# Fortunate number

Let P be the product of the first n primes.  Reo Fortune (once married to anthropologist Margaret Mead) conjectured that if q is the smallest prime greater than P+1, then q-P is prime.  For example, if n is 3, then P is 2.3.5=30, q=37, and q-P is the prime 7.

These numbers q-P are now called Fortunate numbers, and the conjecture has yet to be settled! The sequence of fortunate numbers begins

3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, 101, 103, 233, 223, 127, 223, 191, 163, 229, 643, 239, 157, 167, 439, 239, 199, 191, 199, 383, 233, 751, 313, 773, 607, 313, 383, 293, 443, 331, 283, 277, 271, 401, 307, 331, ....
Paul Carpenter thinks we should similarly define the less-fortunate numbers (or lesser fortunate numbers) by letting q be the greatest prime less than P-1 (the product of the first n primes) and considering the sequence P - q.  This sequence begins
3, 7, 11, 13, 17, 29, 23, 43, 41, 73, ...
He conjectures these numbers are all prime.

Are these conjectures likely to be true? There is good reason to think so.  For suppose the kth Fortunate number is composite, then since it is not divisible by any of the first k primes, we know it is at least the square of the kth prime: pk. By the prime number theorem this is about

(k log k)2.
This is the first prime following P (pk primorial), which is about epk (again by the prime number theorem). So we are looking for a prime gap near P of asymptotically more than
(log P)2.
Such a large gap is thought to be very unlikely!