Loading ...
Sorry, an error occurred while loading the content.

24107Re: Two large consecutive smooth numbers

Expand Messages
  • Kermit Rose
    Mar 4, 2012
      On 3/4/2012 7:59 AM, primenumbers@yahoogroups.com wrote:
      > 1c. Re: Two large consecutive smooth numbers
      > Posted by: "Phil Carmody"thefatphil@... thefatphil
      > Date: Sat Mar 3, 2012 3:03 pm ((PST))
      >
      > Calling Doctor Broadhurst for suggestion of the best metric by which to evaluate such records. A simple log doesn't necessarily tell the whole tale at all.
      >
      > Be warned, Warren - Dr. B is sitting on a corpus of algebraic formulae such that p(x) and p(x)+1 have algebraic factorisations, which makes smoothness measurably (I was going to say immeasurably, and then realised the stupidity of such a word choice) more likely.
      >
      > I'll not play this game, as I have an appointment with 21 farmers in Lithuania (otherwise known as the biggest brewery crawl yet...)
      >
      > Phil

      To construct quadratics,

      x^2 - b x + c and x^2 - b x + c + 1 which are both factorisable,

      we look for integers b, k1, k2 such that

      c = k1 * (b-k1)
      and
      c+1 = k2 * (b-k2)

      1*3 = 3; 2 * 2 = 4

      x^2 - 4 x + 3 = (x-1)*(x-3)
      x^2 - 4 x + 4 = (x-2)^2

      2 * 4 = 8; 3 * 3 = 9

      x^2 - 6 x + 8 = (x-2)*(x-4)
      x^2 - 6 x + 9 = (x-3)^2 which is really the same as the first example
      translated by 1.

      I will guess that maybe the only quadratic polynomial solutions are
      translations of the first example.

      Cubic polynomial solutions should be more prolific.

      Kermit
    • Show all 17 messages in this topic