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Statement

Conjecture 1.2 (p. 276, quoted). "For any $k$, there is a bound on the largest component of Gaussian primes connected by step size at most $k$."

The paper motivates the strengthening (p. 276): the random model predicts arbitrarily large finite components even when no infinite one exists, while the Gaussian primes, restricted to congruence classes, seem not to behave so. The conjecture implies Conjecture 1.1. The paper reports (p. 276) that J. H. Jordan and J. R. Rabung (J. Number Theory 8 (1976), 43--51) proved it for $k=\sqrt2$ and E. Gethner and H. Stark (Experiment. Math. 6 (1997), 289--292) for $k=2$, with details in Section 7. Proposition 6.2 reduces the conjecture for a given step to finding one $N$ with no walk to infinity along the Gaussian integers coprime to $N$ (p. 284).

Source. Ilan Vardi, Prime percolation, Experimental Mathematics 7 (1998), no. 3, 275--289, doi:10.1080/10586458.1998.10504373: p. 276, with the reduction on p. 284. The edition read is identified on the source card.

Read depth. Claims checked: the sentence, the motivation and the report of the known cases were read on the printed page; the cited proofs were not read.

Bears on

• #952: the case $k$ implies the negative answer to the problem for step bound $k$, through Conjecture 1.1. The cases $k=\sqrt2$ and $k=2$ are credited to others, the latter recorded on its claim page; the paper itself deduces only the step-$\sqrt2$ walk result (Theorem 7.1), which it says shows the conjecture for step $\sqrt2$ (p. 285). Theorem 1.1 of the 2026 OpenAI manuscript asserts a bound on every component for every real step bound, without naming the conjecture; its standing is recorded on the claim page.