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Cubic x^3 + x^2 + x + t prime generators

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  • Kermit Rose
    Date: Tue, 28 Mar 2006 03:59:23 -0000 From: Mark Underwood Subject: Re: prime generating quadratic conjecture Hello Patrick, My
    Message 1 of 8 , Mar 28, 2006
      Date: Tue, 28 Mar 2006 03:59:23 -0000
      From: "Mark Underwood" <mark.underwood@...>
      Subject: Re: prime generating quadratic conjecture


      Hello Patrick,
      My guess is that many polynomials of degree 3 and up will be found
      which yield over 40 positive and distinct prime numbers in a row. But
      if the coefficients are made to equal one, like x^3 + x^2 + x + t, I'm
      guessing that none will beat Euler's x^2 + x + 41.

      Mark



      From Kermit < kermit@... >

      In order that x^3 + x^2 + x + t not have any positive prime factors < 43,

      t must be

      2 mod 3

      and

      3 or 4 mod 5

      and

      2 or 5 mod 7

      and

      2 or 3 or 7 or 9 mod 11

      and

      3 or 5 or 9 or 11 mod 13


      and

      2 or 5 or 7 or 8 or 10 or 13 mod 17


      and

      3 or 4 or 7 or 9 or 12 or 13 mod 19

      and

      3 or 4 or 5 or 10 or 11 or 12 or 16 or 22 mod 23


      and

      2 or 5 or 8 or 9 or 13 or 14 or 17 or 20 or 24 or 27 mod 29

      and

      2 or 10 or 13 or 14 or 16 or 19 or 24 or 27 or 29 or 30 mod 31

      and

      2 or 4 or 7 or 10 or 14 or 18 or 19 or 24 or 25 or 29 or 33 or 36 mod 37

      and

      3 or 4 or 7 or 10 or 12 or 15 or 16 or 24 or 25 or 28 or 30 or 33 or 36 or
      37 mod 41
    • Mark Underwood
      ...
      Message 2 of 8 , Mar 28, 2006
        --- In primenumbers@yahoogroups.com, "Kermit Rose" <kermit@...> wrote:
        >
        >
        > In order that x^3 + x^2 + x + t not have any positive prime factors
        < 43,
        >
        > t must be
        >
        > 2 mod 3
        >
        > and
        >
        > 3 or 4 mod 5
        >
        > and
        >
        > 2 or 5 mod 7
        >
        > and
        >
        > 2 or 3 or 7 or 9 mod 11
        >
        > and

        > (snip)


        Kermit, what is funny is that I overlooked the obvious:
        x^3+x^2+x+k contains a factor of two for every other x! Brought down
        by the lowly factor of two.

        However here are some things to chew on:

        As we know, x^2 + x + 41 contains no prime factors below 41.

        But who knew that -x^10 + x^2 + 43 contains no prime factors below
        43? And in a strange twist, x^10 - x^2 + 43 also contains no prime
        factors below 43, even though their factor set appears to be
        different. x^10 - x^2 + 43 has as its lowest factors 43,73,89,107 and
        113.

        Even more fun, x^8 + x^4 + 59 has no prime factors below 59.

        And this is interesting:

        x^8 - x^6 - x^4 -x^2 + 1 has no prime factors under 43.

        And to top it all off:

        x^8 - x^4 + 1 has no prime factors below 73 (!)

        Even more incredible is that its lowest five prime factors are
        73,97,193,241,337. This is such a glaring scarcity of low prime
        factors. I can see that the primes must be of the form 6n+1, but
        these appear to go beyond that, to the form 24n+1.

        Alot to investigate here, such as the existence of similar forms
        which are even poorer in prime factors.

        Mark
      • Phil Carmody
        ... Now do you see why PIES has such the incredible density of primes that it has? The above is just Phi(24). PIES is looking at Phi(49152) and Phi(98304). The
        Message 3 of 8 , Mar 28, 2006
          --- Mark Underwood <mark.underwood@...> wrote:
          > And to top it all off:
          >
          > x^8 - x^4 + 1 has no prime factors below 73 (!)

