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, p**n# + m is a prime number, where the primorial p**n# 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 numbers for the first primorials are: :3, 5, 7, 13, 17, 19, 23, 37, 61, 67, 71, 47, 107, 59, 109, etc. .

The Fortunate numbers sorted in numerical order with duplicates removed: :3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, ... .

Fortune conjectured that no Fortunate number is composite (Fortune's conjecture). 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 p**n# is always above p**n and all its divisors are larger than p**n. This is because p**n# + m is divisible by the prime factors of m not larger than p**n. It follows that if a composite Fortunate number does exist, it must be greater than or equal to p**n+12.

Paul Carpenter defines the less-Fortunate numbers as the differences between p**n# and the largest prime less than p**n# -1. These also are conjectured to be always prime.