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k^2+(k+1)^2 and k^4+(k+1)^4

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  • Robert
    The series k^2+(k+1)^2 and k^4+(k+1)^4, k from 1 to infinity produce may primes (prps), as factors of these series (and all series of form
    Message 1 of 3 , Apr 22, 2009
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      The series k^2+(k+1)^2 and k^4+(k+1)^4, k from 1 to infinity produce may primes (prps), as factors of these series (and all series of form k^((2^n))+(k+1)^((2^n))appear to be similar to those factors in cyclotomic polynomials.

      n=1 and 2 appear very rich, as the formula produces quite small primes and prps, but counting primes (prps) for the series k from 1 to k(x) it appear that n=2 has more than n=1, for low x.

      What is quite fascinating is if we look at the mod 2 values of k that produce prps for n=1,2 and race k=0mod2 against k=1mod2. I looked up to k=1200000 and over 100000 of these are prp, for both n=1 and 2, and the even values of k were in the lead for a goodly proportion of the time, in fact for 99%+. At k=1200000, n=2 there were approx 600 more evens than odds.

      It would be interesting to know if this is a permanent phenomenon or one that will eventually reverse. My guess that it will reverse and that for some value of k>1200000, there are more odds than evens.

      Sorry if this area is well researched, but I always find google is not so good at looking for mathematical polynomial equations, unless they have a name.

      Regards

      Robert Smith
    • David Broadhurst
      ... Since 600/sqrt(100000)
      Message 2 of 3 , Apr 23, 2009
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        --- In primenumbers@yahoogroups.com,
        "Robert" <robert_smith44@...> wrote:

        > over 100000 of these are prp
        ...
        > there were approx 600 more evens than odds

        Since

        600/sqrt(100000) < 2

        this seems to tell us little about what might happen later.

        David
      • Robert
        ... After a good deal of lead swapping at small k, even numbers predominate from k=44886 until k=4245827 Regards Robert Smith
        Message 3 of 3 , Apr 30, 2009
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          --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@...> wrote:
          >
          > The series k^2+(k+1)^2 and k^4+(k+1)^4, k from 1 to infinity produce may primes (prps), as factors of these series (and all series of form k^((2^n))+(k+1)^((2^n))appear to be similar to those factors in cyclotomic polynomials.
          >
          > n=1 and 2 appear very rich, as the formula produces quite small primes and prps, but counting primes (prps) for the series k from 1 to k(x) it appear that n=2 has more than n=1, for low x.
          >
          > What is quite fascinating is if we look at the mod 2 values of k that produce prps for n=1,2 and race k=0mod2 against k=1mod2. I looked up to k=1200000 and over 100000 of these are prp, for both n=1 and 2, and the even values of k were in the lead for a goodly proportion of the time, in fact for 99%+. At k=1200000, n=2 there were approx 600 more evens than odds.
          >
          >
          After a good deal of lead swapping at small k, even numbers predominate from k=44886 until k=4245827

          Regards

          Robert Smith
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