Fortune’s conjecture

(Reo F. Fortune) For any integer n>0, the difference between the primorial


(where π(x) is the prime counting function and pi is the ith prime numberMathworldPlanetmath) 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 primeMathworldPlanetmath.

It is obvious that since n# is divisible by each prime p<pπ(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 numberMathworldPlanetmath m=q-n#, then m would have to have at least two prime factorsMathworldPlanetmath both of which would have to be divisible by primes greater than pπ(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 theoremMathworldPlanetmath, finding a composite Fortunate number is tantamount to searching for a prime gap at least (logn#)2 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
Title Fortune’s conjecture
Canonical name FortunesConjecture
Date of creation 2013-03-22 17:31:17
Last modified on 2013-03-22 17:31:17
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Conjecture
Classification msc 11A41