On a Degree of Primality

Abstract

In this paper we introduce a degree of primality for natural numbers, and hence, a measure of primality for intervals of consecutive numbers. We characterize maximally prime intervals of length $\le 3$ and their primalities. Maximally prime intervals of length 2 are those that contain Mersenne or Fermat primes; maximally prime intervals of length 3 are, but for a few exceptions, those whose midpoints are Dan numbers. There are relatively few maxprimes for larger lengths. We present a heuristic argument for an asymptotic form describing the distribution of maximal primalities. Finally, we mention open problems and directions for further research.