The Mathematics of Nature

The life of a cicada has two phases. The first is spent underground, surviving for years on sap drawn from tree roots. This is the juvenile phase. Then adulthood begins — and it lasts only a few short weeks.

The juvenile phase, though, runs for either 13 or 17 years depending on the species. During all that time the cicadas live below the surface, and when the moment comes — after exactly 13 or 17 years — they emerge almost simultaneously. There is no single species with a 13-year cycle and another with a 17-year cycle. Three known genera exist, and each has both 13- and 17-year races.

What is striking is that the split into 13- and 17-year races happened independently, at least three separate times. Evolution kept landing on 13 and 17. The reason isn’t fully understood, but there is a compelling argument, and it turns on one fact: these numbers are prime. They divide evenly by nothing except 1 and themselves — and that lowers the odds of coinciding with the life cycle of a periodic predator or parasite.

The prime-number argument

Suppose a cicada had a 14-year cycle. A parasite that peaks every 2 or 7 years would sync up with it regularly, because both 2 and 7 divide 14:

$$14 = 2 \times 7$$

Now take a cicada with a 17-year cycle. Seventeen is prime, so a parasite only lines up with it when their cycles share a common factor greater than one — and for a prime $p$ the only divisors are:

$$\{1, p\}$$

In general, two cycles of length $m$ (the cicada) and $k$ (the predator) re-synchronise every

$$\operatorname{lcm}(m, k) = \frac{m \cdot k}{\gcd(m, k)}$$

years. When $m$ is prime and doesn’t divide $k$, the greatest common divisor $\gcd(m, k) = 1$, so the least common multiple is simply $m \cdot k$ — the two only meet once every $m \cdot k$ years. A prime cycle pushes those unlucky encounters as far apart as possible.

Evolution, of course, doesn’t know any of this. It produces an enormous number of variants, and whichever reproduces most successfully in the wild carries on — natural selection. Cicadas whose cycles happened to be prime met their parasites less often, out-reproduced the rest, and the non-prime races were gradually culled. What survives to today is a handful of species with prime-numbered lives.

This is a small corner of mathematics showing through into biology. Evolution may have no intelligence and no awareness — but that it is a mathematical genius is hard to dispute.