Fortunate number
In number theory, 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 numbers for the first primorials are:
The Fortunate numbers sorted in numerical order with duplicates removed:
Fortune conjectured that no Fortunate number is composite. A Fortunate prime is a Fortunate number which is also a prime number., all known Fortunate numbers are prime, checked up to n=3000.
The Fortunate number for pn# is always above pn and all its divisors are larger than pn. This is because pn# + m is divisible by the prime factors of m not larger than pn. It follows that if a composite Fortunate number does exist, it must be greater than or equal to pn+12.
Paul Carpenter defines the less-Fortunate numbers as the differences between pn# and the largest prime less than pn# -1. These also are conjectured to be always prime.