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Re: additive combinations all prime?

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  • Mark Underwood
    ... Took a nice sabbatical from numbers, namely the primes. The things drive me crazy. But in a momement of weakness/boredom, perhaps abit of seasonal
    Message 1 of 10 , Feb 5, 2008
      --- In primenumbers@yahoogroups.com, "Mark Underwood"
      <mark.underwood@...> wrote:
      >
      > Well surprise of surprises:
      >
      > The six numbers (5,30,33,42,60,63) yield primes for all 32 additive
      > combinations of the six numbers.(!) I thought it would have been
      > waaay higher.
      >
      > So we have as first cases:
      >
      > (1,4) has all 2 additive combinations yielding 2 primes.
      > (1,4,8) has all 4 additive combos yielding 4 primes.
      > (3,5,8,13) has all 8 additive combos yielding 8 primes.
      > (3,10,12,15,27) has all 16 additive combos yielding 16 primes.
      > (5,30,33,42,60,63) has all 32 additive combos yielding 32 primes.
      >
      > Interesting that thus far the greatest number in each set does not
      > quite exceed 2^n. Surely that can't hold up for long...or can it?
      >
      > Wonders never cease!
      > Mark
      >


      Took a nice sabbatical from numbers, namely the primes. The things
      drive me crazy. But in a momement of weakness/boredom, perhaps abit of
      seasonal affective disorder, I looked back over some of my
      investigations.


      Very shortly after this post from about a year and a half ago, I found
      a way to greatly speed up the search for additive combinations which
      yeild all primes. For instance: for six numbers, each not exceeding
      500, I found on the order of about 70 different solutions, each
      yeilding (in their 32 additive combinations) all primes. But I noted
      only one which yielded 32 *distinct* primes:


      42,60,75,77,105,108

      in its 32 additive combinations yields the 32 unique primes

      13,17,19,23,41,43,47,53,79,97,
      101,103,107,109,113,131,137,163,167,173,193,197,
      229,233,251,257,263,313,317,347,383,467.


      I tried a similar technique to get 7 numbers with all 64 additive
      combinations yielding only primes, but with no success. I don't think
      there is a solution if all the numbers are below 1000, but it wasn't
      quite an exhaustive search.

      Just throwing this out in case anyone is interested, before I entirely
      forgot about it.

      Mark
    • Mark Underwood
      ... [snip] ... [snip] I meant each number not exceeding 150 (not 500). Mark
      Message 2 of 10 , Feb 5, 2008
        --- In primenumbers@yahoogroups.com, "Mark Underwood" <mark.underwood@...> wrote:

        [snip]
        > For instance: for six numbers, each not exceeding
        > 500, I found on the order of about 70 different solutions, each
        > yeilding (in their 32 additive combinations) all primes.
        [snip]

        I meant each number not exceeding 150 (not 500).

        Mark
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