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/...

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/...

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:)

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#### Comment Preferences

• ##### Tip Jar(2+ / 0-)
Recommended by:
twigg, Mr Robert
• ##### So ....(2+ / 0-)

You just made my brain fall over :)

Tell me ... What happens when you multiply a Ryan with a Boehner, and divide by the deficit?

I hope that the quality of debate will improve,
but I fear we will remain Democrats.

• ##### two negative numbers multiplied (2+ / 0-)
Recommended by:
twigg, wilderness voice

become a positive. After Ryan and Boehner get there way we can reasonably expect a bigger deficit. Then we can expect even more taxes on the rich since the Laffer curve is a joke at these taxation levels.
Or we can go broke first then tax the rich even more.
GOP budget ideas to raise money are negative numbers no matter what they claim. :)

• ##### Sorry, Ryan and Boehner are(2+ / 0-)
Recommended by:
twigg, wilderness voice

not negatives.

They are irrational numbers with imaginary components.    Ryan and Boehner can be negative or positive at the flip of a switch.

• ##### Do you know Hilbert? No?(1+ / 0-)
Recommended by:
twigg

Then what are you doing in his space?

Happy little moron, Lucky little man.
I wish I was a moron, MY GOD, Perhaps I am!
—Spike Milligan

[ Parent ]

• ##### obviously(6+ / 0-)

If a number is prime it is not divisible by three, so it either # and 1/3, or # and 2/3. All prime numbers are indivisible by 3, but not all numbers indivisible by 3 are prime.

I didn't read the second part, because the first part was pretty silly.

"The room was dark as an honest politician's prospects." -- Dashiell Hammett

• ##### There is a pattern to Prime Numbers(0+ / 0-)

is the point and with a known pattern you can skip ahead and predict what numbers will be  prime numbers without finding every prime sequentially.

• ##### i.e. (2^K)-1 and (2^K)+1 often are prime.(0+ / 0-)

(3^K)+1 fits the 0.3333333 pattern the diarist is seems to have 'discovered.'

Happy little moron, Lucky little man.
I wish I was a moron, MY GOD, Perhaps I am!
—Spike Milligan

[ Parent ]

• ##### Sort of, but not this(7+ / 0-)

There is some evidence of structure to the primes.  See, for example, the Ulam spriral.  However, at least so far, no efforts to determine a "pattern" to the primes that holds up over all cases have been successful.  Indeed, the related Riemann hypothesis is arguably the most famous unsolved problem in mathematics today.

What is presented here, while well-meaning, is extremely mathematically naive.

"All opinions are not equal. Some are a very great deal more robust, sophisticated and well supported in logic and argument than others." -Douglas Adams

[ Parent ]

• ##### Axiom of Choice?(0+ / 0-)

(just trying to hijack the thread -- diarist is out of his mathematical depth)

Happy little moron, Lucky little man.
I wish I was a moron, MY GOD, Perhaps I am!
—Spike Milligan

[ Parent ]

• ##### I'm pro-AOC(2+ / 0-)
Recommended by:
polecat, atana

I've always thought the arguments to the contrary are more indicative of the limitations of the ability to think about the mathematics than the math itself.  Even if we assume that Russell's infinite pairs of socks are facially identical, they have to be in some sense distinguishable; otherwise, they'd be the same sock.

Not, mind you, that that leads me to any way to sanely apply the well-ordering principle to the real number set.

"All opinions are not equal. Some are a very great deal more robust, sophisticated and well supported in logic and argument than others." -Douglas Adams

[ Parent ]

• ##### I could never pin down my professor on the AC(0+ / 0-)

(and I appear to not be the only student that tried)

He would always seem to work it both ways.  I, for one, always felt uncomfortable with it but never had a way to articulate how Deus Ex Machina the solutions it would permit could be.

Happy little moron, Lucky little man.
I wish I was a moron, MY GOD, Perhaps I am!
—Spike Milligan

[ Parent ]

• ##### Not just him(2+ / 0-)
Recommended by:
polecat, atana

No one likes to get pinned down on the Axiom of Choice.  Either you're anti-AOC, and wind up arguing that it's impossible to pick up just one sock from a pair of socks in a bin ... or you're pro-AOC, and wind up arguing that there is too a smallest real number in the interval (0,1).

