The Evidence for Fortune's Conjecture

Abstract

Euclid's proof that there are infinitely many primes is based on the observation that if En = Pn + 1 is not itself prime (where Pn =P1P2 p Pn is the product of the first n primes), it must still contain a prime factor larger than pn. Little is known about the values of n for which En is prime. In fact, the only known prime values of En occur for n = 1, 2, 3, 4, 5, and 11. When asked by a student whether E,, is prime for infinitely many values of n, George P6lya is reported to have replied, There are many questions which fools can ask that wise men cannot answer. The anthropologist Reo Fortune (once married to Margaret Mead) conjectured that if Qn is the smallest prime number strictly greater than En, then the difference Fn = QnP is always prime. (This conjecture first appeared in print [2] in 1980, and is discussed further in [3].) To illustrate, E = (2 X 3 X5X .** X 17) + 1= 51051 1, and the next larger prime is Q7= 510529. Sure enough, the difference is F7 = Q7P7 = 510529 510510 = 19, a prime. The sequence (Fn} of fortunate begins: 3, 5, 7, 13, 23 ,17, 19, 23, 37, 61, 67, 61,71,47,107,59,61,109,89,103,79,..., and indeed, all the listed numbers are prime. Is this merely a remarkable coincidence? At first glance, most mathematicians are tempted to dismiss this conjecture as almost certainly false. However, a closer inspection reveals that it is quite likely to be true. Since Q,1 is known to be prime, Fn = Pn cannot be divisible by p 1, p2,. . .,pn. Thus Fn ,> p + I for all n, with equality observed at n= 1,2,3,6,7,8,14,16,17,.... On the other hand, so long as F,<pn2+ Fn must be prime, since the smallest composite number coprime to Pn is pn2+ 1. It is quite unlikely that F,, is ever as large as Pn2+ 1, as we shall now see. Each of the following facts is a consequence of the Prime Number Theorem: (1) pt, nln n as n-oo. n