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prime number question

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  • ansamanta
    does the series 1/p, where p are prime numbers, converge or diverge? please cc the answer to ocsurfer19@yahoo.com keep up the good work, Anand
    Message 1 of 4 , Aug 6, 2003
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      does the series 1/p, where p are prime numbers, converge or diverge?

      please cc the answer to ocsurfer19@...

      keep up the good work,

      Anand
    • umistphd2003@yahoo.co.uk
      Diverges. If you take all the primes p
      Message 2 of 4 , Aug 7, 2003
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        Diverges. If you take all the primes p<=x, and sum 1/p, the value is
        loglog x+C, where C is some number which tends to a fixed value as x
        goes to infinity.

        This formula, and others, are elementary properties of the series of
        primes, and can be found in practically any basic number theory book.

        Andy

        > does the series 1/p, where p are prime numbers, converge or diverge?
        >
        > please cc the answer to ocsurfer19@...
        >
        > keep up the good work,
        >
        > Anand
      • Paul Leyland
        ... It diverges, though very slowly. This fact alone provides a proof that there are an infinite number of primes. Paul
        Message 3 of 4 , Aug 7, 2003
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          > From: ansamanta [mailto:ocsurfer19@...]

          > does the series 1/p, where p are prime numbers, converge or diverge?
          >
          > please cc the answer to ocsurfer19@...
          >
          > keep up the good work,

          It diverges, though very slowly.

          This fact alone provides a proof that there are an infinite number of primes.


          Paul
        • Nathan Russell
          --On Thursday, August 07, 2003 1:21 AM -0700 Paul Leyland ... diverge? ... number of primes. ... And that the proportion of numbers that are prime decreases
          Message 4 of 4 , Aug 7, 2003
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            --On Thursday, August 07, 2003 1:21 AM -0700 Paul Leyland
            <pleyland@...> wrote:

            >
            >> From: ansamanta [mailto:ocsurfer19@...]
            >
            >> does the series 1/p, where p are prime numbers, converge or
            diverge?
            >>
            >> please cc the answer to ocsurfer19@...
            >>
            >> keep up the good work,
            >
            > It diverges, though very slowly.
            >
            > This fact alone provides a proof that there are an infinite
            number of primes.
            >
            >
            > Paul

            And that the proportion of numbers that are prime decreases more
            slowly than n^m, where m < -1.
            I vaguely remember that for m < -1, the sum is approximated by
            (picture an integral sign instead of an "S")

            oo oo
            S n^m dn = [ 1/m * n^(m-1) ]
            1 1

            (note that at m=-1 this is a log instead, and log(oo)= oo , so
            it diverges, and for 0>m>-1 it diverges since each term is
            greater than the corresponding term with m = -1, I'm not sure
            how to justify it in terms of the calculus though).

            Anyhow, we have a converging series for all m < -1, which means
            that the primes decrease in density at a speed only slightly
            less than linear.

            Sorry for rambling, it's been 3 years since I took any formal
            math besides boolean logic and related fields.

            Nathan
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