Recently in this blog: an unsolvable problem? This time the biggest prime number is denoted as q, and p is reserved for the second largest prime number. Here some solutions in the table: for a largest prime number q=7 (p=5) und u=47, u+1 is divisible by 3, u+2 is divisible by 7 etc., or for q=11 (p=7) and u=514 the number u+1 is divisible by 5, u+2 is divisible by 3 etc., and u und v are both not divisible by any of the given primes (u means undivisible or unit). You can recognize a construction pattern, which produces chains of length L=p-1. This cannot be proven, because there are exceptions or anomalies. For q=23 and p=19 there is an anomalous solution of chain length L=19 given in the table. The values of u can be computed with the Chinese remainder theorem.
| q | L | divisibility chain |
u |
|---|---|---|---|
| 7 | 4 | (u,3,7,5,3,v) |
47 |
| 11 | 6 | (u,5,3,11,7,3,5,v) |
514 |
| 13 | 10 | (u,3,7,5,3,11,13,3,5,7,3,v) |
? |
| 17 | 12 | (u,11,3,7,5,3,17,13,3,5,7,3,11,v) |
? |
| 19 | 16 | (u,15,13,11,3,7,5,3,17,19,3,5,7,3,11,13,15,v) |
? |
| 23 | 18 | (u,17,15,13,11,3,7,5,3,23,19,3,5,7,3,11,13,15,17,v) |
? |
| 23 | 19 | (u,3,13,5,3,7,11,3,5,19,3,23,7,15,17,13,3,11,5,21,v) |
? |
to be continued...
