**Conjecture** For arbitrary integers $\ 0 \le k \le m\ $ there exists
integer $\ n\ge m\ $ such that for every natural number $\ s\ $ at
least one of the numbers $\ p(x)+s\ (\text{where}\ k\le x\le n)\ $ is not prime.

Here, $\ p(0)=2, p(1)=3,\ldots\ $ is the strictly increasing sequence of all prime numbers.