Uniformly bounded components of Gaussian-prime graphs
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound D, the graph joining Gaussian primes at distance at most D has uniformly bounded finite component sizes, depending only on D, including primes on the coordinate axes.
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MANUSCRIPT 01
Bounded-Step Walks on Gaussian Primes
We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to every starting prime, including primes on the coordinate axes,…
Full source abstract
We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to every starting prime, including primes on the coordinate axes, and is nonexplicit. The proof constructs a finite periodic sieve obstruction using geometric sampling and information-theoretic estimates.
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