St John’s undergraduate advances longstanding mathematical conjecture

Imagine a group of athletes running around a circular track. The athletes line up at the same point and then simultaneously begin running, each at a unique constant speed. If they were to continue running for long enough, would each athlete eventually find themselves, if only temporarily, at a significant distance from all others?

This is the Lonely Runner Conjecture, first posed by the mathematician Jörg M. Wills in the 1960s. It predicts that regardless of each runner’s speed, each one will at some point become ‘lonely’. Loneliness, in this specific instance, is defined as being at least 1/N of the track away from every other runner, where N is the total number of runners. The Lonely Runner Conjecture is connected to foundational results in the field of Diophantine approximation, which studies how real numbers can be approximated by fractions.

Although the conjecture may seem straightforward when the number of runners is small, the problem becomes increasingly difficult with more runners. Since the 1960s, mathematicians have gradually proved the conjecture for more and more runners. Until last summer, the world record was seven runners, established in 2008.

In the past year, however, significant advances have been made. The conjecture has now been proven to hold for up to ten runners, thanks in part to research undertaken by Tanupat (Paul) Trakulthongchai, a second-year Mathematics student at St John’s.

In 2025, Matthieu Rosenfeld, a mathematician at the Laboratory of Computer Science, Robotics, and Microelectronics of Montpellier, settled the conjecture for eight runners. (Read his paper here.) Rosenfeld developed a new computational approach that analysed what the runners’ speeds would have to look like in order for the conjecture not to hold, that is, what kind of arrangement of speeds would prevent a runner from ever becoming lonely. He then demonstrated that no such arrangement can exist for eight runners. Although Rosenfeld’s approach is computationally intensive – analysing the eight-runner case took tens of computing hours – he posited that it could, eventually, 'lead to proof of the lonely runner conjecture'.

Dr Noah Kravitz, a Visiting National Science Foundation (NSF) Postdoctoral Fellow at St John’s with a longstanding interest in the Lonely Runner Conjecture, introduced Rosenfeld’s work to one of his undergraduate students, Tanupat (Paul) Trakulthongchai.

Kravitz initially proposed resolving the nine-runner case by optimizing the implementation of Rosenfeld’s algorithm. To his surprise, Trakulthongchai soon came back with not only a streamlined computer program but also a new mathematical ‘sieve’ that improved the theoretical side of Rosenfeld’s work. ‘Rosenfeld’s approach is akin to showing that a whole haystack is devoid of needles,’ Kravitz explained, ‘and Paul’s insight was that one can quickly rule out 99% of the haystack before searching carefully through the last 1%.’

Trakulthongchai’s improvement drastically reduced the number of speed combinations to be analysed. Using his refinement of Rosenfeld’s framework, Trakulthongchai was able to extend the proof to nine and then ten runners. (His paper is available here.)

Extending the verified cases of the Lonely Runner Conjecture from seven to ten runners adds to the evidence in favor of the conjecture being true. ‘It also provides new ideas for where to look for a proof of the general conjecture — or for potential counterexamples,’ Kravitz added.

A workshop on recent developments on the Lonely Runner Conjecture has been organised for Autumn 2026. It will bring together mathematicians from across the world and spanning several subfields, including Diophantine approximation, graph theory, geometry, and additive combinatorics. As part of this workshop, Trakulthongchai will be able to discuss his research with Jörg M. Wills, who first posed the problem in the 1960s, bringing the lonely runner full circle.

Congratulations, Paul, on this important contribution to a complex mathematical conjecture.