Lonely runner conjecture

Summary

AI-assisted claims

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Why it is open

Jörg M. Wills raised the question in the 1960s in Diophantine approximation, the study of how well real numbers can be approximated by fractions; Guillem Perarnau and Oriol Serra trace it to his 1965 doctoral thesis and to papers of 1967 and 1968 [1, 2]. T. W. Cusick asked the same question independently in 1973 as a "view-obstruction" problem: how large must small cubes centred at the points of space with positive half-integer coordinates be for every ray from the origin into the positive region to hit one [3, 2]?

Only relative positions matter, so the runner who should become lonely can be treated as standing still at the start. The conjecture then says that for any k non-zero speeds there is a time at which all k moving runners are at least 1/(k + 1) from the start, and the speeds may be taken to be positive whole numbers [2]. Most papers number the cases by k, so their "case k" concerns k + 1 runners; Mónica Blanco, Francisco Criado and Francisco Santos note that papers on the cases k = 6 to 12 speak of seven to thirteen runners in their titles [4]. This page always counts all runners, including the stationary one.

The value 1/n cannot be raised. With speeds 1, 2, …, n − 1 around a stationary runner, Dirichlet's approximation theorem shows that the stationary runner is never more than 1/n from the nearest other runner [2]. A resolution is therefore a proof for every number of runners, or a single set of speeds for which some runner never gets that far from all the others. For each fixed number of runners, Terence Tao showed that the question reduces to finitely many sets of speeds [5]; recent work on eight or more runners bounds the size of a smallest counterexample and checks the remaining cases by computer [6]. The problem carries no prize. Perarnau and Serra ended their 2025 survey: "it seems fair to say that the conjecture remains wide open" [2].

What is known

Small cases (1972–2008). Up to three runners the conjecture is easy [7]. Four runners were settled by U. Betke and Wills in 1972 and again by Cusick; five runners by Cusick and Carl Pomerance in 1984, with a computer check of some cases [2, 8]. Wojciech Bienia, Luis Goddyn and three co-authors gave a simpler proof for five runners in 1998 and observed that the conjecture implies a theorem on nowhere-zero flows in graphs and regular matroids [2, 8]. Tom Bohman, Ron Holzman and Dan Kleitman proved the case of six runners in 2001 [9]. J. Barajas and Serra proved seven runners in 2008 with a tool from the colouring of distance graphs, graphs on the integers in which two numbers are joined when their difference lies in a given set [8, 2]; their argument relies on 7 being prime [5]. Results on the conjecture have also been used to compute chromatic numbers of such graphs [8].

How far apart at least (1967–2025). Write k for the number of runners other than the one that should become lonely. A simple argument with a randomly chosen time, already observed by Wills, shows that it always gets at least 1/(2k) from all of them, about half the conjectured 1/(k + 1) [2, 5]. Later bounds improved this only by terms of order 1/k² [10]. Tao proved 1/(2k) + c log k / (k² (log log k)²) for large k, with c an absolute constant [5], and Benjamin Bedert, in a 2025 preprint not yet peer reviewed, obtains 1/(2k) + 1/k^(5/3 + o(1)) [10].

Special speeds (2012–2018). Sebastian Czerwiński showed that, for a fixed number of runners whose whole-number speeds are chosen at random up to N, with probability tending to one as N grows there is a time when all the others are almost half a lap from the stationary runner, far more than 1/n [11, 2]. Tao proved the conjecture whenever every speed is at most 1.2k, and suggested 2k as a natural target [5]. Perarnau and Serra take such results as a hint that the hard instances are well structured [2].

Finite checking (2015–2025). Tao showed that, once the conjecture holds for fewer runners, it suffices to check whole-number speeds of size at most k^(Ck²) for a computable constant C [5]. Romanos Diogenes Malikiosis, Santos and Matthias Schymura, restating the problem through zonotopes (polytopes that are sums of line segments), lowered the bound to (k(k + 1)/2)^(k − 1), which is at most k^(2k) [12, 2]. In September 2025 Matthieu Rosenfeld combined this kind of bound with a computer verification to prove the case of eight runners, published in Mathematics of Computation in August 2026; his method also covers the known cases of four to seven runners [6].

