What question did this study set out to answer?

This research aims to explore a structured method of walking through prime numbers using Pythagorean triples.

June 9, 2026Open Access

Walking on the Pythagorean Tree

Key Points

This research aims to explore a structured method of walking through prime numbers using Pythagorean triples.
Constructs a walk starting from the prime 3 through primitive Pythagorean triples.
Advances along the hypotenuse when it's prime or detours through the longer leg if composite.
Analyzes the cumulative turning angle along the chain from the initial prime.
Generates a sequence of primes including 3, 5, 13, 37, and others.
Establishes a directed forest structure with chains coalescing after sharing steps.
Empirically identifies a constant turning angle around Δ ≈ 1.51181… with open questions regarding its mathematical nature.

Abstract

This is a preprint of the manuscript: "Walking on the Pythagorean Tree" Abstract:Starting from the prime 3, we construct a deterministic walk through the primes by following primitive Pythagorean triples. At each step we either advance along the hypotenuse when it is prime, or detour through the longer leg of the most recent successful triple when the forward hypotenuse is composite. The procedure generates a sequence of primes (3, 5, 13, 37, 613, 93637, …). The walk organises the forward arrows p → (p² + 1)/2 into a directed forest on the primes. Chains starting from different primes coalesce after sharing a common forward step. Additionally, the cumulative turning angle along the chain from 3 empirically stabilises near a constant Δ ≈ 1.51181…, whose mathematical nature (existence as a limit, closed form) remains open. The manuscript has been submitted to a peer-reviewed journal and is currently under review. Keywords: Pythagorean triples, primitive Pythagorean triples, prime walks, forward forest, coalescence, cumulative turning angle, Gaussian integers

Walking on the Pythagorean Tree

Key Points

Abstract

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