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25478Re: [PrimeNumbers] sum of two p-smooth numbers

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  • marku606
    Feb 15, 2014
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      Good find!  And, your copy and paste looks accurate.  

      From the AMS paper I see that starting at prime = 31 is a first occurrence where the least number solution is not prime.   In that case, the least number solution is  307271 = 109 • 2819, and the least prime solution is 366791. 
      Out of curiosity I decided to continue with my investigation past prime 19, up to prime 31, because I was curious.  Namely, would the least number that can't be written as the sum of two 31-smooth numbers correspond to the least number solution (307271) or the least prime solution (366791) ?
      It turns out, neither!
      The first 5 integers > 1 that can't be written as the sum of two 31-smooth numbers are

      118271, 236542, 307271, 354813, 366791.

      118281 = 101*1171.

      Go figure...

      Mark


      ---In primenumbers@yahoogroups.com, <david.broadhurst@...> wrote:

      From
      D. H. Lehmer, Emma Lehmer and Daniel Shanks
      http://www.ams.org/mcom/1970-24-110/S0025-5718-1970-0271006-X
      R. F. Lukes, C. D. Patterson and H. C. Williams
      http://www.ams.org/mcom/1996-65-213/S0025-5718-96-00678-3
      it seems that the continuation of
      http://oeis.org/A002223/b002223.txt
      is as follows:
      27 33857579279
      28 89206899239
      29 121560956039
      30 328878692999
      31 328878692999
      32 513928659191
      33 4306732833311
      34 4306732833311
      35 4306732833311
      36 8402847753431
      37 70864718555231
      38 317398900373231
      39 501108392233679
      40 501108392233679
      41 501108392233679
      42 501108392233679
      43 5551185799073591
      44 5551185799073591
      45 5551185799073591
      46 7832488789769159
      47 7832488789769159
      48 102097158739597271
      49 102097158739597271
      50 315759454565514431
      51 868116409360316399
      52 3412527725201978759
      53 3546374752298322551
      54 3546374752298322551
      55 3546374752298322551
      56 3546374752298322551
      57 3546374752298322551
      However this needs checking by someone
      better at cutting and pasting than I.
      David (struggling with abominable yahoo interface)
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