Recent claims. On 27 November 2025 Tanupat Trakulthongchai posted "Nine and ten lonely runners", which claims computer-assisted proofs for nine and ten runners (k = 8 and 9), obtained by refining Rosenfeld's approach with a sieve [7]. The paper states: "We employed OpenAI GPT-5 to assist with code generation, especially in low-level optimization" [7]. Its second version, of 18 April 2026, carries the note "Updated theorem numbering to match with journal version; added references", and the paper was published in The Electronic Journal of Combinatorics on 5 June 2026 [7]. On 1 December 2025 Rosenfeld posted a preprint claiming an independent proof for nine runners; it names no AI tools [13]. On 26 April 2026 Touch Sungkawichai and Trakulthongchai posted a preprint claiming "a computer-assisted proof of the Lonely Runner Conjecture for k ∈ {10, 11, 12}", that is for eleven, twelve and thirteen runners, using new sieves and a polynomial-method argument [14]. They write: "The development of our code was assisted by AI-based tools, but the process was closely supervised and the code was manually verified" [14]. On 2 September 2026 Jaan Allikvere posted a preprint claiming fourteen runners, extended to fifteen in a second version of 24 September, with code and certificates archived; he states that he "used large language models to assist with code development, drafting, exploring proof ideas, and searching for errors in the manuscript", and adds: "No model output is treated as mathematical evidence" [15]. On 14 October 2026 none of the three preprints listed a journal reference [13, 14, 15].

Main approaches

Case analysis by hand. Sort the speeds by their remainders modulo a small number, such as 7 for seven runners, and rule out each class of speed sets [2, 8]. Strongest evidence: the proofs up to seven runners [9, 8]. Main difficulty: the case analysis grows quickly; the method for seven runners relies on the number of runners being prime, and for eleven runners its case analysis already "becomes cumbersome" [2, 5].

Finite checking by computer. Bound the speeds, or their product, in a smallest counterexample, then exclude the remaining speed sets by computer, sieving them modulo primes [6, 7, 14]. Strongest evidence: published proofs for eight to ten runners [6, 7] and preprints claiming eleven to fifteen [14, 15]. Main difficulty: the cost rises steeply; Allikvere reports about 3.5 × 10^15 candidates at one stage of the fourteen-runner search modulo the prime 199, and a cost of generating candidates that grows roughly as the sixth power of the prime; moreover the polynomial-method step used for eleven and thirteen runners needs the number of runners to be prime [14, 15].

Widening the gap. Raise the general bound from about 1/(2k) towards 1/(k + 1) with Fourier analysis and additive combinatorics [5, 10]. Strongest evidence: Bedert's claimed improvement by a power of k [10]. Main difficulty: every bound so far is still about half the conjectured value [5, 10], and Tao gives an example suggesting that refining the union bound with estimates for overlaps yields only small gains [5, 2].

Geometry of zonotopes. Restate the conjecture as a question about lattice points in zonotopes, where tools from the geometry of numbers apply [2, 12, 4]. Strongest evidence: the finite-checking bound [12], which the computer proofs use [6]. Main difficulty: the "shifted" version, in which runners start from different points, fits the same framework, but Blanco, Criado and Santos give counterexamples to it for six to eighteen runners [4], so an argument that would also prove the shifted version cannot work.

Open sub-questions

1. Checking the computations. Do independent checks of the code and certificates for eleven to fifteen runners [14, 15] confirm them, and can the sieves reach sixteen runners, where the number of runners is again composite [15]?
2. Tight instances. Besides the families obtained by speeding up single runners, are there other infinite families of speed sets for which 1/(k + 1) is exactly attained [2]?
3. A constant-factor gap. For every ε > 0, is the gap at least (1 + ε)/(2k) for all large k [2, 10]?
4. Bounded speeds. Does the conjecture hold whenever every speed is at most 2k, as Tao asked [5]? It is known up to 1.2k [5] and, for large k, up to a little below 2k [2].
5. Shifted starts. Counterexamples to the shifted version are known for six to eighteen runners [4]; does it fail for every larger number of runners?
6. Translation. If the same whole number is added to every speed, can the gap decrease [2]?

