A Stroll Through the Gaussian Primes

THE MOAT PROBLEM. One cannot walk to infinity on the real line if one uses steps of bounded length and steps on the prime numbers. This is simply a restatement of the classic result that there are arbitrarily large gaps in the primes. The proof is simple: a gap of size k is given by (k + 1)! + 2, (k + 1)! + 3, ... (k + 1)! + (k + 1). But the same problem in the complex realm is unsolved. More precisely, an analogous question asks whether one can walk to infinity in Z[i], the Gaussian integers, using the Gaussian primes (henceforth, G-primes) as stepping stones, and taking steps of bounded length. The Gaussian question is much more complex because of the additional dimension. For example, there are arbitrarily large disks in Z[i] that contain only Gaussian composites (see [A, p. 119]; this also follows from our Theorem 4.1), but that has little impact on a trek to infinity for a walker who can, with luck, simply walk around the obstacle. This problem is sometimes called the Gaussian moat problem, since one way of establishing a walk's nonexistence is to present a sufficiently wide moat (region of composites) that completely surrounds the origin. The literature has often attributed the Gaussian moat problem to Paul Erdos. But in fact, the question was first posed by Basil Gordon in 1962 at the International Congress of Mathematicians in Stockholm. There have been few published references to the problem since then: it seems to have been mentioned in only three books ([G], [Mo], [WI]) and two papers ([JR], [H]). A paper by J. H. Jordan and J. R. Rabung [JR] contains some computational results, and also the comment that Erdos conjectured that a walk to infinity does exist. Other authors have also attributed the problem to Erdos ([G], [H], [Mo]). But Erdos [El] recently confirmed that the problem was not posed by him and offered the opinion that the sought-after walk does not exist. Jordan and Rabung constructed a v'TU-moat; thus, steps of size 3 will not get a Gaussian prime-walker to infinity. In this paper we present two larger moats (4 and 18), as well as a computational proof that a 26 -moat exists. Thus, steps of length 5 are insufficient to reach infinity. The first author and Harold Stark [GS] have shown that, starting anywhere in the complex plane, and taking steps of length at most two, one cannot walk to infinity. Ilan Vardi [V] has shown that some reasonable probabalistic assumptions about the primes allow one to apply percolation theory to obtain heuristic reasons why walks to infinity using steps of bounded size should not exist. In Section 2 we summarize some definitions and facts about the G-primes. Section 3 contains several new Gaussian moats, and Section 4 contains results that were inspired by William Duke and questions of Gaussian prime geometry.