I've always liked Jerry Bona's joking take on the whole thing: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

(For those playing along at home, all three of those are the same thing, in different forms.)

"All opinions are not equal. Some are a very great deal more robust, sophisticated and well supported in logic and argument than others." -Douglas Adams

[ Parent ]

• ##### Least in the well-ordering of (0,1)(1+ / 0-)
Recommended by:
polecat

not in the usual ("<") order of (0,1).

• ##### What is presented here, while well-meaning, is (0+ / 0-)

extremely mathematically naive.

I present 2 different theories to find Prime Numbers here I welcome anyone to take my methodology and prove me wrong.
if you have questions about my methods to find Primes just ask.
Assuming something is wrong without proving it first is not what I expect at a Left Wing blog.

• ##### Sigh.(3+ / 0-)
Recommended by:
polecat, Things Come Undone, Dbug

You're doing a lot with arithmetic modulo 3, although you don't state that directly.  But that's what's going one when you look at the decimal component of numbers divided by 3.  In your first "theory", you note that prime numbers greater than 3 have a decimal component of 0.333... or 0.666...; that's true, but it doesn't provide any useful knowledge.  Why not?  Put another way, that means that every prime number > 3 can be written in the form 3n+1 or 3n+2.  But that's self-evident.  ALL numbers can be written in one of those two forms except for those which are evenly divisible by 3 (which are obviously not prime).

Likewise, when you examine the results of taking prime numbers modulo 3, and then multiply them together, you do not actually gain any insight into the structure of the primes.  "Multiply 2 Primes that [when divided by 3] end in .333 and .666 and you get," you note.  And that's true, but it has nothing to do with whether the numbers are prime.  Any time you multiply a number of the form 3n+1 by a number of the form 3n+2, the result is a number of the form 9n^2+9n+2 ... which, when divided by 3, has a remainder of 2 (and, thus, a decimal component of 0.666...).  None of this is specific to the primes.

"All opinions are not equal. Some are a very great deal more robust, sophisticated and well supported in logic and argument than others." -Douglas Adams

[ Parent ]

• ##### yes (0+ / 0-)

After reading your link I agree that I am  doing a lot with arithmetic modulo 3. I don't state it directly because i did not know what it was before you told me.

In your first "theory", you note that prime numbers greater than 3 have a decimal component of 0.333... or 0.666...; that's true, but it doesn't provide any useful knowledge.  Why not?
I disagree I think it requires less work than  the Sieve of Eratosthenes. to find Prime Numbers this way.
Likewise, when you examine the results of taking prime numbers modulo 3, and then multiply them together, you do not actually gain any insight into the structure of the primes
Agreed this has insight into RSA encryption based on multiplying 2 Primes. I thought that a list of Prime numbers divided by the number 3 would be very easy for a computer hacker to generate much easier than cracking RSA encryption then by knowing that half of the multiples of any 2 Prime numbers ends in .333 and the other half end in .666 that would reduce their work load by half when they try and crack the RSA encryption.
I did not mean to imply that this was a method to find Prime Numbers only it was a side result of my theory.

• ##### Your proposals are very much like the Sieve.(2+ / 0-)
Recommended by:
Things Come Undone, polecat

In the Sieve, when you cross out every 3rd number (the multiples of 3), that is exactly the same as keeping just the numbers that end in .333... or .666... when divided by 3.

Division takes longer than counting, so the Sieve method of scratching off every 3rd number could actually be faster than dividing and looking for repeating decimals.

Primes used for RSA encryption generally range from 100 to 1000 digits in length (100 digits would be about a sextillion sextillion sextillion sextillion sextillion). The number of 10-digit primes is about 455 million. It wouldn't be practical to make a list of the 100-digit primes because there are so many. That's part of why it's hard to find the unknown prime in an RSA key pair.

The Prime Pages might interest you.

