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Is there any way to find the product of prime numbers?

  1. Sep 4, 2011 #1
    According to the prime number theorem, the number of prime numbers that are less than N is approximately N\ln(N) for a sufficiently large N. But can we find the product of prime numbers that are less than N?
    (For example, N=20 then 2x3x5x7x11x13x17x19 although I think 20 isn't large enough haha)
     
  2. jcsd
  3. Sep 4, 2011 #2

    HallsofIvy

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    Of course there is- multiply them!
     
  4. Sep 4, 2011 #3
    If you are looking for a list of semiprimes, it exists. http://oeis.org/A001358/b001358.txt

    If you are looking for a formula pik(n)=(ln(n)/n)*((n^k)/k!). Where k is the number of prime factors not necessarily unique.
     
  5. Sep 5, 2011 #4
    Definition: the producht of the first n primes is called primorial and is writtn as n#

    Theorem: n# < [itex]4^{n}[/itex]

    Definition: [itex]\vartheta(x) := \Sigma_{p_{i}<=x} Log(p_{i})[/itex] (Chebychev)

    Theorem: [itex]\vartheta(x)[/itex] ~ x for x -> [itex]\infty[/itex]
     
  6. Sep 5, 2011 #5
    For the second Theorem above from my 'numerical department':

    i |p[itex]_{i}[/itex] |[itex]\vartheta (p_{i})[/itex] |[itex]\vartheta (p_{i})/p_{i}[/itex]

    5 | 11 | 7.745 | 0.704091
    26 | 101 | 88.344 | 0.874688
    169 | 1009 | 963.162 | 0.954571
    1230 | 10007 | 9905.202 | 0.989827
    9593 | 100003 | 99696.902 | 0.996939
     
  7. Sep 5, 2011 #6
    could you let me know the name of the first theorem so i can google it?
     
  8. Sep 5, 2011 #7
    I think, there is no sprecific name for it; but go to:

    //en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate

    and there look for Lemma 4
     
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