(where is the prime counting function and is the th prime number) and the nearest prime number above (excluding the possible primorial prime ) is always a prime number. That is, any Fortunate number is a Fortunate prime.
It is obvious that since is divisible by each prime , then each will also be divisible by that same and thus not prime. If there is a prime such that there is a composite number , then would have to have at least two prime factors both of which would have to be divisible by primes greater than .
Despite verification for the first thousand primorials, this conjecture remains unproven as of 2007. Disproof could require finding a composite Fortunate number. Such a number would have to be odd, and indeed not divisible by the first thousand primes. Chris Caldwell, writing for the Prime Pages, argues that by the prime number theorem, finding a composite Fortunate number is tantamount to searching for a prime gap at least long immediately following a primorial, something he considers unlikely.
- 1 S. W. Golomb, “The evidence for Fortune’s conjecture,” Math. Mag. 54 (1981): 209 - 210. MR 82i:10053