My first theory about Prime Numbers was that every Prime divided by the number 3 to the third decimal point ends in a repeating decimal of either .333 or .666 except of course the number 3. This was explained in my post

http://www.dailykos.com/story/2011/12/05/1042553/-Perhaps-a-simpler-way-to-find-out-if-a-number-is-prime

The problem with that theory I later realized was what I call False Primes which are numbers that are not Prime that when divided by the number 3 also end in a repeating decimal of .333 or .666.

Problem solved as far as odd numbers are concerned some numbers that end in the number 5 when divided by the number 3 also end in a repeating decimal ending in .333 or .666. However numbers ending in 5 are not Prime so I now ignore them.

Next Problem there is an other cause of odd numbers that are not Prime the other source of False Prime numbers are the squares of Prime Numbers starting at the number 7 and going up ( why that is I don’t know )

and numbers that are the result of a Prime Number 7 and above and a Prime number larger than it. Examples

7*7 = 49,    49/3 = 16.333

7*11 = 77,  77/3 = 25.333

7*15 = 105   Multiple of 5 it won’t be Prime anyway

7*17 =  119,  119/3 = 39.666

Since the next Prime to be squared would be 11 and 11 squared is 121 and is larger than 100 then by removing the numbers 49, 77, we can then find any Prime number below 100 just by multiplying any of the remaining odd numbers by 3 and seeing if they end in a repeating decimal.

So if you just want to find out if one number is Prime below 100 this method saves time compared to the Sieve of Eratosthenes.

My second theory of finding Prime Numbers was on this post.

http://www.dailykos.com/story/2012/01/26/1058765/-Perhaps-a-simpler-way-to-Find-Prime-Numbers-2

I tried to use a chart to show it but for some reason the chart did not come out like I typed it when I posted here is the theory without the chart.

The numbers 1 and 2 are assumed to be Prime however once we get to the number 3 and multiply it by itself we find that the numbers inbetween 3 and 9 are Prime. Then we multiply those numbers by 3 and by themselves to find more Prime Numbers ( for some reason the number 9 must be included even though it is not a Prime Number to make this system work why I don’t know.

examples below

3 * 3= 9 the odd numbers inbetween 3 and 9 are the prime numbers 5 and 7

3 * 5= 15 the  odd numbers inbetween 9 and 15 are the prime numbers 11 and 13

3 * 7= 21 the odd numbers in between 15 and 21   are the prime numbers 17, and 19

5 * 5 =  25 odd numbers inbetween 21 and 25 that are Prime 23

3 * 9 = 27 the odd numbers inbetween 25 and 27 are 0

3 * 11 = 33 odd numbers inbetween 27 and 33 are 31

5 * 7 = 35 odd numbers inbetween 33 and 35 that are Prime 0

3 * 13 = 39 odd numbers inbetween 35 and 39 that are Prime 37

5 * 9 = 45 odd numbers inbetween 39 and 45 that are Prime 41 and 43

7 * 7 = 49 odd numbers inbetween 45 and 49 that are Prime 47

3 * 17 = 51 odd numbers inbetween 49 and 51 that are Prime 0

5 * 11 = 55 odd numbers inbetween 51 and 55 that are Prime 53

3 * 19 = 57 odd numbers inbetween 55 and 57 that are Prime 0

9 * 7 = 63 odd numbers inbetween 57  and 63  that are Prime 59, 61

5 *13 = 65 odd numbers inbetween 63 and 65 that are Prime 0

23 * 3 = 69 odd numbers inbetween 65 and 69 that are Prime 67

5 * 15 = 75 odd numbers inbetween 69 and 75 that are Prime 71, 73

7 * 11 = 77 odd numbers inbetween 75 and 77 that are Prime  0

9 * 9 = 81 odd numbers inbetween 77 and 81 that are Prime 79

5 * 17 =  85 odd numbers inbetween 81 and 85 that are Prime 83

29 * 3 = 87 odd numbers inbetween 85 and 87 that are Prime 0

7 *13 = 91  odd numbers inbetween 87 and 91 that are Prime 89

31 * 3 = 93 odd numbers inbetween 91 and 93 that are Prime 0

5 * 19 = 95 odd numbers inbetween 93 and 95 that are Prime 0

33 * 3 = 99 odd numbers inbetween 95 and 99 that are Prime 97

This method has a few draw backs notice below that after you multiply 5 by itself you don’t them multiply 7 by itself instead  3 * 9 gives you the smaller number that you need to find the next Primes but there is no way of knowing that ahead of time without first multiplying all the numbers realizing that fact and then place the numbers in order of smallest answer first sequentially later then looking for the odd numbers inbetween.

7 * 7 would 5 * 5 =  25 odd numbers inbetween 21 and 25 that are Prime 23

3 * 9 = 27 the odd numbers inbetween 25 and 27 are 0

3 * 11 = 33 odd numbers inbetween 27 and 33 are 31

25 equations to find all the Primes to 100 is not bad. But not as good as my first theory now that it has been reworked.

In my research on Primes finding out the answer when two Primes have been multiplied together has been mentioned in the comments of my posts perhaps because they want to break RSA encryption.

My new methods to solve Prime numbers might be only slightly faster than current ones a computer would be needed to test this however as far as solving whether 2 prime numbers were multiplied  together goes this might help them.

My first method uses dividing any Number by 3 to find Prime numbers any number ending in  .333 or .666.

it is important to remember that any time  we add 2 even numbers together we get an even number examples

2 + 2 = 4,     8 + 6 = 14

any time we add to odd numbers we get

3 + 1 = 4, 7 + 9 = 16

however any time an odd and an even number is added together we get an odd number

5 + 4 = 9,                  7 + 3 = 10

We get similar results when we Multiply 2 Prime numbers that end in .666 when divided by 3 together like 5 and 11 and you get a number that ends in .333

5 / 3 = 1.666      11 / 3 = 3.666

5 * 11 = 55,

55 / 3 = 18.333

Multiply 2 Primes  that end in .333 when divided by 3 together like 7 and 13 and you get a number that ends in .333

7 / 3 = 2.333,        13 / 3 = 4.333

7 * 13 = 91

91 / 3 = 30.333

Multiply 2 Primes that end in .333 and .666 and you get

7 * 11 = 77

77 / 3 = 25.666

If one wanted to crack a code based on two Prime Numbers being multiplied together and we knew what the result of those two Prime Numbers was then ussing this method we could cut the work to find that number by 50%.

I think the NSA might want to offer me a new computer with the latest in virus protection/s.   I am now working on a new theory to further reduce the work to find the multiples of two Prime numbers i don’t know if it will pan out.

Even better they can offer me a job:)