(Reo F. Fortune) For any integer $n > 0$ , the difference between the primorial $$n\# = \prod_{i = 1}^{\pi(n)} p_i$$ (where $\pi(x)$ is the prime counting function and $p_i$ is the $i$ th prime number) and the nearest prime number above (excluding the possible primorial prime $n\# + 1$ ) is always a prime number. That is, any Fortunate number is a Fortunate prime.

It is obvious that since $n\#$ is divisible by each prime $p < p_{\pi(n)}$ , then each $n\# + p$ will also be divisible by that same $p$ and thus not prime. If there is a prime $q > n\# + 1$ such that there is a composite number $m = q - n\#$ , then $m$ would have to have at least two prime factors both of which would have to be divisible by primes greater than $p_{\pi(n)}$ .

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 $(\log n\#)^2$ long immediately following a primorial, something he considers unlikely.

Bibliography

1

S. W. Golomb, ``The evidence for Fortune's conjecture,'' Math. Mag. 54 (1981): 209 - 210. MR 82i:10053