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Re: [PrimeNumbers] Can someone help explain this pattern

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  • Jens Kruse Andersen
    ... Phil probably thought you were plotting numbers in a spiral but that is not the case. You are plotting them like this (possibly mirrored/rotated and I
    Message 1 of 3 , Sep 22, 2008
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      Phil Carmody wrote:
      > --- On Mon, 9/22/08, Gav C <g4v83@...> wrote:
      >> For background, I wrote a small java program that plots
      >> prime numbers on a grid. I tried a few different grid sizes.
      >> First I tried 100 by 100, I found lots of straight lines -
      >> but realized that these corresponded to the even numbers and
      >> multiples of 5.
      >>
      >> Now I've got a grid that's 546 across, and I find 2
      >> straight lines that are 9 numbers wide, at these widths 241
      >> to 249 and then 295 to 303. These lines contain no primes
      >> at all.
      >
      > What you're seeing is almost certainly related to the lines in Ulam's
      > Spiral.
      > I'm sure there's a Visualisation section in the Prime Links part of
      > http://www.primepages.org/

      Phil probably thought you were plotting numbers in a spiral but that is not
      the case.
      You are plotting them like this (possibly mirrored/rotated and I guess you
      start at 2):
      2 3 4 5 6 7
      8 9 10 11 12 13
      14 15 16 17 18 19
      ....
      This example is 6 across. Yours is 546 = 2*3*7*13. This means your column
      with x at the top contains numbers of the form x + 546*n = x + 2*3*7*13*n.
      If any one of 2, 3, 7, 13 divides x then it will divide all numbers of form
      x + 2*3*7*13*n.

      I guess you start at 2 because that would mean your prime-free columns 241
      to 249 and 295 to 303 correspond to x = 242 to 250 and x = 296 to 304. These
      x values are all divisible by 2, 3, 7 or 13, so there will never be primes
      in those columns.

      x and a are called relatively prime if there is no prime which divides both
      of them.
      x + a*n for fixed x and a with n running through integers is called an
      arithmetic progression with common difference a.
      The x values of your prime-free columns are those x which are *not*
      relatively prime to the common difference a (546 in your case).

      Dirichlet's theorem says that if x and a *are* relatively prime then there
      are infinitely many primes in the arithmetic progression x + a*n.
      So the number of primes in your columns is either 1 (for x = 2, 3, 7, 13) or
      0 (for other x divisible by 2, 3, 7 or 13), or infinite (for x not divisible
      by 2, 3, 7 or 13).

      p#, called p primorial, is the product of all prime numbers <= p. You can
      try making grids which are a primorial across, for example 210 = 2*3*5*7.
      This gives especially many prime-free columns.

      --
      Jens Kruse Andersen
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