Problem 952

../

claims/: The 4 claim pages of Problem 952, one per claimant's result; the problem's standing derives from them.

---

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

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

Formulation. The site's wording (page last edited 08 April 2026). $|x_{n+1}-x_n|\ll1$ means that the steps are bounded by an absolute constant $C$: the question asks for an infinite walk along distinct Gaussian primes with steps of length at most $C$. A bounded region contains finitely many Gaussian primes, so such a sequence is unbounded; this is the question whether one can walk to infinity on the Gaussian primes with bounded steps, the Gaussian moat problem. Erdős's 1977 print asks for "an infinite sequence of distinct Gaussian primes satisfying $|y_{n+1}-y_n|<C$" and his 1980 print for one "for which $|P_{k+1}-P_k|<C$ for some absolute constant $C$". Jordan and Rabung's 1970 title calls it "a conjecture of Paul Erdős", the attribution Erdős corrects in 1977 (below). The formal-conjectures file bounds the norm (the squared modulus) of the steps by an integer $C$, the same condition. The statement prescribes no starting point; the computational "moats" around the origin (below) concern walks that start there.

Status. 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or proof claim; two arXiv preprints claiming a negative answer, by Das (2019) and Stumpenhusen (2024), had been withdrawn by their authors, as their arXiv records state.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). 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.

Source. erdosproblems.com/952, accessed 2026-09-18: the problem page (OPEN, with the site's note that no finite computation can settle it; last edited 08 April 2026; source keys [Er77c] and [Er80, p. 114]), its five-comment discussion thread (5 December 2025) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #952, https://www.erdosproblems.com/952, accessed 2026-09-18.

References.

• [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_problems_results_combinatorial_number_theory_iii.
• [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_survey_problems_combinatorial_number_theory.
• [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.
• [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_prime_percolation, 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, 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.
• 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_bounded_step_walks_gaussian_primes (result pages theorem_1_1, theorem_1_2 and proposition_2_1). The accepted claim; see its page.

Formalization. Proved in Lean by the release and built here. The declaration OAI.GaussianMoat.fullMain of [OAI26]'s Lean development, pinned by the release's comparator challenge ComparatorChallenges/GaussianMoat.lean, states that for every real $D$ no injective sequence of irreducible Gaussian integers has all consecutive Euclidean distances at most $D$, together with a uniform bound on component sizes; the corpus's verification built it and checked that it uses only propext, Classical.choice and Quot.sound. The claim page records the statement audit. Before that, the statement alone was formalized: the file ErdosProblems/952.lean of formal-conjectures, at the commit the link pins (main, 2026-09-18), declares erdos_952 : answer(sorry) ↔ ∃ (x : ℕ → GaussianInt) (C : ℤ), Function.Injective x ∧ ∀ n, Prime (x n) ∧ (x (n + 1) - x n).norm < C under category research open, with proof sorry; the norm of a Gaussian integer is its squared modulus, so the bound is the site's $\ll1$. The community database (teorth/erdosproblems, 2026-09-18) records the problem open (31 August 2025), the statement formalized since 21 November 2025, the comment "Gaussian moat problem" and no formal proof. The statement file was not built.

Current assessment

The question (site formulation of 2026-09-18). The statement above; OPEN; last edited 08 April 2026. The site's commentary identifies the problem as the Gaussian moat problem, says that it is not Erdős's own and was misattributed to him, quotes his 1977 recollection of hearing it from Motzkin (the passage below), and notes his 1980 expectation of a negative answer. The thread (5 December 2025) has five comments: one on [Ts05], which cites a general encyclopedia's summary of the computational moat of width $6$ separating the origin from infinity and remarks that the computations fit a random model of the Gaussian primes under which no bounded infinite sequence exists; one stating Jordan and Rabung's form of the conjecture, an $M$ and a sequence of Gaussian primes $(\gamma_j)$ with $|\gamma_1|<M$, $|\gamma_j-\gamma_{j-1}|<M$ and $|\gamma_j|\to\infty$, with their report that a true conjecture needs $M\ge4$; and replies settling that [JoRa70] did not prove the conjecture but reported a computer search that made it look plausible to its authors, that a sentence in a paper of Tuza describing an analogous version as solved by Jordan and Rabung refers to the moat problem itself, and that a misprint in the comment had been corrected. The proof-claim tab was empty on 2026-09-18. The community database record says open.