• ##### sorry for the long wait(0+ / 0-)

In the Sieve, when you cross out every 3rd number (the multiples of 3), that is exactly the same as keeping just the numbers that end in .333... or .666... when divided by 3.

No It is not the the numbers and multiples of 7, 11, 13 etc will not be removed when you remove the 3's unless 3 * 7 = 21 is one example.

Division takes longer than counting, so the Sieve method of scratching off every 3rd number could actually be faster than dividing and looking for repeating decimals.

yes but if you notice the part about

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

by removing those numbers you can find any Prime up to 100 without going in sequence to find the Primes.
Divsion is harder than counting though:)

The number of 10-digit primes is about 455 million. It wouldn't be practical to make a list of the 100-digit primes because there are so many.

I got more than a few comments my last few posts about how if my ideas were so great I could not find these 100 digit numbers.
So I included the information about how to reduce the work load by half hoping to satisfy them.
I agree without a super computer the problem is still very hard however foreign countries looking to crack our codes do have super computers and can do as I suggest.
THe NSA should investigate my idea to see if it works on a super computer before some other country gets the idea and starts reading our messages to our armies in Iraq, Afghanistan and maybe Iran.
But for the record I am against war. Maybe Anonymous will find this post useful to read Karl Rove's email.

• ##### I'm sure the NSA would like to have a(0+ / 0-)

list of all of the 100-digit primes.

Let's say NSA cryptanalysts are trying to decode an RSA-encrypted message. They have obtained a 200-digit number that they know is the product of two 100-digit secret key primes. Using their list of all of the 100-digit primes,  they can do a bunch of trial divisions until they find a prime that divides the 200-digit number evenly, with a remainder of 0. That prime, and the quotient, are the two parts of the secret key, with which they can quickly read the message.

How many 100-digit primes are there?

A mathematician named Riemann came up with a formula that can give a very accurate estimate. There are about 4.362 x 1097 primes of 100 digits or less. If we exclude the primes of 99 digits or less (about 4.406 x 1096 of them), we are left with about 3.921 x 1097 primes of exactly 100 digits.

How long will it take to make a list of all of the 100-digit primes with a really good supercomputer?

Suppose one processor can find a billion 100-digit primes per nanosecond (that is 1018 or a quintillion per second), and the NSA's supercomputer has a quintillion processors. And suppose the Things Come Undone techniques reduce the workload by half, and with a little more thought, reduce the workload by another factor of a quintillion to one.  Then the list will be finished in about 1.961 x 1043 seconds, which is about 6.2 x 1032 millennia, or about 45 million quintillion times the present age of the universe.

How big is the list of 100-digit primes?

Suppose we can write a 100-digit prime very compactly onto a single atom. Since the observable universe contains an estimated 1082 atoms, we'll need about 4 trillion universes to store all those primes.

Since the list takes up so much space, the NSA's cryptanalysts better hope that one of the secret primes occupies a nearby atom, hopefully no farther than the Andromeda galaxy, because if they have to go farther, the message will be old news by the time they can decode it.

Nevertheless, in 2005, researchers successfully factored a 200-digit number into its two previously secret 100-digit primes. It took less than 2.5 years. The largest RSA number known to have been successfully factored is a 232-digit product of two 116-digit primes.  How did they do it?  I'd like to know.  Obviously they have a better idea that doesn't require an impossibly large list of primes.

• ##### Use of prime numbers(1+ / 0-)
Recommended by:
Things Come Undone

Prime numbers are most famously useful in cryptography.

I have heard that design of internal combustion engines can make use of prime numbers in optimizing fuel burning.

Finally, mechanical rotors ....  sry -- got to go.  use google.

• ##### Prime Numbers (0+ / 0-)

Can be used to improve combustion in an engine do you have a link this sounds very interesting.

• ##### Why don't you work in base 3 instead of base 10?(1+ / 0-)
Recommended by:
Things Come Undone

Or a nice beautiful base like 210 (divisible by 2*3*5*7) and spare yourself some trouble.

That .3333 pattern is an artifice of base 10 (2*5) being unable to represent numbers that have a 1/3 remainder (of course).

