prime numbers

• Nov 2nd 2007, 08:07 PM
Singular
prime numbers
I'm sorry if I ask a stupid question.....

I know that I don't have any capability in this field (numbre theory)(Tmi)
I just wonder....

1. Is the distribution of prime numbers random?

2. Is it possible that the distribution is pseudo-random?

3. Is it possible that in the distribution of prime numbers, it takes a lot of counts before it repeats a same pattern?

4. Does it(the distribution of prime numbers) have a pattern?

Thank you
• Nov 2nd 2007, 09:36 PM
kalagota
Quote:

Originally Posted by Singular
I'm sorry if I ask a stupid question.....

I know that I don't have any capability in this field (numbre theory)(Tmi)
I just wonder....

1. Is the distribution of prime numbers random?

2. Is it possible that the distribution is pseudo-random?

3. Is it possible that in the distribution of prime numbers, it takes a lot of counts before it repeats a same pattern?

4. Does it(the distribution of prime numbers) have a pattern?

Thank you

this is what i think..
As long as the Riemann Hyp has not yet proven, it is random....

like what my professor who is a number theorist (Dr. Fidel Nemenzo), told us, the set of prime numbers is both chaotic and organize.. Ü
• Nov 3rd 2007, 10:29 AM
CaptainBlack
Quote:

Originally Posted by Singular
I'm sorry if I ask a stupid question.....

I know that I don't have any capability in this field (numbre theory)(Tmi)
I just wonder....

1. Is the distribution of prime numbers random?

In what sense? Algorithmic information theory?

In a intuitive sense no, the n-th prime is the n-th prime and so is deterministic.

The shortest progam to calculate the n-th prime is k*n+C bits long where
k>0 and C are constants? No because, because the progam is of fixed length.

Quote:

2. Is it possible that the distribution is pseudo-random?
Again depends on what you mean

Quote:

3. Is it possible that in the distribution of prime numbers, it takes a lot of counts before it repeats a same pattern?
What do you think this means?

Quote:

4. Does it(the distribution of prime numbers) have a pattern?
See the prime number theorem.

RonL
• Nov 3rd 2007, 04:20 PM
ThePerfectHacker
The Riemann Hypothesis is not going to give a formula to produce prime numbers*. In its original statement, RH is more of a complex analysis problem than a number theory problem. However, if it is true it can strengthen the prime number theorem which is very important in analytic number theory.

*)That is nothing what the problem is about. If you are looking for a way to see if a number is prime or not you can use Wilson's theorem which always works. But is highley inefficient.
• Nov 3rd 2007, 06:15 PM
kalagota
Wilson's Thm will only tell you if a number is a prime, it won't tell you the next prime.. And what does Prime number theorem say? It only gives you the number of prime numbers below a certain integer but it also does not give you what the next prime is.. Ü
Anyways, has anyone read "The Music of the Prime"? it is amazing!
• Nov 3rd 2007, 08:12 PM
ThePerfectHacker
Quote:

Originally Posted by kalagota
Wilson's Thm will only tell you if a number is a prime, it won't tell you the next prime.. And what does Prime number theorem say? It only gives you the number of prime numbers below a certain integer but it also does not give you what the next prime is.. Ü
Anyways, has anyone read "The Music of the Prime"? it is amazing!

You want a formula? Okay. How about $n^2+n+41$ for $n\geq 0$. :D
• Nov 3rd 2007, 08:50 PM
kalagota
Quote:

Originally Posted by ThePerfectHacker
You want a formula? Okay. How about $n^2+n+41$ for $n\geq 0$. :D

is it proven? hehe
• Nov 3rd 2007, 08:57 PM
DivideBy0
Quote:

Originally Posted by ThePerfectHacker
You want a formula? Okay. How about $n^2+n+41$ for $n\geq 0$. :D

Wow nice find, gives primes up to n = 39
• Nov 3rd 2007, 10:36 PM
angel.white
Quote:

Originally Posted by ThePerfectHacker
You want a formula? Okay. How about $n^2+n+41$ for $n\geq 0$. :D

Unless I misunderstand, 41 should divide this equation when n=41
• Nov 4th 2007, 12:40 AM
kalagota
Quote:

