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Math Help - prime numbers

  1. #1
    Junior Member Singular's Avatar
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    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)
    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
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Singular View Post
    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
    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..
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Singular View Post
    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?
    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.

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

    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?

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

    RonL
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  4. #4
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    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.
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  5. #5
    MHF Contributor kalagota's Avatar
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    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..
    Anyways, has anyone read "The Music of the Prime"? it is amazing!
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  6. #6
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    Quote Originally Posted by kalagota View Post
    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..
    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.
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  7. #7
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    You want a formula? Okay. How about n^2+n+41 for n\geq 0.
    is it proven? hehe
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  8. #8
    Senior Member DivideBy0's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    You want a formula? Okay. How about n^2+n+41 for n\geq 0.
    Wow nice find, gives primes up to n = 39
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  9. #9
    Super Member angel.white's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    You want a formula? Okay. How about n^2+n+41 for n\geq 0.
    Unless I misunderstand, 41 should divide this equation when n=41
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  10. #10
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by angel.white View Post
    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..
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  11. #11
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    Quote Originally Posted by DivideBy0 View Post
    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.
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  12. #12
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    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.
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  13. #13
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    Quote Originally Posted by letx=x View Post
    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.
    Yes I read about this in my number theory textbook many years ago.
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  14. #14
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    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
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  15. #15
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Skinner View Post
    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

    i've seen this before.. have you tried extending this up to at least 1million or larger?? 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..
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