          Now do you see why PIES has such the incredible density of primes that it has?
          The above is just Phi(24). PIES is looking at Phi(49152) and Phi(98304). The
          super-fruit subprojects can have even higher densities.

          One puzzle I set the PIES guys was the following:
          <<<
          Mathematicians are invited to calculate the relative density of primes
          of the form Phi(24576,715*b^2) compared to arbitrary numbers of the
          same size.
          >>>
          Just finding out what its smallest possible factor is should be an eye-opener.

          Phil

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        • Mark Underwood
          ... that it has? ... (98304). The ... primes ... eye-opener. ... Phil, Something has gone whizzing over my head, and it had to do with Pies and Phi s and such.
          Message 4 of 8 , Mar 29, 2006
            --- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...>
            wrote:
            >
            > --- Mark Underwood <mark.underwood@...> wrote:
            > > And to top it all off:
            > >
            > > x^8 - x^4 + 1 has no prime factors below 73 (!)
            >
            > Now do you see why PIES has such the incredible density of primes
            that it has?
            > The above is just Phi(24). PIES is looking at Phi(49152) and Phi
            (98304). The
            > super-fruit subprojects can have even higher densities.
            >
            > One puzzle I set the PIES guys was the following:
            > <<<
            > Mathematicians are invited to calculate the relative density of
            primes
            > of the form Phi(24576,715*b^2) compared to arbitrary numbers of the
            > same size.
            > >>>
            > Just finding out what its smallest possible factor is should be an
            eye-opener.
            >

            Phil,
            Something has gone whizzing over my head, and it had to do with Pies
            and Phi's and such. But it looks totally fascninating and I hope to
            learn more along those lines.

            My last result with x^8 - x^4 + 1 producing primes of the form 24n+1
            got me to realize that there might be application to the "Web of
            Ones" thread, and without introducing factors of five into the
            coefficients. And sure enough,

            x^32 - x^24 + x^16 - x^8 + 1

            yields only prime factors ending in one.
            (Except when x=0) Furthermore the prime factors are all of the form
            80n + 1, and there are just 4 different prime factors below 1000:
            241,401,641 and 881.

            Mark
          • Phil Carmody
            ... Phi s are cyclotomic polynomials. They are the primitive parts of the family of polynomials f(x) = x^n-1. Equivalently they are the minimal polynomial
            Message 5 of 8 , Mar 30, 2006
              --- Mark Underwood <mark.underwood@...> wrote:
              > > > x^8 - x^4 + 1 has no prime factors below 73 (!)
              > >
              > > Now do you see why PIES has such the incredible density of primes
              > that it has?
              > > The above is just Phi(24). PIES is looking at Phi(49152) and Phi
              > (98304). The
              > > super-fruit subprojects can have even higher densities.
              > >
              > > One puzzle I set the PIES guys was the following:
              > > <<<
              > > Mathematicians are invited to calculate the relative density of
              > primes
              > > of the form Phi(24576,715*b^2) compared to arbitrary numbers of the
              > > same size.
              > > >>>
              > > Just finding out what its smallest possible factor is should be an
              > eye-opener.
              > >
              >
              > Phil,
              > Something has gone whizzing over my head, and it had to do with Pies
              > and Phi's and such. But it looks totally fascninating and I hope to
              > learn more along those lines.

              Phi's are cyclotomic polynomials. They are the primitive parts of the family of
              polynomials f(x) = x^n-1. Equivalently they are the minimal polynomial which
              vanishes at each of the primitive n-th roots of unity.
              e.g.
              Phi(2) = x+1, as -1 is the only primitive 2nd root of unity,
              Phi(4) = x^2+1, as +/-i are the only two primitive 4th roots of unity.

