A Generalization of Fortune’s Conjecture
A. Dinculescu *
Esperion at Western Michigan University, Kalamazoo, MI 49008, USA and 4148 NW 34th Drive, Gainesville, Florida, 32605, USA.
*Author to whom correspondence should be addressed.
Abstract
Fortune’s Conjecture is extended from a relatively short interval after each primorial to an infinite numbers of similar intervals on both sides of primorials, where n is a positive integer.
In addition, it is shown that for every prime in the interval ,there is a number in the interval that is also a prime or 1, such that.
Since n can take infinitely many values, it is highly probable that the reverse of the above theorem is also true. Accordingly, it is conjecture that for every prime , there exist a prime in the interval that gives a much larger prime when added to or subtracted from the primorial multiplied by an integer n.
Keywords: Primes, distribution of primes, sieves.