Thread: Covering sets View Single Post
2016-01-08, 16:38   #3
axn

Jun 2003

2·32·269 Posts

Quote:
 Originally Posted by robert44444uk 1mod2, 1mode3, 3mod5, 4mod7, 5mod11, 9mod13, 1mod17, 1mod19 0,2,4,5,6,10,2,2 1,0,0,6,7,11,3,3 0,2,3,4,5,9,16,3 0,2,4,5,6,10,0,4 0,1,1,0,8,12,4,4 1,0,0,6,7,11,1,5 0,1,1,0,8,12,2,6
These 8 solutions are not unique. They represent one "family" of solution.

These are the 8 solutions paired up
1,1,3,4,5,9 ,1,1 : 1,1,3,4,5,9,16,3 (you had it as 0,2,3,4,5,9,16,3. is that correct?)
0,2,4,5,6,10,2,2 : 0,2,4,5,6,10,0,4
1,0,0,6,7,11,3,3 : 1,0,0,6,7,11,1,5
0,1,1,0,8,12,4,4 : 0,1,1,0,8,12,2,61

All the lines differ from the next one by 1. i.e. They just represent shifting the starting point by 1. And within a pair, the first five modular classes are same - i.e these are primes that hit more than one point in the 30. 2,3, and 5 together eliminate 22 out of every 30 consecutive numbers (always). Of the remaining 8, the best we can do is to hit 2 each with 7,11, and 13. That leaves 2 points. Any prime > 15 can only hit at most 1 point. Therefore the last two requires two primes (any two primes > 15), which yields a family of solutions.