--- 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