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Updated Oct 8

Problem 1212

claims/: The 1 claim page of Problem 1212, one per claimant's result; the problem's standing derives from them.

Statement. Let $G$ be the graph with vertex set those pairs $(x,y)\in \mathbb{N}^2$ with $\mathrm{gcd}(x,y)=1$ , in which we join two vertices if the differ in only one coordinate, and there by $\pm 1$ .

Is there a path going to infinity on $G$ , say $P$ , such that for all $(x,y)\in P$ both $\min(x,y)>1$ and at least one of $x$ or $y$ is composite?

Formulation. Erdős first asked the question without the composite condition ([Er80], printed p. 114; library card: erdos_1980_survey_problems_combinatorial_number_theory): in a talk at Michigan State University Erdős asked whether there is an infinite path through visible lattice points no coordinate of which is $1$ , and offered a prize for a proof. That question has the answer yes. The same evening Stewart joined $(p_k,p_{k+1})$ to $(p_{k+1},p_{k+2})$ , where $p_k$ is the $k$ th prime, by Chebyshev's theorem; the site's commentary writes the route out, through $(p_k,p_{k+2})$ , and it stays in the graph when no multiple of $p_k$ lies in $[p_{k+1},p_{k+2}]$ , which holds for $k\ge4$ since then $p_{k+2}<2p_k$ . The path passes through the pairs $(p_k,p_{k+1})$ , both of whose coordinates are prime. Erdős then wrote that perhaps the question should have asked for a path going to infinity that avoids the points both of whose coordinates are prime and the points with a coordinate $1$ ; that is the Statement, since for coprime pairs with both coordinates above $1$ , avoiding pairs of primes is the same as having a composite coordinate. The passage also asks for a monotone path, every step increasing the distance from the origin, and for one that changes direction after a bounded number of steps; the site's commentary repeats both questions, which are not part of the Statement.

Status. OPEN, in the site's label (page last edited 08 April 2026), with the explanation that the question cannot be settled by a finite computation. The derived standing departs from the label because of a pending proof. The site's proof-claims tab carries one entry, a full claim by Alex Chengyu Li, published on Zenodo on 2026-09-06 and filed on the tab on 2026-09-07 with a tools line naming Proof Engine, ChatGPT 5.6 and ChatGPT 6 Astra, and with a Lean development this corpus has not built. The claim answers the Statement yes; the site has not accepted it, and it is recorded as pending on Li's claim page. A pending full claim gives the derived standing claimed with claim proved, while the site's label, which records no accepted proof, is OPEN.

Source. erdosproblems.com/1212, accessed 2026-09-04; its discussion thread accessed 2026-10-07. Cite as: T. F. Bloom, Erdős Problem #1212, https://www.erdosproblems.com/1212.

References.

[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[HeSt71] Herzog, Fritz and Stewart, B. M., Patterns of visible and nonvisible lattice points. Amer. Math. Monthly (1971), 487-496.

Formalization. Statement in formal-conjectures. The claimant's Lean development is linked from Li's claim page above; this corpus has not built it.

Current assessment

The site's wording carries one misprint, "the differ" for "they differ", flagged in the thread on 21 July 2026; "there by $\pm1$ " follows Erdős's wording in [Er80] (p. 114) and means by $\pm1$ in that coordinate. The graph is thus the one in which two coprime pairs are adjacent when they differ by $1$ in exactly one coordinate. The site's commentary reports, after [Er80], that Herzog and Stewart studied this graph of visible lattice points and proved that it has a single infinite component, while noting that [HeSt71] does not address the question. Stewart's answer to Erdős's first question, recorded under Formulation, does not bear on the Statement, since its path runs through pairs of primes.

Li's claim is the only proof claim; its page records the route and its standing. The thread also holds observations that are not claims and have no claim page: a comment of 2026-07-21 reports Lean-checked structural facts, that any qualifying infinite path has unbounded distance from the diagonal, that finite qualifying paths exist, that the subgraph has isolated vertices such as $(3,4)$ , and computational evidence of percolation at coordinates around $10^3$ , and calls the problem open; a comment of 2026-08-23 records a determinant observation on points satisfying modular constraints and says that it is not a solution. The 1.0.3 revision of Li's Zenodo record (2026-09-08) credits that determinant observation as sharing the local arithmetic mechanism of its interpolation step, and credits the observations of 2026-07-21, as Li's page records. A Zenodo preprint of Zijian Zeng, Finite-prime obstructions for Erdős Problem 1212 (2026-09-04, doi:10.5281/zenodo.22293614), proves a necessary condition on any qualifying path: for every bound $B$ , the path meets, at bounded gaps of path index, points whose two coordinates both have no prime factor below $B$ , and, with the drift obstruction, does so arbitrarily far from the diagonal; it states that the question remains open, so it is an adjacent obstruction result and not a claim, and has no claim page. The coprime-percolation literature linked below, and the wider literature on known results, are not assessed on this page.

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.

Linked from (26)

Li's infinite path in the composite-restricted visible lattice problems/primes/E1212/claims Primes number_theory/erdos_1980_survey_problems_combinatorial_number_theory Primes primes/florez_2019_distribution_generalized_greatest_common_divisor_visibility Corollary 5 (p. 13): b-visible points surrounded by b-invisible points exist, so G_b is not connected Theorem 6 (p. 4), proved as Theorem 11 (p. 12): a b-pattern is realizable iff its circles contain no complete rectangle modulo (p, p^b) for any prime p Theorem 8 (p. 10): a lattice point of N x N has on average 4/zeta(b+1) b-visible neighbors primes/fourn_2025_percolative_properties_random_coprime_colouring primes/herzog_1971_patterns_visible_nonvisible_lattice_points Corollary 2 (p. 493): patterns with one, two or three circles are realizable, so the plane has arbitrarily lonesome visible points Theorem 1 (p. 490): a planar pattern of visible and nonvisible points is realizable iff its circles contain no complete square modulo any prime primes/martineau_2022_coprime_percolation_visibility_graphon_local_limit Proposition 2.3 (p. 5): coprime colouring along a Følner sequence with coprime density 1/zeta(d)Remark, p. 13 (Section 2.3): percolation of the limit coprime colouring, from Vardi's theorems Theorem 1.1 (p. 4): local limit of the GCD profile of Z^d seen from a uniform point of a dilated convex body Theorem 2.1 (p. 5): local limit of the coprime colouring of Z^d seen from a uniform point of a dilated convex body primes/vardi_1998_prime_percolation Question (p. 277, item (c)): whether the unit-distance graph of coprime pairs has a nonzero limiting density primes/vardi_1999_deterministic_percolation Lemma 7.1 (p. 58): the line at height 1, the prime columns and Heath-Brown--Iwaniec corridors at prime heights lie in the infinite component Proposition 3.1 (p. 50): the coprime lattice points with distance-1 adjacency have a unique infinite component Theorem 3.2 (p. 51): the infinite component of the coprime lattice points has an asymptotic density Theorem 3.3 (p. 51): the asymptotic density of the infinite component of the coprime lattice points is not zero Theorem 3.4 (p. 51): almost every coprime-lattice site is enclosed by a small rectangle whose edges lie in the infinite component

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