Originally Posted by angel.white
Unless I misunderstand, 41 should divide this equation when n=41

yeah, $41^2 + 41 + 41 = (41)(43)$

and so the same with the multiples of 41..
• Nov 4th 2007, 05:44 AM
ThePerfectHacker
Quote:

Originally Posted by DivideBy0
Wow nice find, gives primes up to n = 39

Quote:

Originally Posted by kalagota
is it proven? hehe

Quote:

Originally Posted by angel.white
Unless I misunderstand, 41 should divide this equation when n=41

It is a very famous polynomial in from number theory. It is amazing on how many primes it produces. In fact, since its discovery over 250 years (stupid) people actually believed that it only produces primes. They asked Euler to prove it, of course, Euler looked at them like a bunch of idiots and came up with the counterexample at 41.

It can be easily proven that there is no non-constant polynomial that produces only prime numbers.
• Nov 26th 2007, 11:18 AM
letx=x
Formula producing only primes
Hi guys

f(n) = floor{a^3^n} only produces primes for some number a. Trouble is, it only produces very large primes and the exact value of the constant is unknown! It is approximately 1.3063...and is known as Mills' constant. There are a few other similar functions.
• Nov 26th 2007, 06:25 PM
ThePerfectHacker
Quote:

Originally Posted by letx=x
Hi guys

f(n) = floor{a^3^n} only produces primes for some number a. Trouble is, it only produces very large primes and the exact value of the constant is unknown! It is approximately 1.3063...and is known as Mills' constant. There are a few other similar functions.

• Dec 3rd 2007, 06:37 PM
Skinner
I am a lowly college freshman, but I have come up with some ideas about the distribution of primes myself.

How do we look at the distribution of primes? If you're thinking on a one dimensional number line like this:

http://aycu40.webshots.com/image/369...5293260_fs.jpg
You're probably not going to find a pattern of the distribution, The problem of primes seems like it's not a linear problem.

I'm sure that there are alot of mathematicians more seasoned than me who have heard of the Ulam Spiral, where you start drawing a grid of numbers starting with one in the middle like this:

http://aycu20.webshots.com/image/355...0328293_fs.jpg

It gives you diagonal patterns of primes like this:

http://aycu14.webshots.com/image/333...3401392_fs.jpg

There are even other similar types of spirals done which give you surprising patterns. What if this idea could be projected somehow in 3 dimensions? It would give us a reallt cool pattern. I still don't think that would be it though, it is still limited and based on an arithmetic approach. I think that it will take something like complex numbers or something nonlinear and non arithmetic to to determine the true distribution. This just gives us a glimpse. It's fun to think about though.

I hope that I am not too terribly misinformed to make myself look foolish :D
• Dec 5th 2007, 04:53 AM
kalagota
Quote:

Originally Posted by Skinner
I am a lowly college freshman, but I have come up with some ideas about the distribution of primes myself.

How do we look at the distribution of primes? If you're thinking on a one dimensional number line like this:

http://aycu40.webshots.com/image/369...5293260_fs.jpg
You're probably not going to find a pattern of the distribution, The problem of primes seems like it's not a linear problem.

I'm sure that there are alot of mathematicians more seasoned than me who have heard of the Ulam Spiral, where you start drawing a grid of numbers starting with one in the middle like this:

http://aycu20.webshots.com/image/355...0328293_fs.jpg

It gives you diagonal patterns of primes like this:

http://aycu14.webshots.com/image/333...3401392_fs.jpg

There are even other similar types of spirals done which give you surprising patterns. What if this idea could be projected somehow in 3 dimensions? It would give us a reallt cool pattern. I still don't think that would be it though, it is still limited and based on an arithmetic approach. I think that it will take something like complex numbers or something nonlinear and non arithmetic to to determine the true distribution. This just gives us a glimpse. It's fun to think about though.

I hope that I am not too terribly misinformed to make myself look foolish :D

i've seen this before.. have you tried extending this up to at least 1million or larger?? Ü it gives nice diagonal patterns everywhere.. actually, i've been thinking it before that i should have made it as my background on my website.. Ü