The Gaussian moat: why bounded steps cannot go forever

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Family 028 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

On the ordinary number line, arbitrarily long gaps between primes prevent an infinite journey through distinct primes with a fixed maximum step length. In the plane, a traveler can go around an obstacle. Does that extra freedom allow an endless walk through the Gaussian primes?

Gaussian integers are complex numbers $a+bi$ with integer coordinates. Their prime elements form a scattered set of points in the plane. The Gaussian moat problem asks whether one can visit infinitely many distinct such primes while keeping every step below some fixed distance.

The manuscript claims that the answer is no, in a stronger uniform form: for each fixed step bound, every connected group of reachable Gaussian primes contains at most a certain finite number of vertices. The bound works everywhere in the plane, including on the coordinate axes. It is not given explicitly.

The challenge is genuinely two-dimensional. Finding enormous empty disks does not suffice: a path might travel around every one of them. The proposed argument instead constructs a finite collection of divisibility restrictions that collectively prevent any infinite bounded-step route.

2. Background and essential definitions

The Gaussian integers form the ring

$$\mathbb Z[i]=\{a+bi:a,b\in\mathbb Z\},\qquad i^2=-1.$$

Their norm is $N(a+bi)=a^2+b^2$, and their Euclidean length is $|a+bi|=\sqrt{N(a+bi)}$. Norms multiply, which makes factorization compatible with planar geometry.

A Gaussian prime is a nonzero nonunit element that cannot be factored into two nonunits. The units are $1,-1,i,-i$; multiplying by a unit rotates a point without changing its prime character. These four rotations are called associates.

For example, $1+i$ is prime and has norm two. The ordinary prime five splits as

$$5=(2+i)(2-i).$$

In general, a rational prime $p\equiv1\pmod4$ splits into conjugate Gaussian prime factors, each of norm $p$. Rational primes congruent to three modulo four remain Gaussian primes and appear on the axes, together with their associates. If both coordinates are nonzero, $a+bi$ is Gaussian prime exactly when $a^2+b^2$ is a rational prime.

Turn walks into a graph

Fix a finite distance $D\ge0$. Make a graph $G_D$ whose vertices are Gaussian primes, joining two distinct vertices if their Euclidean distance is at most $D$. A connected component is a maximal group of vertices linked by finite chains of such edges.

An infinite walk must visit distinct vertices to be meaningful here; otherwise bouncing forever between two nearby primes would qualify. Such a walk is also called a self-avoiding path.

Since a bounded disk contains only finitely many lattice points, each vertex has finitely many possible neighbors. In a graph with this property, an infinite connected component contains an infinite self-avoiding path. However, the absence of infinite components alone need not bound the sizes of all finite components. A disjoint collection of finite chains of lengths one, two, three and so on illustrates the distinction. The manuscript claims the stronger uniform bound.

3. Development history and previous partial results

The historical attribution has more than one strand. Gethner and Stark trace the problem to Basil Gordon at the 1962 International Congress of Mathematicians. Erdős later credited Gordon and Motzkin, recalling a discussion with Motzkin in 1963. Both accounts are recorded in the manuscript’s introduction.

Earlier work established substantial local obstacles. Gethner, Wagon and Wick constructed arbitrarily large Gaussian-prime-free disks, with centers constrained to suitable lines. They also constructed arbitrarily isolated Gaussian primes on the real axis, crediting an independent argument to Vardi. These facts show that prime-free regions can be large, but leave open whether a walk can avoid them.

Computational work explored components reachable from particular starting regions. Tsuchimura bounded the distance reachable with steps at most six from the origin, with the origin adjoined as an initial vertex. This is useful evidence about a specific component, not a theorem simultaneously covering every starting prime and every fixed step bound.

A closer predecessor is periodic sieving. Gethner and Stark established periodic obstructions for step bounds $2$ and $\sqrt2$, and proposed using small Gaussian prime divisors to handle larger bounds. Vardi studied the corresponding periodic coprimality graphs and the relationship between the absence of infinite walks and uniform component bounds. The present manuscript builds on this arithmetic and graph-theoretic viewpoint.

Its proposed new ingredient is a finite sieve obstruction for every fixed $D$. To produce that obstruction, it combines planar geometry with information-theoretic estimates. The use of approximately translation-invariant sampling and bounds on accumulated information is related methodologically to Tao’s entropy-decrement argument, although the residue coverage and walk estimates needed here are specific to this problem.

4. What the new manuscripts claim

The main theorem states that for every finite step bound $D$, there exists a finite number $B_D$ such that

$$|C|\le B_D \quad\text{for every connected component }C\text{ of }G_D.$$

Consequently, every sequence of distinct Gaussian primes with consecutive distances at most $D$ has at most $B_D$ terms. The same bound applies to every starting prime and includes primes on the axes. For $D<1$, distinct lattice points cannot be adjacent, so one can take $B_D=1$.

The main arithmetic statement is a finite sieve theorem. Given $D\ge1$, the manuscript claims that one can choose a finite set $P_D$ of rational primes congruent to one modulo four. For each $p\in P_D$, let $\pi_p,\overline{\pi_p}$ be its conjugate Gaussian factors, and define

$$A(P_D)=\{z\in\mathbb Z[i]: \pi_p\nmid z,\ \overline{\pi_p}\nmid z \text{ for every }p\in P_D\}.$$

The theorem says that this set admits no infinite self-avoiding walk with steps at most $D$. The set of sieving primes depends only on the step bound, not on a hypothetical walk or its starting point.