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Key references

1. Wills, J. M. (1968). Zur simultanen homogenen diophantischen Approximation. I. Monatshefte für Mathematik 72(3), 254–263. 10.1007/BF01362551 · Checked 14 Oct 2026
2. Perarnau, G. & Serra, O. (2025). The Lonely Runner Conjecture turns 60. Computer Science Review 58, 100798. 10.1016/j.cosrev.2025.100798 · Checked 14 Oct 2026
3. Cusick, T. W. (1973). View-obstruction problems. Aequationes Mathematicae 9(2–3), 165–170. 10.1007/BF01832623 · Checked 14 Oct 2026
4. Blanco, M., Criado, F. & Santos, F. (2026). Coloopless zonotopes and counterexamples to the Shifted Lonely Runner Conjecture. Preprint, arXiv:2603.24784 (version 3, 2 October 2026, revised after referee comments). Source · Checked 14 Oct 2026
5. Tao, T. (2018). Some remarks on the lonely runner conjecture. Contributions to Discrete Mathematics 13(2), 1–31. 10.55016/ojs/cdm.v13i2.62728 · Checked 14 Oct 2026
6. Rosenfeld, M. (2026). The lonely runner conjecture holds for eight runners. Mathematics of Computation, published online 10 August 2026; arXiv:2509.14111 (version 1, 17 September 2025). 10.1090/mcom/4243 · Checked 14 Oct 2026
7. Trakulthongchai, T. (2026). Nine and ten lonely runners. The Electronic Journal of Combinatorics 33(2), P2.46; arXiv:2511.22427 (version 1, 27 November 2025; version 2, 18 April 2026). 10.37236/14972 · Checked 14 Oct 2026
8. Barajas, J. & Serra, O. (2008). The lonely runner with seven runners. The Electronic Journal of Combinatorics 15(1), R48. 10.37236/772 · Checked 14 Oct 2026
9. Bohman, T., Holzman, R. & Kleitman, D. (2001). Six lonely runners. The Electronic Journal of Combinatorics 8(2), R3. 10.37236/1602 · Checked 14 Oct 2026
10. Bedert, B. (2025). Riesz products and the Lonely Runner Conjecture: a wider gap of loneliness. Preprint, arXiv:2511.16636 (version 1, 20 November 2025), not peer reviewed. Source · Checked 14 Oct 2026
11. Czerwiński, S. (2012). Random runners are very lonely. Journal of Combinatorial Theory, Series A 119(6), 1194–1199. 10.1016/j.jcta.2012.02.002 · Checked 14 Oct 2026
12. Malikiosis, R. D., Santos, F. & Schymura, M. (2025). Linearly exponential checking is enough for the lonely runner conjecture and some of its variants. Forum of Mathematics, Sigma 13, e164. 10.1017/fms.2025.10107 · Checked 14 Oct 2026
13. Rosenfeld, M. (2025). The lonely runner conjecture holds for nine runners. Preprint, arXiv:2512.01912 (version 1, 1 December 2025; version 2, 27 January 2026), not peer reviewed. Source · Checked 14 Oct 2026
14. Sungkawichai, T. & Trakulthongchai, T. (2026). Eleven, twelve, and thirteen lonely runners. Preprint, arXiv:2604.23906 (version 1, 26 April 2026; version 2, 1 September 2026), not peer reviewed. Source · Checked 14 Oct 2026
15. Allikvere, J. (2026). Fourteen and fifteen lonely runners. Preprint, arXiv:2609.02604 (version 1, "Fourteen lonely runners", 2 September 2026; version 2, 24 September 2026), not peer reviewed. Source · Checked 14 Oct 2026

Page reviewed 14 Oct 2026

Source

Open, as listed by Wikipedia · Unsolved problems in mathematics. Checked 08 Oct 2026.

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