Opinion 163: Prime Numbers are Often Red Herrings

By Doron Zeilberger

Written: Dec. 31, 2017

The great French mathematical columnist, Jean-Paul Delahaye, recently (Pour La Science (Sept. 2017), 80-85) posed the following brain-teaser, adapting a beautiful puzzle, of unknown origin, popularized by Peter Winkler in his wonderful book "Mathematical Mind-Benders". Here is a free translation from the French.

Enigma: Nine Beetles and prime numbers

One places nine beetles on a circular track, where the nine distances, measured in meters, between two consecutive beetles are the first nine prime numbers, 2,3,5,7,11,13,17,19 and 23. The order is arbitrary, and each number appears exactly once as a distance. At starting time, each beetle decides randomly whether she would go, traveling at a speed of 1 meter per minute, clockwise or counter-clockwise. When two beetles bump into each other, they immediately do "U-turn", i.e. reverse direction (like billiard balls). We assume that the size of the beetles is negligible. At the end of 50 minutes, after many collisions, one notices the distances between the new positions of the beetles. The nine distances are exactly as before, the first nine primes numbers! How to explain this miracle?

Before looking at the answer, try to do it yourself.


Note that the length of the circular track is 2+3+5+7+11+13+17+19+23 = 100 meters. Let each beetle carry a flag, and whenever they bump into each other, let them exchange flags. Since the flags always move in the same direction, and also move at a speed of one meter per minute, after 50 minutess, each flag is exactly at the ``antipode" of its original location; hence, the distances are the same! Of course, this works if the original distances were any sequence of numbers: All that they have to obey is that their sum equals 100, or more generally, that half the sum of the distances divides the product of the speed 1 meter/minute in this puzzle) and the elapsed time (50 minutes in this puzzle).

This variation, due to Delahaye, is much harder than the original version posed in Peter Winkler's book, where also the initial distances were arbitrary. In Delahaye's rendition, the solver is bluffed into trying to use the fact that the distances are primes. Something analogous happened to the great Paul Erdős, the patron saint of combinatorics and number theory, who introduced covering systems and posed a problem that was open since 1950 until two years ago, when Bob Hough found a great refutation. A close reading of his proof reveals that essentially all you need is the asymptotic property of the primes, namely that pn is asymptotic to n times log(n), and in hindsight, Erdős' guess (that there are distinct covering systems with arbitrarily large minimal moduli) was unlikely to be true by a healthy heuristic argument.

Also the recent breakthroughs in the twin prime conjectures (that are still very far from the true state of affairs), use very clever sieving arguments, but the same arguments should work if one replaces the sequence of primes by more general sequences, with similar asymptotic properties.

Hence, who knows? perhaps even the Riemann Hypothesis, that is equivalent to a statement about primes, would be made tractable if one can find a more general statement, valid for a wide class of sequences, that since it is more general, would be (as often is the case) much easier to prove. That far more general statement should reduce to RH (and even GRH) when one specializes to the primes. Good luck!

Opinions of Doron Zeilberger