# How many ways can a number be Fortunate?

An integer $m$ is Fortunate if it can be written as $q-P$, where $P$ is a primorial and $q$ is the smallest prime greater than $P+1$. It is conjectured that Fortunate numbers are always prime.

It is easy to see that there are only finitely many possible primorials $P$ for which a given $m$ can be decomposed in the above manner (this is because $m$ must be greater than the largest prime dividing $P$).

QUESTION: Is the number of such representations of an integer $m$ uniformly bounded above?

• What largest prime? Gerhard "Ask Me About System Design" Paseman, 2012.09.04 – Gerhard Paseman Sep 5 '12 at 3:19
• I think you probably meant to say "smallest prime greater than $P+1$." For anyone who, like me two minutes ago, doesn't know what a primorial is: that's a portmanteau of "prime" and "factorial", i.e., multiply the first n primes together to get the $n^{th}$ primorial. – Todd Trimble Sep 5 '12 at 3:23
• oh....so some primes are unfortunate, so unfair @('_')@ – Suvrit Sep 5 '12 at 10:26
• Thanks for the correction; I fixed the question so that it doesn't contradict Euclid. – Jon Cohen Sep 6 '12 at 15:36
• The Fortunate numbers are tabulated at oeis.org/A005235. The numbers are actually named after R F Fortune, who is credited with the conjecture that they are all prime. A number of references are given at the oeis page. 23 is the smallest "doubly Fortunate" number; 61, the smallest triply Fortunate. 2 and 11 are the smallest unFortunate primes. – Gerry Myerson Sep 6 '12 at 23:58