20080215, 03:30  #1 
May 2004
New York City
108B_{16} Posts 
Prime Sums
Call an ordered set of odd primes "pgood" if:
it contains p elements, all prime, which sum to a prime, and every subset of q consecutive elements in the set sum to a prime, where q is any odd prime less than p. Thus for example: {3,5,11}, {5,7,11}, {3,7,13} and {5,11,13} are 3good sets, {3,7,13,17,31}, {3,11,17,19,23} and {5,7,11,13,17} are 5good sets, {13,17,23,31,43,53,61}, {5,13,19,29,31,47,53} are 7good sets. A pgood set is "pexcellent" if it minimizes the largest element in the set. A pgood set is "pswell" if it minimizes the set sum. These two measures may conflict, but if a pgood set is both pexcellent and pswell it's "pbest". Even this theoretically may not be punique. The problem is to construct a pbest (or at least pexcellent) set for every prime < 100. 
20080215, 03:52  #2 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{2}×3^{2}×5^{2}×7 Posts 
Repetition allowed? {3,3,5} valid?

20080215, 13:54  #3 
May 2004
New York City
5×7×11^{2} Posts 
It's a set, so the elements are distinct.
However, allowing repetitions may make the problem easier (smaller primes), so that could be considered a different case. Last fiddled with by davar55 on 20080215 at 13:55 
20080215, 17:19  #4 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{2}·3^{2}·5^{2}·7 Posts 
With only a few available primes <100 this can be a brute force problem. If I feel too lazy to think about a smart solution I may write a small proggy to enumerate all values just to exercise the coding muscles a bit.
But if you made it something like all primes <500 then the brute force thing is probably nullified, and then it requires some proper thinking about it. 
20080215, 18:43  #5  
Jun 2003
143B_{16} Posts 
Quote:


20080215, 18:57  #6 
"Brian"
Jul 2007
The Netherlands
2·3·5·109 Posts 
Indeed, it seems to me an enormously difficult problem. Finding any pgood sets at all seems to me beyond simple bruteforce searches for p greater than about 50. I don't know what sort of number theory could be used to refine the search.

20080215, 19:12  #7 
May 2004
New York City
5×7×11^{2} Posts 
Yes, I may have set the sights too high at all primes p < 100,
but I figured that what was beyond my hardware resources and software and theory skills might be just a challenge here. However, I didn't want to scale the problem down and make it too easy, and wasn't sure where to draw the line. Since 100 is arguably arbitrary, perhaps a lower limit would be more appropriate. 
20080215, 19:15  #8  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{2}×3^{2}×5^{2}×7 Posts 
Quote:
Quote:


20080217, 08:33  #9 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{2}×3^{2}×5^{2}×7 Posts 
The best I can find for the original problem without repetition:
3best;{3,5,11} 5best;{5,7,11,13,17} 7best;{5,7,11,19,29,31,37} 11excellent;{5,19,23,37,43,59,61,71,79,89,101};{5,13,29,37,43,59,61,71,79,89,101} 11swell;{3,7,13,17,29,37,43,47,83,97,103};{5,7,11,13,29,41,43,53,67,103,107};{5,7,11,13,29,41,43,47,73,103,107} 13best;{3,5,11,13,23,37,43,47,83,103,107,127,149};{3,5,11,13,23,37,43,47,89,97,107,127,149} 17best;{7,23,37,43,47,83,97,103,131,149,151,167,223,229,239,251,271} 
20080217, 11:35  #10  
"Brian"
Jul 2007
The Netherlands
2×3×5×109 Posts 
Quote:
I'm interested about your reason for finishing with the 17best. Was it (1) that 19good sets are too difficult to find at all? or (2) that they are findable but too difficult to search exhaustively for the 19best? The problem of finding a pgood set for largest possible p is interesting too. Did you find larger pgood sets than 17good? Has anyone else tried? (Otherwise I might try myself, but I'm lazy.) 

20080217, 19:25  #11 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
189C_{16} Posts 
retina was wrong
Okay, everybody just forget what you saw above, it is wrong, some of the internal subtotals are not prime.
Here is new set 3best;{3,5,11} 5best;{5,7,11,13,17} 7best{5,7,11,19,29,31,37} 11swell;{5,7,17,19,23,31,47,53,79,97,191} 11swell;{5,7,17,19,23,31,47,53,79,107,181} 11swell;{5,7,17,19,23,37,41,53,79,97,191} 11swell;{5,7,17,19,23,37,41,53,79,107,181} 11excellent;{7,13,17,29,37,43,47,83,97,113,127} 11excellent;{7,13,17,29,37,43,47,83,103,107,127} 11excellent;{11,13,17,29,37,43,47,83,103,107,127} 13swell;{7,13,17,29,37,43,47,83,97,113,127,181,239} 13excellent;{7,11,13,29,41,43,53,67,103,107,157,197,223} I have checked all sets of primes up to 800 and found no pgood sets for p>13 
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