Resolution (accepted claim). [OAI26] proves, as its Theorem 1.1, that for every finite real $D$ some finite $B_D$ bounds the number of vertices of every component of the graph on all Gaussian primes with edges at distance at most $D$, so that no sequence of distinct Gaussian primes with steps at most $D$ has more than $B_D$ terms, for any constant and any starting point. The bound is not explicit. The engine is Theorem 1.2, a finite sieve obstruction: for every $D\ge1$ a finite set of rational primes $p\equiv1 \pmod 4$, depending only on $D$, is such that the Gaussian integers divisible by neither Gaussian factor of any of them contain no infinite sequence of distinct points with steps at most $D$; its proof is an entropy argument over sampled times of a fixed walk. Proposition 2.1 turns the periodic obstruction into the component bound, since all but finitely many Gaussian primes avoid the sieved residues. This is the content of Vardi's Conjectures 1.1--1.2 in [Va98]; the manuscript cites [Va98] for the periodicity reduction (Section 6, Proposition 6.2) but does not name those conjectures. The claim is accepted on formalized evidence: the corpus's verification built the release's Lean declaration OAI.GaussianMoat.fullMain and checked its axioms, and its first conjunct, pinned by the release's comparator challenge, is exactly the question above with every real step bound, every starting point, and associates and axis primes included. No refereed publication or review independent of the release is recorded, and the site's label is unchanged. The statement audit and the links are on the claim page; the manuscript's statements are on the library card cited in the References, with no step of their proofs checked here.

Origin and attribution. [Er77c], printed p. 69: Erdős apologizes for occasionally inaccurate references, recalls that the problem had once been attributed to him, states it as "Does there exist an infinite sequence of distinct Gaussian primes satisfying $|y_{n+1}-y_n|<C$?", and continues: "The conjecture was told me by Motzkin at the Pasadena number theory meeting 1963 November and it was apparently raised by Basil Gordon and Motzkin." He adds that E. Straus cleared the matter up, that he had told the problem to many people while attributing it to Motzkin, which was later forgotten, and ends: "Thus the problem is returned to its rightful owners." [Er80], printed pp. 114--115: "The following beautiful conjecture is due to Gordon and Motzkin: Is there a sequence of distinct Gaussian primes $P_1,P_2,\ldots$, for which $|P_{k+1}-P_k|<C$ for some absolute constant $C$. The answer is almost certainly negative." Neither passage records a result.

What is known (leads by identifier, known here by their records). [JoRa70] (1970): the conjecture in the form above, with the report that a true conjecture needs $M\ge4$ (as the thread quotes it) and a computer search. [GWW98] (1998): moat computations; the arXiv abstract of 1901.04549 credits "Gethner et al." with a moat of value $\sqrt{26}$. [Ts05] (2005): the width-6 moat separating the origin from infinity, per the thread. [Va98] (1998): a Cramér-type random model of the Gaussian primes in which walks of step size $k\sqrt{\log|z|}$ percolate exactly for $k$ above a critical constant, and the conjecture (its Conjectures 1.1--1.2, p. 276) that no bounded-step infinite component of Gaussian primes exists; the digest also records the paper's Section 7 mention of Gethner--Stark's and Jordan--Rabung's bounded-component results. The moat computations show that walks from the origin, for the step lengths computed, stay in a bounded region. Gethner and Stark's periodic moats [GeSt97] prove, from any starting point, that no infinite sequence of distinct Gaussian primes has all steps of length at most $2$; this includes step $\sqrt2$, which Vardi credits to Jordan and Rabung. The question is thus settled negatively for every step bound up to $2$, not for larger bounds; and the heuristic (the density $2/(\pi\log|z|)$ of Gaussian primes near $z$ falls below every percolation threshold) is Vardi's and the thread's reason for expecting a negative answer, not a proof. The two withdrawn preprints claimed the negative answer and retracted it, as their records quoted above say; their claim pages are Das's and Stumpenhusen's. The Semantic Scholar citation lists of [JoRa70] (21 records) and [GWW98] (41 records) contain, by title, computational and expository items (a 2007 Monthly article on stepping to infinity along Gaussian primes; 2014--2016 papers in Integers on divisibility and unavoidable obstructions in Gaussian walks; 2020--2022 walks on Gaussian lines and on primes in $\mathbb Z[\sqrt2]$) and no claimed resolution. All of this predates [OAI26], which proves the conjecture these sources state or model.