Happy little moron, Lucky little man.
I wish I was a moron, MY GOD, Perhaps I am!
—Spike Milligan

• ##### You're not a mathematician or programmer, are you?(2+ / 0-)
Recommended by:
Things Come Undone, kurt

Happy little moron, Lucky little man.
I wish I was a moron, MY GOD, Perhaps I am!
—Spike Milligan

[ Parent ]

• ##### Nope (1+ / 0-)
Recommended by:
kurt

Philosophy Major but  unemployed with plenty of time on his hands who does this to keep his mind occupied because it is hard:)

• ##### There is another pattern you should consider:(1+ / 0-)
Recommended by:
Things Come Undone

As N increases, then number of primes less than N is somewhat proportional to the log of N.

Happy little moron, Lucky little man.
I wish I was a moron, MY GOD, Perhaps I am!
—Spike Milligan

• ##### Cidadas are better at finding(3+ / 0-)
Recommended by:
Things Come Undone, Mr Robert, kurt

primes than some humans.

• ##### Very interesting simulation n/t(1+ / 0-)
Recommended by:
Things Come Undone

The only trouble with retirement is...I never get a day off!

[ Parent ]

• ##### The Sieve of Eratosthenes works very well(1+ / 0-)
Recommended by:
atana

Once you find a prime, you know all multiples of that prime will be composite (non-prime) numbers, by definition. For example, 3 is prime, which means 6, 9, 12, 15, etc. are not primes. So let’s say you want to see if 31 is prime. You only have to test the primes up through the square root of 31. The square root of 31 is between 5 and 6 (because 5 squared is 25 and 6 squared is 36).  You only have to check the integers up through multiples of 5.

Here’s how the sieve works. Write down all 31 numbers (or if you’re programming a computer, use an array):

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

1 isn’t prime. The next number is 2, which is prime. Keep that number and cross out all multiples of 2 (4, 6, 8, 10, etc.). The letter x means the number isn’t prime:

x 2 3 x 5 x 7 x 9 x 11 x 13 x 15 x 17 x 19 x 21 x 23 x 25 x 27 x 29 x 31

The next number is 3. Not crossed out, so it’s prime. Keep it and cross out multiples of 3 (note that 6, 12, 18, are already crossed out because they’re also multiples of 2):

x 2 3 x 5 x 7 x x x 11 x 13 x x x 17 x 19 x x x 23 x 25 x x x 29 x 31

Almost done. 5 is the next prime, so cross out multiples of 5. Actually, you can start with 25 (5 squared) because all numbers below that (10, 15, and 20) were already crossed out. Cross out 25…

x 2 3 x 5 x 7 x x x 11 x 13 x x x 17 x 19 x x x 23 x x x x x 29 x 31

There’s the list of the first primes up to 31, using the Sieve of Eratosthenes.

If you can program on the bit level, you can fit this into four eight-bit bytes. 1 is prime, 0 is composite.

01101010 00101000 10100010 00001010 (the least significant bit on the left, which is backwards, so it really should be 01010000 01000101 00010100 01010110). But I’ll shut up in a minute.

Is 161 a prime number? I’m not sure. You only have to check the numbers to the square root of 161. 12 squared is 144. 13 squared is 169. So you only need to check the primes up to 12. Is 161 divisible by 2? No. 3? No. 5? No. 7? No. 11? No. So 161 is a prime.

“If you misspell some words, it’s not plagiarism.” – Some Writer

• ##### I used to assign that as a programming exercise(1+ / 0-)
Recommended by:
Dbug

in my Mathematics for Computer Science class.

• ##### Yeah, a long time ago (1980s)(1+ / 0-)
Recommended by:
atana

I wrote an assembly language program for the Commodore 64 that used most of its memory to find prime numbers and print them out. About 38K of usable memory, one bit per number, so 8x38x1024 was the highest number. Later I figured out I could double the number by skipping even numbers, but it got more complicated. Then I figured out a way to extend that further, but that's another story.

“If you misspell some words, it’s not plagiarism.” – Some Writer

[ Parent ]