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## Re: additive combinations all prime?

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• ... 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

Mark
• ... [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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