Fortunate number


A Fortunate number, named after Reo Fortune, is the smallest integer m > 1 such that, for a given positive integer n, pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers.
For example, to find the seventh Fortunate number, one would first calculate the product of the first seven primes, which is 510510. Adding 2 to that gives another even number, while adding 3 would give another multiple of 3. One would similarly rule out the integers up to 18. Adding 19, however, gives 510529, which is prime. Hence 19 is a Fortunate number. The Fortunate number for pn# is always above pn and all its divisors are larger than pn. This is because pn#, and thus pn# + m, is divisible by the prime factors of m not larger than pn.
The Fortunate numbers for the first primorials are:
The Fortunate numbers sorted in numerical order with duplicates removed:
Reo Fortune conjectured that no Fortunate number is composite. A Fortunate prime is a Fortunate number which is also a prime number., all the known Fortunate numbers are prime.