The Surprising Unproven Mathematical Conjecture About the Lone Runner

He always looked forward to this moment. Everything freezes, waiting for the starting pistol to fire. The sounds and colors of the outside world fade into the background. The whole body freezes like a tightly compressed spring. Time becomes long and drawn out, flows more and more slowly and reluctantly. An eternity passes before the sharp starting shot is finally heard. The brain does not yet have time to understand and realize anything, but the trained body already habitually straightens and rushes forward. Rivals are flying on adjacent tracks. They are always there, too close, you can’t escape them, you can’t tear yourself away in a desperate dash. Usually the forces are almost equal. Milliseconds, millimeters and photo finishes decide everything. But not this time…

Stuck nut

Mathematicians are amazing people. They love unsolvable problems and unprovable hypotheses. Don’t feed them bread, just let them come up with some tricky problem and give it some amazing name. And it would be fine if these tasks were just abstract exercises for the mind. But somehow it turns out that many of them are of fundamental importance for the development of science. It’s like a stuck nut—until you unscrew it, you won’t get any further.

So the hypothesis about the lone runner turned out to be one of these problems. It was formulated in 1967 by mathematician J. M. Wills. True, at that time this hypothesis had not yet received its poetic name. This was taken care of in 1998 by L. Goddyn.

The essence of the hypothesis

Let's assume that n runners run along a ring track. The length of the track is conditionally equal to 1. All runners start at the same point, but everyone’s speed is different. A runner is considered lonely if at some point in time he is more than 1/n from any other runner.

Under such conditions, is it possible to say that each of the runners will sooner or later find himself alone? It seems like, yes, it is possible. But no one has yet been able to prove this statement for any number of runners. In general, the famous lines “Beauty! Among those running: there are no first and no lagging behind” – this is not our case.

The strict formulation of the hypothesis states that the speeds of the runners must be expressed in whole numbers that are not divisible by the same prime number. A lone runner will have zero speed. If D is an arbitrary set of positive integers that contains exactly k−1 number with greatest common divisor equal to 1, then:

$\exists t\in \mathbb{R}\quad \forall d\in D\quad ||td|| \geq \frac{1}{k}$

Here's the entry ||x|| means distance from number x to the nearest integer.

Strange experiment

…In this strange experimental race, everything was not as usual. Take, for example, the corridor along which he had to run. Not a single straight section where it would be possible to accelerate properly. Just a smoothly curving giant circle. The turn is almost imperceptible, but it is always there. And it was very annoying – like a small dead pixel in the middle of a perfectly clean screen. And the choice of opponents for the race was, to put it mildly, strange. The crowd of athletes at the start consisted of representatives of all ages and population groups: from junior schoolchildren to cheerful pensioners. There were also runners here who were well known to him, whom he had met more than once at international starts. He knew well that it was too early for him to compete with them in speed.

After the starting pistol fired, this entire motley crowd rushed along the corridor and quickly stretched out into a long tail. Well-trained athletes pulled far ahead, weak runners immediately fell behind. He knew that, according to the conditions of the experiment, he would have to run more than one lap, so he moved at a comfortable, calm pace – he saved his energy, but also did not fuss in vain. Soon he made an almost complete circle and caught up with those hopelessly lagging behind. But by this time, one after another, nimble champions began to outstrip him…

There are only seven of them so far

As usual, for such hypotheses there is a number of confirmations for individual particular conditions. The lone runner hypothesis is confirmed for cases where the number of runners n less than 8. More precisely, for n=1 there is no need to confirm it at all.

The only sad runner is lonely by definition right at the start.

For n=2 The time of onset of loneliness for a runner can be calculated using the formula:

$t = \frac{1}{2 (v_1-v_0)}$

For example, if the speed of one runner is 1, and the other is 2, then sooner or later they will find themselves on opposite parts of the track and will be, so to speak, mutually lonely.

For n=3 Mathematicians have a cunning “excuse”. It is believed that any evidence for n > 3 automatically confirms the correctness of the hypothesis for n=3.

But then things began to move more slowly than desired. Here is a timeline of emerging evidence for different meanings n:

4 – 1972;
5 – 1984;
6 – 2001;
7 – 2008.

All these proofs are quite voluminous works with many calculations. For n > 7 There is still no evidence that the hypothesis is correct. However, in 2011, scientists obtained a formula for determining the speeds at which the hypothesis is confirmed for any sufficiently large number of runners. However, the formula for specific speeds is not yet proof of the entire hypothesis. After all, in the end you need to prove that the hypothesis is true for any n without any conditions or additional reservations.

Sudden disappearance

…This time no one was running nearby, all the opponents had different speeds. But at the same time, there was always someone ahead and behind him. The width of the corridor allowed athletes to easily overtake each other. Quite quickly he got used to it and methodically outstripped his opponents one after another. He began to recognize some of them by sight, and a couple of times he even greeted one of them on the next lap. Sometimes he stood aside and allowed more experienced athletes to overtake him.

But suddenly something strange happened. He suddenly realized that there was not a single runner left ahead. Throughout its entire visible part, the corridor was completely empty. There was nothing special about it, but he suddenly felt extremely uncomfortable. The sudden disappearance of the other runners was so unusual that he swallowed nervously and looked back. There was no need to do this – he already knew that there wouldn’t be a soul behind him either. Suddenly he found himself completely alone in this strange, gently curving corridor. The environment became unnaturally quiet. He mechanically continued to run, and the sound of his steps echoed off the walls…

Why all this running around?

Someone already has a completely logical question in their head: “Why is all this necessary?” Why is it so important to prove some abstract hypothesis about runners running laps in completely unnatural conditions? The thing is that often such mathematical problems are a beautiful presentation of serious problems that lie at the root of more complex problems. Mathematics is like this: if some part of it seems useless, then we simply don’t yet know how it can be used.

Mathematics must then be taught, because it puts the mind in order.
M. V. Lomonosov

…A few painful seconds passed and the obsession passed. More trained athletes caught up with him from behind, and another slow runner appeared around the bend in front. He was again among people, again participating in an endless race. The experiment was over. Scientists have made their conclusions: no matter how many runners participate in the race, each of them may sooner or later find themselves alone. But the lone runner knew for sure that there was always someone there, around the bend.