Problem 952

Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that

\lvert x_{n+1}-x_n\rvert \ll 1?

Source: erdosproblems.com/952

Number theory

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

Disproved. The site labels the problem OPEN (page last edited 08 April 2026; accessed 2026-09-18). The accepted claim is the 2026 OpenAI manuscript, which answers the question negatively for every step bound and every starting point, and whose Lean proof of exactly this statement the corpus's verification built and checked for axioms; the problem's standing is therefore solved with a negative answer, ahead of the site's label. The search whose scope the Current assessment records had found no proof, disproof or proof claim; two arXiv preprints claiming a negative answer, by Das (2019) and Stumpenhusen (2024), had been withdrawn by their authors, as their arXiv records state. Gethner and Stark's refereed theorem [GeSt97] settles the question negatively for step bounds up to $2$ (its claim page (Gethner and Stark, 1997)). Erdős's own expectation (1980) was that "the answer is almost certainly negative", and the random model of Vardi (1998; its library home is under Problem 1212) predicts the same.

Er77c: Erdős, P., Problems and results on combinatorial number theory. III. Number Theory Day (Proc. Conf., Rockefeller Univ., New York, 1976), Lecture Notes in Mathematics 626, Springer (1977), 43--72. Printed p. 69. Library home: Erdos 1977.
Er80: Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115. Printed pp. 114--115; the site's locator is p. 114, where the sentence begins. Library home: Erdos 1980.
JoRa70: Jordan, J. H. and Rabung, J. R., A conjecture of Paul Erdős concerning Gaussian primes. Math. Comp. 24 (1970), 221--223, doi:10.1090/S0025-5718-1970-0256976-8 (Crossref record; the paper is in the journal's open back file that the thread links). A lead by identifier; the thread reports its statement (below).
GeSt97: Gethner, E. and Stark, H. M., Periodic Gaussian moats. Experiment. Math. 6 (1997), no. 4, 289--292, doi:10.1080/10586458.1997.10504616. Recorded on its claim page (Gethner and Stark, 1997).
GWW98: Gethner, E., Wagon, S. and Wick, B., A stroll through the Gaussian primes. Amer. Math. Monthly 105 (1998), no. 4, 327--337, doi:10.1080/00029890.1998.12004889 (Crossref record). Not held; a lead by identifier, known here by its record.
Ts05: Tsuchimura, N., Computational results for Gaussian moat problem. IEICE Trans. Fundamentals E88-A (2005), no. 5, 1267--1273, doi:10.1093/ietfec/e88-a.5.1267 (Crossref record). Not held (the thread links a technical-report copy); a lead by identifier, known here by its record.
Va98: Vardi, I., Prime percolation. Experiment. Math. 7 (1998), no. 3, 275--289, doi:10.1080/10586458.1998.10504373 (Crossref record). Library home, under Problem 1212: Vardi 1998, whose digest records the random model and the conjecture that no bounded-step infinite component of Gaussian primes exists (Conjectures 1.1--1.2 on p. 276 and Theorem 1.1 on p. 277). - Withdrawn preprints (arXiv records accessed; each has a claim page): arXiv:1908.10392 (Das, 2019; version 2 of 6 September 2024 marks it withdrawn: "The claim in Theorem 3 is incorrect. The defined paths P_i (for i=1,2,3,...) do not cover all the Gaussian primes. Additionally, the paths include Gaussian integers, not just primes. Without a proper error term computation, the claim does not hold"), recorded on Das's claim page (2019), and arXiv:2401.08441 (Stumpenhusen, 2024; version 2 of 17 January 2024: "The width of the moat was not correctly computed"), recorded on Stumpenhusen's claim page (2024). - Adjacent, not the problem: arXiv:1901.04549 (2019; an algorithm for computing moats, whose version 3 of 6 September 2024 withdraws its Section 8 claims as based on a probabilistic model), arXiv:2411.06783 (2024; a numerical bound $O(\log^2|p_n|)$ for a "boxcar-metric" gap statistic, not a path result) and arXiv:2011.07386 (2020; random walks on primes in $\mathbb Z[\sqrt2]$ , a different ring).
OAI26: OpenAI, Bounded-Step Walks on Gaussian Primes. OpenAI Math Release preprint, 26 September 2026 (release folder preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026; Theorem 1.1 and its deduction on pp. 1--3, the proof of Theorem 1.2 on pp. 4--28). Library home: OpenAI 2026 (result pages OpenAI 2026, Theorem 1.1, OpenAI 2026, Theorem 1.2 and OpenAI 2026, Proposition 2.1). The accepted claim; see its page.

Problem 952

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