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Ulam Spiral

  1. Apr 16, 2008 #1
    I've was reading about it [http://mathworld.wolfram.com/PrimeSpiral.htm] and found it intriguing, has there been a great deal of study devoted to it or is it thought of as some kind of quark?

    [P.S. I don't really know much about number theory, just curious]
     
  2. jcsd
  3. Apr 16, 2008 #2

    CRGreathouse

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    I think it's fairly well-understood. The diagonals are related to prime-generating polynomials (which you can also read about on MathWorld).
     
  4. Apr 17, 2008 #3
    but ALL the diagonals of Ulam spiral generate Polynomials that have prime values or only a few special diagonals of Ulam spiral only generate primes.
     
  5. Apr 17, 2008 #4
    Be careful, no polynomial generates only primes indefinitely. I think you meant --only generate primes up to a large value of n --!
     
  6. Apr 17, 2008 #5
    Uh.. sorry , then i meant what are the diagonals that generate primes up to a large value of 'n' or perhaps a bit harder , given a certain prime what is the Polynomial in Ulam Spiral that for a certain integer the Polynomial gives you the prime 'p' are there SERIOUS studies with calculations for Ulam spiral.
     
  7. Apr 17, 2008 #6

    CRGreathouse

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    I'm working through that one, give me a bit.


    Parse tree:

    Code (Text):
    (Uh.. sorry)
    (then i meant
     (
      (
       (
        what are the diagonals
        (that generate primes up to a large value of 'n')
       )
      or perhaps a bit harder ,
       (
        (
         (given a certain prime)
        what is the Polynomial in Ulam Spiral
         (
          (that for a certain integer)
          the Polynomial gives you the prime 'p'
         )
        )
       )
      )
     are there SERIOUS studies with calculations for Ulam spiral.)
    )
    Semantic re-formation:

    Are there serious computational studies addressing:
    1. What diagonals generate only primes up to large values of n?
    2. Harder: given a prime p, what polynomials P have P(n) = p for some n?

    Answer:

    There are studies addressing #1. There is a close relationship between prime-generating polynomials and such serious topics as Heegner numbers (about which I know little). UPNT and MathWorld have lots of references.

    #2 does not seem difficult. Do you have more conditions, or have I perhaps misunderstood you?
     
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