Search scope. None of the routes below found a proof, disproof or proof claim; [OAI26] postdates the search.

• The site: problem page, discussion thread and proof-claim tab; formal-conjectures at the pinned commit; the community database on 2026-09-18.
• The primary sources: [Er77c] p. 69 and [Er80] pp. 114--115.
• arXiv API: all:"Gaussian moat" sorted by date (6 records, 2014--2024, by abstract and comments: the two withdrawn claims, the algorithm preprint, the $\mathbb Z[\sqrt2]$ model and a 2014 note on walks on primes in imaginary quadratic fields); abs:"Erdős problem" AND (abs:951 OR abs:952 OR abs:972 OR abs:981) (no records).
• Crossref: the records of [JoRa70], [GWW98], [Ts05] and [Va98] by bibliographic query.
• Semantic Scholar: the citing records of [JoRa70] and [GWW98], by title as stated.
• The 1976 Erdős--Hall paper on subset sums in abelian groups (eight pages; it does not mention Gaussian primes).

Not searched: MathSciNet, zbMATH, Google Scholar, X. [JoRa70], [GWW98] and [Ts05] are known here by their records, and [Va98] by pp. 276--277.

Remaining gaps. (1) The question is settled negatively by the accepted claim; the site's label is OPEN, and no refereed publication or review independent of the release is recorded. The bound $B_D$ of [OAI26] is not explicit, so the size of the largest component for a given step length remains undetermined. (2) [JoRa70], [GWW98] and [Ts05] are known here by identifier only; [JoRa70] is in the journal's open back file. Jordan and Rabung's step-$\sqrt2$ result, which Vardi credits and the thread describes as a computer search rather than a proof, has no claim page because its statement is not recorded here; [GeSt97] covers that step bound. (3) Compiled coverage: the manuscript's Theorems 1.1 and 1.2 and Proposition 2.1 are carded at statement level, with no step of the proofs checked by hand; the acceptance evidence is the Lean proof, which the corpus's verification built and axiom-checked, and whose pinned statement the claim page audits against the question. (4) The attribution is settled by Erdős's own 1977 correction; the 1970 title's "conjecture of Paul Erdős" is the misattribution he describes.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1977_problems_results_combinatorial_number_theory_iii
• erdos_1980_survey_problems_combinatorial_number_theory
• openai_2026_bounded_step_walks_gaussian_primes
• openai_2026_bounded_step_walks_gaussian_primes / proposition_2_1
• openai_2026_bounded_step_walks_gaussian_primes / theorem_1_1
• openai_2026_bounded_step_walks_gaussian_primes / theorem_1_2
• vardi_1998_prime_percolation
• vardi_1998_prime_percolation / conjecture_1_1
• vardi_1998_prime_percolation / conjecture_1_2
• vardi_1998_prime_percolation / conjecture_1_3
• vardi_1998_prime_percolation / proposition_6_1
• vardi_1998_prime_percolation / proposition_6_2
• vardi_1998_prime_percolation / theorem_1_1
• vardi_1998_prime_percolation / theorem_7_1