I'm sorry if I ask a stupid question.....
I know that I don't have any capability in this field (numbre theory)
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
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.
Again depends on what you mean2. Is it possible that the distribution is pseudo-random?
What do you think this means?3. Is it possible that in the distribution of prime numbers, it takes a lot of counts before it repeats a same pattern?
See the prime number theorem.4. Does it(the distribution of prime numbers) have a pattern?
RonL
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.
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!
Originally Posted by kalagotaIt 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.Originally Posted by angel.white
It can be easily proven that there is no non-constant polynomial that produces only prime numbers.
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.
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:
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:
It gives you diagonal patterns of primes like this:
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