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Correction on conjecture.

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  • Kermit Rose
    Kermit Rose wrote: This is correction of my previous post. I mistakenly said an upper bound on the number of primes between (n+1) and (2 n) when I mean to say
    Message 1 of 2 , Sep 5, 2009
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      Kermit Rose wrote:

      This is correction of my previous post.

      I mistakenly said an upper bound on the number of primes between (n+1)
      and (2 n)
      when I mean to say a lower bound on the number of primes between (n+1)
      and (2 n).


      >
      > In order to get an intuitive notion of the number
      > of primes between (n+1) and (2n) for large n,
      > I considered the following.
      >
      > As a result of these observations I made a
      > conjecture about the
      > numerical difference ,
      > number of primes between 1 and n,
      > minus
      > the number of primes between (n+1) and (2n).
      >
      >
      > Between 1 and 5 there are 2 even integers.
      > Between 6 and 10 there are 3 even integers.
      >
      > Between 1 and 6 there are 3 even integers.
      > Between 7 and 12 there are 3 even integers.
      >
      > Between 1 and 7 there are 3 even integers.
      > Between 8 and 14 there are 4 even integers.
      >
      > Between 1 and 15 there are 5 multiplies of 3.
      > Between 16 and 30 there are 5 multiplies of 3.
      >
      > Between 1 and 16 there are 5 multiplies of 3.
      > Between 17 and 32 there are 5 multiplies of 3.
      >
      > Between 1 and 17 there are 5 multiplies of 3.
      > Between 18 and 34 there are 6 multiplies of 3.
      >
      > Between 1 and 18 there are 6 multiplies of 3.
      > Between 19 and 36 there are 6 multiplies of 3.
      >
      >
      > Between 1 and 15 there are 3 multiplies of 5.
      > Between 16 and 30 there are 3 multiplies of 5.
      >
      > Between 1 and 16 there are 3 multiplies of 5.
      > Between 17 and 32 there are 3 multiplies of 5.
      >
      > Between 1 and 17 there are 3 multiplies of 5.
      > Between 18 and 34 there are 3 multiplies of 5.
      >
      > Between 1 and 18 there are 3 multiplies of 5.
      > Between 19 and 36 there are 4 multiplies of 5.
      >
      >
      > Speculation:
      >
      > The number of primes between 1 and n
      > minus the number of primes between (n+1) and (2n)
      >
      > is LESS than or equal to the number of primes p,
      > 1 < p < n such that
      >
      > n mod p > p/2.
      >
      >
      >

      The rational for this conjecture is that

      for p < n, if n mod p is > p/2,

      then there is one more multiple of p between n +1and 2n than between 1
      and n.

      There cannot be more than one extra multiple of p between n+1 and 2 n

      This means that there is at most one extra composite number due to p
      between
      n+1 and n/2.


      This means that there is 1 or zero less primes due to p, between n+1 and
      (2n)

      Thus there are two factors limiting the loss of primes between n+1 and (2n).

      (1) The number of primes such that n mod p < p/2

      (2) Each composite number between n+1 and (2n) is the product of 2 or
      more primes.


      Kermit
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