Every Gaussian prime lies in this avoiding set except for associates of the finitely many chosen factors. Thus the finite sieve obstructs almost all prime vertices at once. The manuscript handles the exceptional vertices when passing back to the full graph.

5. Why the advance matters

Importance: 4/5, provisionally. The claim resolves a longstanding and accessible problem at the intersection of prime distribution, planar geometry and graph connectivity. Its uniform component conclusion is stronger than merely prohibiting one infinite route.

The distinction from one-dimensional prime gaps is instructive. A local obstacle blocks a line, whereas a plane admits detours. The proposed proof must therefore constrain the entire supply of possible routes, not just exhibit empty regions.

The finite-sieve formulation is also stronger structurally than a description based on isolated Gaussian-prime accidents. It says a finite periodic pattern of composite residue classes already forces the obstruction. The information argument supplies a new route from arithmetic avoidance to geometric disconnection, although the absence of explicit bounds limits its immediate computational use.

6. The main idea

Periodicity gives uniformity

Let $P$ be a finite sieving set and write $Q=\prod_{p\in P}p$. The set $A(P)$ repeats under translation by $Q$ in either coordinate. Suppose its bounded-step graph has no infinite self-avoiding path. Every component is then finite.

If one component contained two points differing by a nonzero vector in $Q\mathbb Z^2$, translating the component by that vector would give a component intersecting the original one. They would therefore be the same component. But a nonempty finite set cannot be invariant under a nonzero translation. Thus each component contains at most one point from each residue class modulo $Q$, giving at most $Q^2$ vertices.

Only finitely many exceptional Gaussian primes were removed by the sieve. Each has finitely many neighbors, so adding them back can join only a bounded number of these finite pieces. This produces a bound uniform in location. The hard part is constructing the sieving set in the first place.

Sample a deterministic path

Assume there is an infinite self-avoiding path avoiding the selected zero residue classes. The path itself is deterministic. Randomness is introduced by sampling times along it and by auxiliary choices between the two factors above a split rational prime.

The entropy of a finite random variable $Z$, with probabilities $p_z$, is

$$H(Z)=-\sum_z p_z\log p_z.$$

It measures uncertainty: a uniform choice among $m$ possibilities has entropy $\log m$. Since a bounded step has only finitely many possible lattice increments, a single increment can carry only a bounded amount of information.

The paper compares this finite information supply with the information needed to keep avoiding many arithmetic residue classes.

Force broad residue coverage

Long self-avoiding planar segments have many distinct displacement vectors. The construction samples such differences and their endpoints. Geometric and divisibility estimates show that randomly selected Gaussian factors can distinguish enough of those differences to increase the joint entropy of sampled residue positions.

High joint entropy alone does not ensure enough mass in most individual residue classes. The manuscript therefore proves a separate coverage step. Repeated continuations from a common later checkpoint produce one displacement shared across the residue coordinates. If coverage failed too often, that displacement would locate the true residue on a short list with excessive success. The required information would exceed what the displacement can provide.

The resulting schedules use a common terminal sampling law for all prime batches. An independent time offset makes this law approximately invariant under the shifts needed in the later comparisons. This coordination prevents the argument from counting information measured in incompatible sampling experiments.

Accumulate incompatible information costs

Short words of consecutive increments test which starting residues could avoid zero classes. Residue coverage and self-avoidance force these tests to reveal information. For prime batches near a large scale $T$, the manuscript obtains a cost of at least a constant times $1/\log T$ per increment.

The batches are arranged so their accumulated costs become arbitrarily large. Entropy inequalities, using nested residue data and different word lengths, bound the total by the finite information available in one bounded step. This contradiction supplies the finite obstruction. Auxiliary random factor choices serve the estimates; the final sieve uses both conjugate factors and is independent of the path.

7. Scope, limitations, and open questions

The theorem is qualitative in its dependence on $D$. It does not give a practical value of $B_D$, a smallest possible sieving set or an efficient way to compute the largest component for a chosen step size. Understanding quantitative growth would be a further problem.

A fixed bound is essential. The claim does not forbid infinite sequences of Gaussian primes whose allowed step lengths grow. Repeated vertices are also excluded: finite components can support walks that loop forever without visiting infinitely many distinct primes.

The result applies to Gaussian integers with Euclidean distance. It is not automatically a theorem about prime elements in every number ring or about arbitrary higher-dimensional prime graphs.

The digest reports the manuscript’s assertions without assessing its proof. The key scope distinction is between local empty regions, the absence of infinite components and a uniform bound on all components; the manuscript claims the strongest of these three conclusions.

8. Sources and further reading

Keith Conrad’s The Gaussian Integers supplies undergraduate background on norms and factorization. Gethner and Stark’s Periodic Gaussian moats and Vardi’s Prime percolation are primary sources for the periodic viewpoint.

For the earlier geometric constructions, see Gethner, Wagon and Wick, A stroll through the Gaussian primes. Tsuchimura’s computational results concern a specified starting component. M1 below gives the finite-sieve claim and its information-theoretic construction.

Source manuscripts

• M1. OpenAI, Bounded-Step Walks on Gaussian Primes (September 26, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.