Let P(n) be the product of four consecutive integers, starting with n, minus one:

 
	P(n) = n* (n+1)* (n+2) *(n+3) - 1. 

Clearly, P(n) can be either prime or not. e.g. P(1) = 23 is prime and P(2) = 119 = 7*17 is not.

Let us call "kc-tuple" a string of k consecutive values : { n, n+1, n+2, ..., n+k-1} such that : P(n), P(n+1), ... , P(n+k-1) are all prime.

There is a significant number of twins (kc = 2) or triplets (kc = 3) but the quadruplets (kc = 4) are rather sparse, and 5 & 6-tuples even sparser.

1) Find  the 1st four examples for kc=2,3,4, ( 5 & 6 optional) .
2) Prove there are no kc-tuples for kc>6.

after a bit of computer time I got
6985660, 22083548, 26203678, 30014408

Still waiting for kc=6, based on the sizes for 5, it could be a while

Posted by Daniel on 2010-12-21 21:15:51