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18024Re: additive combinations all prime?

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  • Patrick Capelle
    May 9, 2006
      --- "Patrick Capelle" <patrick.capelle@...> wrote:
      Something interesting with the product of the numbers ?
      (1,4)--> 4 = 2^2
      (1,4,8)--> 32 = 2^5
      (3,5,8,13)--> 1560 = 2^3 * 3 * 5 * 13
      (3,10,12,15,27)--> 145800 = 2^3 * 3^6 * 5^2
      (5,30,33,42,60,63)--> 785862000 = 2^4 * 3^6 * 5^3 * 7^2 * 11

      In each case the product of the n numbers gives a number whose
      number of different prime factors (in the factorization) is smaller
      or equal to n. Is it always the case ?
      ---------------------------------------------------------------------
      --- "Patrick Capelle" <patrick.capelle@...> wrote:
      It holds for n = 6 :
      ( 5,30,33,42,60,63)--> 785862000 = 2^4 * 3^6 * 5^3 * 7^2 * 11
      (30,33,35,60,63,72)--> 9430344000 = 2^6 * 3^7 * 5^3 * 7^2 * 11
      (30,33,47,60,72,75)--> 15075720000 = 2^6 * 3^6 * 5^4 * 11^ * 47
      (30,42,47,60,63,75)--> 16788870000 = 2^4 * 3^6 * 5^4 * 7^2 * 47
      (15,42,48,57,70,75)--> 9049320000 = 2^6 * 3^5 * 5^4 * 7^2 * 19
      (30,33,42,60,75,77)--> 14407470000 = 2^4 * 3^5 * 5^4 * 7^2 * 11^2
      (15,33,42,57,70,90)--> 7465689000 = 2^3 * 3^6 * 5^3 * 7^2 * 11 * 19
      (15,35,42,57,72,90)--> 8144388000 = 2^5 * 3^7 * 5^3 * 7^2 * 19
      ---------------------------------------------------------------------
      --- "Mark Underwood" <mark.underwood@...> wrote:
      I assumed you meant it would hold for first cases. Afterall it
      certainly doesn't hold for some cases after the first case. For
      instance

      14 + 9 = 23
      14 - 9 = 5

      And 14*9 has three different prime factors.

      I'm going to try to look for solutions for n=7 and all 64 additive
      combinations yielding primes. My computer is so slow however it does
      not look promising.

      kind regards,
      Mark
      ---------------------------------------------------------------------

      Important precision.
      Thank you Mark.
      I only started with the examples that you gave, without thinking to
      other cases.
      When we look at each n it is possible that it holds only for the
      smallest set(s).
      Or for all the sets when n is not too small ?

      Best regards,
      Patrick Capelle.
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