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Semi-prime twins

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  • Jens Kruse Andersen
    I have found the largest known semi-prime twins - not that hard considering I may be the first to search for it. If you don t have enough skill, luck or
    Message 1 of 2 , Mar 26, 2003
      I have found the largest known semi-prime "twins" - not that hard considering
      I may be the first to search for it. If you don't have enough skill, luck or
      computing power for coveted records then make up your own records :-)
      I got a little unknowing help from contributors to an old top 5000. Also
      thanks to David Broadhurst who noted there were enough factors for an easy
      proof in a similar problem with triplets.

      p=1171440861*2^80025+1 was found by David Underbakke and Yves Gallot.
      q=2704029*2^98305+1 was found by Phil Carmody.
      Both were in 2000 so the exponents may not seem familiar to them.
      r=(p*q+1)/2 was found by me using PrimeForm/GW, after testing many
      combinations of known primes. I trial factored each combination with my own
      program.
      p, q and r are primes, so p*q and 2*r are semi-prime twins, i.e. two
      consecutive integers which are both the product of two primes.
      The twins have 53699 digits where the twin prime record is 51090 digits, but
      the semi-prime problem is far easier with the right algorithm. I set my target
      to "beat" the twin prime record anyway.

      The proof of r uses the factor 2^80024 of r-1:
      > Running N-1 test using base 13
      > Calling Brillhart-Lehmer-Selfridge with factored part 44.86%
      > ((1171440861*2^80025+1)*(2704029*2^98305+1)+1)/2 is prime!

      p and q are factors of 2r-1, not r-1, and did not have to be primes. Therefore
      Underbakke, Gallot and Carmody are not credited in the top 5000 submission of
      r - not that they would care, considering the number of submissions they have.
      > 1051a ((1171440861*2^80025+1)*(2704029*2^98305+1)+1)/2
      > 53698 p97 2003
      --
      Jens Kruse Andersen
    • David Broadhurst
      Neat work, Jens. The complexity is a bit like that of finding an AP3 by picking over enough pairs of old prime carcasses. But at 53k-digit size it s still
      Message 2 of 2 , Mar 26, 2003
        Neat work, Jens. The complexity is a bit like that of finding
        an AP3 by picking over enough pairs of old prime carcasses.
        But at 53k-digit size it's still impressive!
        David
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