Suppose $m=7919$# which is the product of the first $1000$ prime numbers.
It is easy to find primes of the form $m\cdot n+1$ , where $n$ is a positive integer.
But what is the smallest positive integer $n$ such that $m\cdot n+1$ and $m\cdot (n+1)+1$ are both prime ? Sieving out small prime factors still lets many candidates remain , so I assume this positive integer $n$ exists. According to my calculations , we have $n\ge 3\cdot 10^5$ , but a doublecheck is welcome.
With the number theory tool PFGW this can be attacked using this input :
ABC2 7919#*($a+1)+1&7919#*$a+1
a:from 1 to 10000000
Since trial division does not accelerate the search much , has anyone a better idea how to attack such problems avoiding the extreme tough brute force ?
Update : The number $$n=681028$$ does the job , but I am not sure it is the smallest solution.