Finding prime pairs with particular gaps

Abstract:

By a prime gap of size $g$, we mean that there are primes $p$ and $p+g$ such that the $g-1$ numbers between $p$ and $p+g$ are all composite. It is widely believed that infinitely many prime gaps of size $g$ exist for all even integers $g$. However, it had not previously been known whether a prime gap of size $1000$ existed. The objective of this article was to be the first to find a prime gap of size $1000$, by using a systematic method that would also apply to finding prime gaps of any size. By this method, we find prime gaps for all even integers from $746$ to $1000$, and some beyond. What we find are not necessarily the first occurrences of these gaps, but, being examples, they give an upper bound on the first such occurrences. The prime gaps of size $1000$ listed in this article were first announced on the Number Theory Listing to the World Wide Web on Tuesday, April 8, 1997. Since then, others, including Sol Weintraub and A.O.L. Atkin, have found prime gaps of size $1000$ with smaller integers, using more ad hoc methods. At the end of the article, related computations to find prime triples of the form $6m+1$, $12m-1$, $12m+1$ and their application to divisibility of binomial coefficients by a square will also be discussed.