              > My last result with x^8 - x^4 + 1 producing primes of the form 24n+1

              That 24 comes from the fact that the above is Phi(24)
              (In Pari/GP, us "polcyclo" to get the n-th cyclotomic polynomial.)

              It's instructive to prove that if p|Phi(n,x) then, with a few exceptions,
              p == 1 (mod n).

              > got me to realize that there might be application to the "Web of
              > Ones" thread, and without introducing factors of five into the
              > coefficients. And sure enough,
              >
              > x^32 - x^24 + x^16 - x^8 + 1

              Phi(80,x)

              > yields only prime factors ending in one.
              > (Except when x=0) Furthermore the prime factors are all of the form
              > 80n + 1, and there are just 4 different prime factors below 1000:
              > 241,401,641 and 881.

              Yup, this is why cyclotomics are rich hunting grounds for primes, as they can't
              have small divisors. My GEFs are typically 2-3 times as dense as Yves' GFNs, as
              I reject even more small factors.

              Phil

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            • Robert
              ... I don t know if this helps, but the following series will take some beating, albeit it is not cubic: k^36673176307-k^36673176306 from k =1 to infinity has
              Message 6 of 8 , Mar 30, 2006
                --- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> wrote:
                >
                > --- Mark Underwood <mark.underwood@...> wrote:

                > > > <<<
                > > > Mathematicians are invited to calculate the relative density of
                > > primes
                > > > of the form Phi(24576,715*b^2) compared to arbitrary numbers of the
                > > > same size.
                > > > >>>
                > > > Just finding out what its smallest possible factor is should be an
                > > eye-opener.
                > > >
                > >

                I don't know if this helps, but the following series will take some
                beating, albeit it is not cubic:

                k^36673176307-k^36673176306 from k =1 to infinity has no factors less
                than 22664022957727 !!!!!

                The problem with these polys is that, whilst the factors are of a
                certain form and are a quite often a minuscule subset of the primes,
                their relative frequence of occurrence is quite high therefore the
                density is less attractive than at first glance.

                Maybe it is different for cubics but their derivation, described by
                Phil in his reply to you, are similar for the poly k^n-k^(n-1)

                Regards

                Robert Smith
              • Mark Underwood
                ... the family of ... polynomial which ... Thank you Phil, Jose and Jack for shedding some light on the theory and showing some pretty brilliant methodologies
                Message 7 of 8 , Mar 30, 2006
                  --- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...>
                  >wrote:>
                  > Phi's are cyclotomic polynomials. They are the primitive parts of
                  the family of
                  > polynomials f(x) = x^n-1. Equivalently they are the minimal
                  polynomial which
                  > vanishes at each of the primitive n-th roots of unity.
                  > e.g.
                  > Phi(2) = x+1, as -1 is the only primitive 2nd root of unity,
                  > Phi(4) = x^2+1, as +/-i are the only two primitive 4th roots of
                  >unity.

                  Thank you Phil, Jose and Jack for shedding some light on the theory
                  and showing some pretty brilliant methodologies to boot. And showing
                  the capabilities of GP Pari. I have alot to mull over.

                  Robert, k^n-k^(n-1) has of course factors of k so I assume you did a
                  typo somewhere?

                  Mark
                • Robert
                  ... Oops, how did I get this muddled? Too many beers when I wrote the response most probably. I meant to say: k^p-(k-1)^p, where p is prime Regards Robert
                  Message 8 of 8 , Apr 1, 2006
                    --- In primenumbers@yahoogroups.com, "Mark Underwood"
                    <mark.underwood@...> wrote:
                    >

                    >
                    > Robert, k^n-k^(n-1) has of course factors of k so I assume you did a
                    > typo somewhere?
                    >
                    > Mark
                    >

                    Oops, how did I get this muddled? Too many beers when I wrote the
                    response most probably. I meant to say:

                    k^p-(k-1)^p, where p is prime

                    Regards

                    Robert Smith
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