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Statement

Conjecture 1.1 (p. 276, quoted). "For any $k$, there is no infinite component of Gaussian primes connected by step size at most $k$."

Two Gaussian primes are joined when their distance is at most $k$. The paper's ground for the conjecture (p. 276) is a heuristic: the density of Gaussian primes in a disk of radius $x$ is about $2/(\pi\log x)$, so the "probability" that a lattice point is prime becomes smaller than any fixed $\rho$. The paper states the conjecture for every $k$ and proves it in general for none; it deduces the case $k=\sqrt2$ as Theorem 7.1 (p. 285); Conjecture 1.2 implies it, and p. 276 reports that Conjecture 1.2 was proved for $k=\sqrt2$ by Jordan and Rabung (1976) and for $k=2$ by Gethner and Stark (1997).

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

Read depth. Claims checked: the sentence and the heuristic before it were read on the printed page.

Bears on

• #952: for each $k$ the conjecture is equivalent to the negative answer to the problem with step bound $k$ (an observation of this page): an infinite sequence of distinct Gaussian primes with steps at most $k$ lies in one component of the distance-$k$ graph, which is then infinite; conversely that graph is locally finite, so an infinite component contains such a sequence. The paper offers the conjecture on heuristic grounds and deduces only the step-$\sqrt2$ case (Theorem 7.1). Theorem 1.1 of the 2026 OpenAI manuscript asserts this content for every real step bound, without naming the conjecture; its standing is recorded on the claim page.