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19820Re: [PrimeNumbers] Inaugural puzzle

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  • Jens Kruse Andersen
    Jan 16, 2009
      Spoiler warning.

      David Broadhurst wrote:
      > p1 is the smallest titanic prime of the form
      > sqrt(3*x^2-2) with integer x.
      > p2 is the smallest titanic prime of the form
      > sqrt(5*y^2-4) with integer y.
      > p3 is the smallest prime such that
      > 2*p1*p2*p3-1 is prime.
      > Find a metrical connection between p3 and liberty.

      I suck at diophantine equations so I didn't even try but just
      bruteforced the first few x values which gave integer square root.
      A quick lookup of the x sequences in OEIS and there they were.
      http://www.research.att.com/~njas/sequences/A001835 has comment:
      Terms are the solutions to: 3x^2-2 is a square.
      http://www.research.att.com/~njas/sequences/A001519 has comment:
      Terms for n>1 are the solutions to : 5x^2-4 is a square.

      A little PARI+PFGW later and I get a 1673-digit p1,
      a 1223-digit p2, and p3 = 12241.
      The metrical connection is left as an exercise for the reader
      (actually, I don't know and didn't look hard).
      I haven't run Primo on p1 and p2. Knowing David there might be
      easy proofs but I'm not looking. OK, I'm lazy.
      Assuming they are prime, PFGW proved 2*p1*p2*p3-1.

      Jens Kruse Andersen
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