pattern in prime numbers?

[alibaba]

Hi all,
I was searching for any pattern in the prime numbers' sequence. After long nights of trying, I am growing tired of that. Does anybody know of a proof or a reason why one can not find a formula to find any prime number? Or is it there but no one has found it yet?

 

[ceptimus]

Tough question. I wouldn't even begin to look for such patterns, on the basis that there have been plenty of genius mathematicians, who never found such things, so why would an idiot like me find any.

Of course there are plenty of rules that all prime numbers fulfil, like being odd (except for 2) and the digits added together not being divisible by 3.

All primes after 2 and 3, are either one more or one less than a multiple of six.

There are the mersenne primes, of the form 2ⁿ-1, like 2¹⁷-1 and 2¹⁹-1. In this case the power, n, must itself be prime, but that is not sufficient to guarantee that the number generated is prime.

So there are plenty of known patterns, but none known that are able to always generate a fresh prime.

 

[Skeptoid]

You'd be amazed how much you can learn about the primes by Googling on "prime numbers". You can spend days following links to interesting facts about primes and number theory.

 

[El Greco]

There was this very interesting thread last year in the Win32Asm forum. You'll need a lot of reading though to get something useful from it.

 

[Matabiri]

You'd be amazed how much you can learn about the primes by Googling on "prime numbers". You can spend days following links to interesting facts about primes and number theory.

The one I like is the Prime Number Theorem, which states that the count of primes up to a number, x, tends to x/ln(x) as x becomes very large.

 

[MRC_Hans]

Ah, the memories. How long since I played with prime finder programs. --- Hey, I wonder how they'll perform on my 2.66gig PC .

Hans

 

[Ladewig]

I don't know if it will help, but you may have fun playing with Ulam's Spiral

spiral

 

[MRC_Hans]

Ahh, yes. Nice! Prime patterns are a bit like double-talk: There is an intriguing amount of near-patterns in them, but it never gets really consistent.

Hans

 

[davidhorman]

Here's a nicer (but non-interactive) image of Ulam's Spiral (or Ulam's Rose, apparently):

http://www.abarim-publications.com/plaatjes/Ulam's Rose.JPG

Edit: the validity of said image might be doubtful - the front page is called "Understand the Bible through Quantum Mechanics and Chaos Theory" It was linked in the thread that's linked to in an earlier post so I assumed it was something relevant.

David

 

[WanderingKnight]

Of course there are plenty of rules that all prime numbers fulfil, like being odd (except for 2) and the digits added together not being divisible by 3.

Er...that's because all numbers who's digits add up to a multiple of three are themselves a multiple of three and therefore not prime. No primes' digits add up to a multiple of nine either, because those numbers are multiples of nine. (2x9=18 -> 1+8=9 12x9=108 -> 1+0+8=9 55x9=495 -> 4+5+9=18) There's a few tricks like that for divisibility.

All primes after 2 and 3, are either one more or one less than a multiple of six.

Again, this sounds really neat, but isn't really a mysterious pattern so much as it is a result of the number (six) you chose. If a number was 2 or 4 more (or less) than a multiple of 6, it would be even and not prime. 3 more would make it a multiple of 3. 5 more is the same case as one less, and so on.

 

[jayrev]

Again, this sounds really neat, but isn't really a mysterious pattern so much as it is a result of the number (six) you chose. If a number was 2 or 4 more (or less) than a multiple of 6, it would be even and not prime. 3 more would make it a multiple of 3. 5 more is the same case as one less, and so on.

I once thought I had found something profound when I discovered a similar thing...the square of any prime > 3 mod 12 = 1, but it works by a similar easily proven means.

 

[Soapy Sam]

You need chaos theory and quantum mechanics to understand the bible? Good grief- that explains why I kept failing Sunday School.

Is the definition of primes not rather negative / exclusive? (A prime number is NOT divisible by two or by any whole factors, etc).Seems to me (Non math argument follows, look away, JJ) that since numbers tend to form patterns and groups, just as the utensils in a kitchen , or any collection of objects do, if we EXCLUDE sufficient subgroups, whatever remains will have similarities merely as a result of that filtering process. Humans, being innate pattern seekers, may well see Patterns in the Numbers as well as faces on Mars and the Moon, where really there are just numbers and moondust.

 

[alibaba]

I have already got to the point where prime numbers should exclude multiples of primes. This way you can exclude what is not a prime as you find a prime, but you have to go all the way from 2 to the p to find p. This way is good for a computer program, but not feasible for more than 32bit or 64 bit numbers in some computers. Maybe 128 bit number for super computers.

I am trying to find a formula that has complexity of o(1) for getting any prime from 2 to infinity.

Thanks to all of you who shared thier ideas

 

[alibaba]

Tough question. I wouldn't even begin to look for such patterns, on the basis that there have been plenty of genius mathematicians, who never found such things, so why would an idiot like me find any.

probably the reason such thing is not found yet is that all the living genius mathematicians think the way you do

The prime numbers have been very fascinating. No one has found a formula for them, and no one has proven such formula doesn't exist! I noticed there are lots of smart poeple in this forum, that is why I trew my problem here

 

[alibaba]

Here's a nicer (but non-interactive) image of Ulam's Spiral (or Ulam's Rose, apparently):

http://www.abarim-publications.com/plaatjes/Ulam's Rose.JPG

Edit: the validity of said image might be doubtful - the front page is called "Understand the Bible through Quantum Mechanics and Chaos Theory" It was linked in the thread that's linked to in an earlier post so I assumed it was something relevant.

David

Dazzling! I can see some pattern there... just have to stare at it a little more

 

[alibaba]

I don't know if it will help, but you may have fun playing with Ulam's Spiral

spiral

Something I don't seem to understand: why isn't the density of primes dropping as we travel away from the center of spiral?

 

[alibaba]

Something I don't seem to understand: why isn't the density of primes dropping as we travel away from the center of spiral?

Oh, I see now, they only start thining when I get close to:
26,500,000,000,000,000. That is probably due to the logarithmic complexity of the spiral.

 

[Chupacabras]

FWIW, I remember to have arrived at a web site on distributed computing a while ago (probably was this ). They had a big prize for a new prime, if your computer should find it (or run into it?). On retrospective, this strikes me as a tip that there is no such pattern.

But then again, I like to dream

 

[KonTiki]

This link provides examples of what you asked for, but such formulas are not horribly useful.

 

[xouper]

jayrev: I once thought I had found something profound when I discovered a similar thing...the square of any prime > 3 mod 12 = 1, but it works by a similar easily proven means.

Here's something to keep you busy for another three and a half minutes. See if you can extend that to <nobr>p² mod 24 = 1,</nobr> for any prime p>3.

This can be visualized in terms of a 24 hour clock, as in the follow diagram, where the squares of primes greater than three will all on be on the one o'clock radial:

The above diagram is from page 156 of Peter Plichta's book God's Secret Formula.

This kind of "one way" statement about primes only works if it is already known that p is prime. A "two way" statement about primes would be something like Wilson's Theorem, which says that (p-1)! mod p = -1 if and only if p is prime.