A Fortunate number, named after Reo Fortune, is the smallest integer m > 1 such that, for a given positive integern, pn# + m is a prime number, where the primorialpn# 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 (2, 3, 5, 7, 11, 13 and 17), 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. If a composite Fortunate number does exist, it must be greater than or equal to pn+12. [citation needed]
The Fortunate numbers for the first primorials are:
Fortune conjectured that no Fortunate number is composite (Fortune's conjecture).[1] A Fortunate prime is a Fortunate number which is also a prime number. As of 2017[update], all known Fortunate numbers are prime, checked up to n=3000.
