Innumber theory, aFortunate number, named afterReo Fortune, is the smallest integerm > 1 such that, for a given positiveintegern,pn# +m is aprime number, where theprimorialpn# is the product of the firstn 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 numbers for the first primorials are:
Fortune conjectured that no Fortunate number iscomposite (Fortune's conjecture).[1] AFortunate 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.
The Fortunate number forpn# is always abovepn and all its divisors are larger thanpn. This is becausepn# +m is divisible by theprime factors ofm not larger thanpn. It follows that if a composite Fortunate number does exist, it must be greater than or equal topn+12.[2]
Paul Carpenter defines theless-Fortunate numbers as the differences betweenpn# and the largest prime less thanpn# -1. These also are conjectured to be always prime.[2]
