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RE: [PrimeNumbers] A new proposition

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  • Didier van der Straten
    Allow me to add something here to underscore a close relationship between those searches of polynomials tending to generate high density of primes and
    Message 1 of 5 , Dec 21, 2003
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      Allow me to add something here to underscore a close relationship between
      those searches of polynomials
      tending to generate high density of primes and observations I suggested as
      explanation of the Ulam's spiral
      phenomenon :

      For instance the polynom n^2 + n + k, depending on the value of k (odd), can
      generate infinite series of numbers that are not
      divisible by a range of primes (not necessarily consecutive). The most
      prominent example is n^2 + n + 41
      which is not divisible by any prime from 2 to 37, which is enough to
      increase the prime density for this series
      by a factor of about 6.

      More reading on my pages at

      Appreciate any feedback.

      -----Message d'origine-----
      De : mikeoakes2@... [mailto:mikeoakes2@...]
      Envoye : samedi 20 decembre 2003 19:31
      A : primenumbers@yahoogroups.com
      Objet : Re: [PrimeNumbers] A new proposition

      In a message dated 19/12/03 16:52:47 GMT Standard Time,
      decio@... writes:

      > I'd say this is another example of the law of small numbers. I performed a
      > computation with PARI/GP with l,m,n taking all prime values up to 5000,
      > meaning about 300 million numbers were tested for primality. My script did
      > check for trivial cases (i.e. gcd(l,m,n) != 1) since there are so few of
      > anyway. The result was that only 14.36% of numbers of that form are prime.
      > doesn't seem particularly deserving of notice to me.
      > I'll let Paul Leyland and Mike Oakes chime in with the theory now (:

      (How flattering to be asked:-)

      I think all that's going on is the following:-

      If l = m, G = m*(m+2n) i.e. G is composite; so l, m, n should be distinct

      G mod n = l*m <> 0, so n does not divide G; and similarly l, m.

      If d is any factor of (l+m), then d cannot be l or m,
      so d does not divide l*m, and so does not divide G;
      and similarly for any factor of (m+n) or (l+n).

      So, by construction, G is free of any factors of l, m, n, (l+m), (l+n),

      [This explains why the first factor of Suresh's counterexample is 19.]

      But that is probably /all/ that this construction guarantees: the removal of
      a handful from the (as l, m, n increase in magnitude) very large number of
      potential factors; so one would not expect any spectacular asymptotic
      in the probability of G being prime relative to the probability of an
      arbitrary number of that magnitude being prime.

      The simpler formula H(m) = 2*m+1 gives a 2-fold enhancement in the
      probability of H being prime.
      Suresh's would seem to be of the same type.

      [These comments are only my 2c, rather than "theory"...]

      -Mike Oakes

      [Non-text portions of this message have been removed]

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