. Ex: 21 = 2+1=3

5.    Number Reduction 6 remainder –  An odd integer above 9 when reduced cannot equal a remainder of 6. Ex: 51 = 5+1=6

6.    Number Reduction 9 remainder – An odd integer above 9 when reduced cannot equal a remainder of 9. Ex: 27 = 2+7=9

The Laws of Prime Exclusion

###### Part II

In The Laws of Prime Exclusion Part I, I gave 6 simple laws to exclude numbers from being Prime.

The laws are

1.    Even number rule - All even numbers above 2 are not prime.

2.    Divisible by 5 rule - All numbers above 5 that end with a 5 are not prime.

3.    Repeating odd integer rule -  A prime cannot have repeating odd integers ex: 777 etc.

4.    Number Reduction 3 remainder – An odd integer above 9 when reduced cannot equal a remainder of 3. Ex: 21 = 2+1=3

5.    Number Reduction 6 remainder –  An odd integer above 9 when reduced cannot equal a remainder of 6. Ex: 51 = 5+1=6

6.    Number Reduction 9 remainder – An odd integer above 9 when reduced cannot equal a remainder of 9. Ex: 27 = 2+7=9

In the next 30 Column table, you can see how every prime neatly falls into one of 8 rows.

## Red number = Prime

### Green number = excluded number in Prime column

 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 130 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 160 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 190 91 92 93 94 95 96 97 98 99 200 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 220 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 250 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

This table also shows exactly where every TWIN PRIME will fall. It is the simplest Euclidian proof to show that Twin Primes repeat infinitely.

A proof for if Twin Primes are random and or infinite is another mathematical question that man has been searching Millenniums for. The Prime Mod 30 Chart clearly shows Twin Prime Candidates make up 75% of the possible Prime Candidates in the Prime Mod 30 Chart.

The 7th Law of Prime Exclusion is the whole square root exclusion rule. If an odd integer has a square root that is a whole integer (which by the way must be a lower prime) that number is excluded. Ex: 49 the square root is a whole integer that being 7. 121 the square root is 11 and 169 the square root is 13.

The 8th Law of Prime Exclusion is the Law of Exclusion for all non-primes that don't fall within the Mod 30 algorithm for Primes. This law is based on the fact that all primes above 7 are located in a Mod 30 field of distribution. The only non Primes left in this small field of prime number distribution are prime number candidates that are within the Mod 30 Prime algorithm, but are factors of smaller prime numbers and therefore excluded. These are a small amount of numbers compared to the large amount of numbers that are generally excluded with this rule!

The 8th Law of Prime Exclusion is stated as:

If you subtract NINE from any whole integer above NINE, and then divide by 30 and the result is a remainder of 2, 4, 8, 10, 14, 20, 22, 28 such numbers are prime candidates. If a number doesn't have one of these 8 numbers as a remainder, you can EXCLUDE it 100% from being prime.

The 9th Law of Prime Exclusion is the final test to determine if a number is prime. The purpose of the first 8 Prime Exclusion Laws is to enable a person to quickly determine the likelihood that a number is a possible prime candidate and to QUICKLY EXCLUDE most numbers as being possible prime candidates. The first 8 Laws cannot determine 100% that a number is prime. The final exclusion rule of primes determines absolutely if a number is prime. The final rule is testing an odd number above 89 for equal prime factoring by primes below the square root of a number and greater than 5.  Ex: 91 is the first number that is not easily disqualified from being prime by the first 8 Laws of Prime Exclusion. The first prime factor test we need to try is factoring by 7. Since 91 is a factored by 7 and 13, the number is not prime.

 Prime Compressed Twin Prime Compressed Twin Prime Compressed Twin 11 2 T 101 2 T 191 2 T 13 4 103 4 193 4 17 8 T 107 8 T 197 8 T 19 1 109 1 199 1 23 5 113 5 203 5 (NP) 29 2 T 119 2 (NP) 209 2 (NP) 31 4 121 4 (NP) 211 4 37 1 127 1 217 1 (NP) 41 5 131 5 221 5 (NP) 43 7 133 7 (NP) 223 7 47 2 137 2 T 227 2 T 49 4 (NP) 139 4 229 4 53 8 143 8 (NP) 233 8 59 5 T 149 5 T 239 5 T 61 7 151 7 241 7 67 4 157 4 247 4 (NP) 71 8 T 161 8 (NP) 251 8 73 1 163 1 253 1 (NP) 77 5 (NP) 167 5 257 5 79 7 169 7 (NP) 259 7 (NP) 83 2 173 2 263 2 89 8 179 8 T 269 8 T 91 1 (NP) 181 1 271 1 97 7 187 7 (NP) 277 7

Addendum

Fibonacci Number Sequence

I am including a chart that shows how the famous Fibonacci Number Sequence has a similar algorithm to the Prime Mod 90 Chart. The hidden algorithm within the Fibonacci sequence repeats every 24th number. This is the same number in which the Prime Mod 90 Chart algorithm repeats. I find it interesting how these hidden algorithms both share the number 24 in their length.

 Sequence Fibonacci Reduction Sequence Fibonacci Reduction 1 1 1 25 75025 1 2 1 1 26 121393 1 3 2 2 27 196418 2 4 3 3 28 317811 3 5 5 5 29 514229 5 6 8 8 30 832040 8 7 13 4 31 1346269 4 8 21 3 32 2178309 3 9 34 7 33 3524578 7 10 55 1 34 5702887 1 11 89 8 35 9227465 8 12 144 9 36 14930352 9 13 233 8 37 24157817 8 14 377 8 38 39088169 8 15 610 7 39 63245986 7 16 987 6 40 102334155 6 17 1597 4 41 165580141 4 18 2584 1 42 267914296 1 19 4181 5 43 433494437 5 20 6765 6 44 701408733 6 21 10946 2 45 1134903170 2 22 17711 8 46 1836311903 8 23 28657 1 47 2971215073 1 24 46368 9 48 4807526976 9