Strings of Strongly Composite Integers and Invisible Lattice Points

Abstract

Many of us are familiar with the result that there are arbitrarily large gaps between successive primes. The proof runs as follows: For any positive integer w, consider the sequence of n consecutive numbers (? + 1)! + 2, (n 41)! 43,..., (n 4 1)! 4 (n 4-1). Since j\(n 4 1)! 4y for each j = 2,..., n 4 1, we have constructed a string of n composite numbers. But is this sequence the best one we can construct? In our sequence above, some of the numbers may be just barely composite, i.e., a product of two distinct primes or the square of a prime. Also, the construction does not preclude the possibility of another string of n composites made up of smaller positive integers. Recently I discovered a construction that generates arbitrarily long strings of strongly composite numbers; that is, each number has at least a prescribed number of prime divisors. Furthermore, if the primes are specified, the method ensures that the numbers generated are as small as possible. Let us state the result formally: