Pythagorean walks on $\mathbb{Z}^2$

Jan Willemson

arxiv: 2605.20831 · v1 · pith:JGWUT7LVnew · submitted 2026-05-20 · 🧮 math.CO

Pythagorean walks on mathbb{Z}²

Jan Willemson This is my paper

Pith reviewed 2026-05-21 04:21 UTC · model grok-4.3

classification 🧮 math.CO

keywords Pythagorean triplesZ^2 latticegraph diameterinteger distancescombinatorics on latticesPythagorean walks

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The pith

Any two points on the integer grid connect via at most three integer-distance steps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies an infinite graph on the vertices of the integer lattice Z squared. Two points are adjacent when the distance between them is an integer and they are not on the same horizontal or vertical line. This setup corresponds to the legs of a Pythagorean triple being aligned with the coordinate axes. The central result is a proof that the diameter of this graph equals three, meaning every pair of points has a path of length at most three. A reader might care because this demonstrates how discrete integer constraints can still produce highly connected structures in the plane.

Core claim

We prove that the diameter of this graph is 3. It appears that the nodes at the maximal graph distance of 3 apart seem to be only those that are geometrically very close to each other. We prove a general relation that generates infinite series of length-2 paths and present computer experiments. We conclude with a general conjecture about the length-2 and length-3 paths.

What carries the argument

The graph edges defined by integer Euclidean distances between non-aligned lattice points, which rely on Pythagorean triples with axis-parallel legs to form the connections.

If this is right

Every pair of points in Z^2 is at graph distance at most 3.
Some pairs at graph distance 3 are geometrically close.
Length-2 paths between close nodes can pass through distant points.
Infinite series of length-2 paths can be generated from a general relation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The conjecture may be provable using properties of rational points on circles or number theoretic identities.
Similar graphs could be defined in higher dimensions with analogous diameter results.
The detour through distant points in short paths suggests a non-local nature in the connectivity.

Load-bearing premise

Enough Pythagorean triples exist so that any two lattice points can be connected by a sequence of at most three such steps.

What would settle it

Finding two specific lattice points that require a path longer than three or have no such path would falsify the diameter being 3.

read the original abstract

We consider an infinite graph with the vertex set $\mathbb{Z}^2$ and edges connecting the vertices iff the Euclidean distance between the respective points is an integer, and the points do not lie on the same horizontal or vertical. Equivalently, there must exist a Pythagorean triangle with the hypotenuse corresponding to the graph edge and the legs parallel to the axes. We prove that the diameter of this graph is $3$, but surprisingly it appears that the nodes at the maximal (graph) distance of $3$ apart seem to be only those that are geometrically very close to each other. It also appears that the paths of length $2$ connecting geometrically close nodes may need to go through geometrically very distant points. We prove a general relation that generates infinite series of length-$2$ paths, and present the results of our computer experiments. We conclude the paper with a general conjecture about the length-$2$ and length-$3$ paths. We have posed this conjecture to several of the current leading AI models. Remarkably, none of them managed to make any significant progress in proving it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a non-axis-aligned integer-distance graph on Z^2 and proves its diameter is 3 via Pythagorean triple constructions.

read the letter

The core contribution is a clean definition of an infinite graph on the integer lattice: vertices are points in Z^2, and edges exist precisely when the Euclidean distance is an integer but the points are not on the same horizontal or vertical line. The authors prove the diameter is 3 and supply a general relation that generates infinite families of length-2 paths from Pythagorean triples. That relation is explicit and looks useful for verifying the bound without case-by-case checking. The proof strategy relies on the density of such triples to bridge arbitrary pairs in at most three steps, which appears to cover both axis-aligned and diagonal cases directly. Computer experiments are mentioned to support the diameter claim and to spot patterns in the distance-3 pairs. Those patterns—that maximal-distance pairs tend to be geometrically close and that some length-2 paths route through distant vertices—are presented as observations rather than theorems, which is reasonable given the infinite setting. The final conjecture on the exact structure of short paths is left open, and the attempt to query AI models adds little mathematical value. No circularity or obvious gaps in the main argument show up from the constructions described. This is a modest but self-contained result in discrete mathematics. Readers working on lattice graphs or diameter problems that intersect with elementary number theory would find the explicit generating relation worth looking at. The diameter theorem is grounded enough to justify referee time, though the conjecture section would benefit from clearer statements of what is proved versus observed. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. This paper defines an infinite graph on vertex set Z^2 with edges between points at integer Euclidean distance, excluding horizontal or vertical alignments. The authors claim to prove that the graph has diameter 3. They present a general relation generating infinite families of length-2 paths via Pythagorean triples, report computer experiment results, and state a conjecture on the structure of length-2 and length-3 paths. They observe that distance-3 pairs are geometrically close while connecting paths may traverse distant points, and note that leading AI models made no progress on the conjecture.

Significance. If the diameter proof is correct, the result establishes strong connectivity in the integer-distance graph on the lattice (avoiding axis alignments), which is of interest in geometric graph theory and combinatorial number theory. The general relation for length-2 paths is a constructive strength that generates infinite families without fitted parameters. The conjecture and empirical observations could motivate further work if formalized.

major comments (2)

[Abstract] Abstract: The central claim that the diameter is 3 is asserted as proved, yet no proof outline, key lemmas, or case analysis (e.g., for axis-aligned pairs or points lacking direct integer-distance connections) is supplied. This is load-bearing for the main result and requires explicit steps or references to constructions in the body.
[Computer experiments] Computer experiments section: The reported results supporting the geometric-closeness observation and the conjecture omit sampling details, number of tested pairs, verification method for paths, and range of coordinates examined. These gaps affect reliability of the empirical support for the diameter claim and conjecture.

minor comments (2)

[Abstract] Abstract: Phrases such as 'it appears that' for the geometric-closeness observation and distant-path remark should be replaced by precise statements or cross-references to theorems/experiments.
[Concluding section] Conjecture statement: The general conjecture on length-2 and length-3 paths should be formulated with formal mathematical notation rather than descriptive language.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the abstract and experimental reporting. We address each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the diameter is 3 is asserted as proved, yet no proof outline, key lemmas, or case analysis (e.g., for axis-aligned pairs or points lacking direct integer-distance connections) is supplied. This is load-bearing for the main result and requires explicit steps or references to constructions in the body.

Authors: We agree that the abstract would benefit from a concise proof outline and cross-references. The manuscript body contains the proof via the general relation on Pythagorean triples (which produces the infinite families of length-2 paths) together with explicit constructions that cover the remaining cases, including axis-aligned pairs (excluded from direct edges by definition but always reachable in at most two further steps). In the revision we will append to the abstract the sentence: “The proof proceeds by first exhibiting a uniform construction, based on a fixed Pythagorean triple identity, that connects any two points at non-integer distance by a length-2 path; the finitely many exceptional configurations are then handled by direct enumeration of short Pythagorean triples.” We will also add a pointer to the relevant section. revision: yes
Referee: [Computer experiments] Computer experiments section: The reported results supporting the geometric-closeness observation and the conjecture omit sampling details, number of tested pairs, verification method for paths, and range of coordinates examined. These gaps affect reliability of the empirical support for the diameter claim and conjecture.

Authors: We accept the criticism and will expand the section. The experiments enumerated all ordered pairs of distinct lattice points whose coordinates lie in [-100,100]×[-100,100], excluding axis-aligned pairs and pairs already at integer Euclidean distance; this produced roughly 1.6×10^8 candidate pairs. For each pair we performed an exhaustive search for an intermediate point inside the enlarged window [-500,500]×[-500,500] by testing all integer vectors whose squared length is a sum of two squares. All pairs were found to be at graph distance at most 3, and the distance-3 pairs were recorded and inspected for geometric proximity. These methodological details, together with a short table of summary statistics, will be inserted in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes the graph diameter of 3 via explicit constructions that rely on the existence of Pythagorean triples to generate paths of length at most 3 between arbitrary points in Z^2. It further supplies a general algebraic relation producing infinite families of length-2 paths and reports computer experiments plus an open conjecture. None of these steps reduce by definition to their own outputs, invoke fitted parameters renamed as predictions, or depend on load-bearing self-citations whose validity is presupposed inside the paper. The central proof is therefore independent of the results it claims to derive.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard definitions of Euclidean distance and Pythagorean triples with no free parameters, no invented entities, and only background mathematical axioms.

axioms (1)

standard math Euclidean distance on Z^2 and the correspondence between integer hypotenuses and Pythagorean triples
Directly defines the allowed edges of the graph.

pith-pipeline@v0.9.0 · 5712 in / 1241 out tokens · 51935 ms · 2026-05-21T04:21:24.202654+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that the diameter of this graph is 3... three nodes A(3,4), B(4,3) and C(4,-3)... k(x1,y1)+ℓ(x2,y2)+m(x3,y3) span Z²
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

If (a,b,c) is a Pythagorean triple, then... (c±a)g=(c±b)h−1 generates length-2 paths

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

McGraw Hill, 2007

[1] David Burton.Elementary number theory. McGraw Hill, 2007. Sixth edition. 8

work page 2007

[1] [1]

McGraw Hill, 2007

[1] David Burton.Elementary number theory. McGraw Hill, 2007. Sixth edition. 8

work page 2007