Prime Desert n-Tuplets

Abstract

Although the number of primes is infinite, one can still find many consecutive composite integers as one pleases; 100, 500, 1,000, 1,000,000, or whatever. That is, there are regions of consecutive integers in the set N = {1, 2, 3,... ) of positive integers where no prime is to be found and the number of integers in such a region can be many we please. For example, if we wish to point out a region of 1,000 consecutive composite integers in N we may consider the numbers 1001! + 2, 1001! + 3, ..., 1001! + 1001 and observe that the first integer is divisible by 2, the second is divisible by 3, and so on, and that the last integer is divisible by 1001. In general, to find k consecutive positive integers in N, none of which is a prime, we need only consider the numbers (k + 1)! + 2, (k + 1)! + 3,...,(k + 1)! + (k + 1). It is to be pointed out, of course, that these numbers may not form the first such region in N. For example, the numbers 22! + 2, 22! + 3. ... , 22! + 22 form a region of 21 consecutive integers where no prime is present, but the integers 1130,1131,...,1150 form the first such region. It is also to be pointed out that one or both of the numbers (k + 1)! + 1, (k + 1)! + (k + 2) may not be prime, meaning that the region can be extended in one direction or the other (perhaps both) to include additional consecutive composite integers. This is not the case for the region containing the integers from 1130 to 1150, inclusive, since 1129 and 1151 are both prime numbers. When a region of k consecutive composite integers cl, c2,..., Ck cannot be extended in either direction to include additional composites because the integers c1 and ck + 1 are both primes, we shall call the region a prime desert of length k. Thus the subset {1130 + 1,1131, .. .,1149,1150) of N is a prime desert of length 21 (and it is the first prime desert of length 21). Several questions immediately arise: