Fortune's conjecture is a conjecture about prime numbers made by the New Zealander social anthropologist Reo F. Fortune (1903–1979), who had a reputation for unstable behavior bordering on the psychotic. According to one report, Fortune once tried to settle an academic dispute with a University of Toronto colleague, Thomas McIlwraith, by challenging him to a duel with any weapon of his choice from the collections of the Royal Ontario Museum. (Alternatively, it may be that Fortune had actually challenged McIlwraith to a contest of identifying artifacts in the ROM, both men having had museum backgrounds.)

 

Fortune proposed that if q is the smallest prime greater than P + 1, where P is the product of the first n primes, 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 known as Fortunate numbers. The conjecture remains unproven but is generally thought to be true.

 

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, ...