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Re: RE 40 prime polynomials

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  • Mark Underwood
    ... A modification is in order. Any prime sequence which is found in ax^2 + sx + t where s =2a can be found in the simpler ax^2 + bx + c where b
    Message 1 of 12 , Apr 5 8:13 AM
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      --- In primenumbers@yahoogroups.com, "Mark Underwood"
      <mark.underwood@...> wrote:
      >
      >
      > Put differently, any sequence which is found in
      > ax^2 + sx + t
      > where s >= 2a,
      > can be found in the simpler
      > ax^2 + bx + c
      > where b < 2a.
      >

      A modification is in order. Any prime sequence which is found in
      ax^2 + sx + t where s>=2a
      can be found in the simpler
      ax^2 + bx + c
      where b <= a and both a and b are positive.


      For instance, consider 5 polys which generate 40 consecutive primes
      from x=0 to x=39.

      x^2 + x + 41
      4x^2 - 154x + 1523
      4x^2 - 158x + 1601
      9x^2 - 231x + 1523
      9x^2 - 471x + 1603


      The second and third polys generate the same sequence (in reverse),
      and so does the fourth and fifth polys.

      So really there are three:
      x^2 + x + 41
      4x^2 - 154x + 1523
      9x^2 - 231 + 1523.

      But these three can be simplified to
      x^2 + x + 41 (prime from x=0 to x=39)
      4x^2 + 2x + 41 (prime from x=-20 to x=19)
      9x^2 + 3x + 41 (prime from x=-20 to x=19)


      Here's another 40 I stumbled upon, but which dips into the negatives
      in the middle of the sequence:

      8x^2 + 6x - 661 (prime from x=-19 to x=20)

      Anyways, the point I wanted to reiterate is that if one is searching
      for prime polys of the form
      ax^2 + bx + c
      such that x roams in a certain range, one need only consider positive
      b's from 0 to a. (Otherwise there is redundancy.)


      Mark
    • Mark Underwood
      ... Another example: Consider the (already known) prime poly 36x^2 - 810x + 2753 which produces 45 consecutive primes (including negatives) from x=0 to x=44.
      Message 2 of 12 , Apr 5 10:50 AM
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        --- In primenumbers@yahoogroups.com, "Mark Underwood"
        <mark.underwood@...> wrote:
        >
        >
        > Anyways, the point I wanted to reiterate is that if one is searching
        > for prime polys of the form
        > ax^2 + bx + c
        > such that x roams in a certain range, one need only consider positive
        > b's from 0 to a. (Otherwise there is redundancy.)
        >

        Another example: Consider the (already known) prime poly
        36x^2 - 810x + 2753
        which produces 45 consecutive primes (including negatives) from x=0 to
        x=44.

        This poly can be reduced to
        36x^2 + 18x - 1801
        which is prime from x=-33 to x=11.
        Notice how b is positive and no more than a. Also notice how b contains
        all the prime factors of a. It must, or else the poly is forced to
        contain the prime factors of a which b doesn't have.

        Mark
      • Mark Underwood
        ... wrote: ... Correction: 9x^2 + 3x + 41 is prime from x=-13 to x=26 Mark (no alcohol required to goof up) Underwood
        Message 3 of 12 , Apr 5 12:29 PM
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          --- In primenumbers@yahoogroups.com, "Mark Underwood"
          <mark.underwood@...> wrote:>
          > But these three can be simplified to
          > x^2 + x + 41 (prime from x=0 to x=39)
          > 4x^2 + 2x + 41 (prime from x=-20 to x=19)
          > 9x^2 + 3x + 41 (prime from x=-20 to x=19)
          >

          Correction: 9x^2 + 3x + 41 is prime from x=-13 to x=26

          Mark (no alcohol required to goof up) Underwood
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