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21868Prime Mine Update

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  • Aldrich
    Oct 6, 2010
      The program to find large numbers of primes
      appearing on A = 5x^2 + 5xy + y^2 seems to work
      pretty well. It is based, as you may recall from
      some of my earlier postings on the topic, upon the
      observation that primes and squares of primes ending
      in one or nine seem to appear only once as values of
      A, whereas composites will appear more than once.
      A simple stepwise iteration and storage system finds
      and records eligible integers, and the composites among
      them are eliminated when they appear a second time.

      For a sample of small 13 digit numbers, the program
      found 18086 primes in 5.5 seconds. As the sample
      numbers grew the rate slowed for 18 digit numbers
      to 4,037,5243 primes located in 2500 seconds. The
      results for 13 and 14 digit numbers were all verified
      by trial division. Only a few of the the larger
      numbers were verified this way.

      It has occurred to me that some of you out there
      might be able to use similar methods to find 100 times
      as many primes per second as I have on my old Pentium3
      running Pascal software. I have heard that some low-end
      encryption systems may be vulnerable to this blizzard of
      primes even at these low levels (just above 10^18). Of
      course, encryption need not depend on large primes to
      be effective. If you're so inclined, check out my
      programming toy at the US copyright office (enxxxxx4a.rtf)
      for a really different concept.Still, medium term, I don't
      think our friends at BlowCash or SloShark systems have
      anything to worry about.