Why Do Cicadas Spend Years Underground?
– Prime Numbers as a Survival Strategy
In summer, cicadas sing with a deafening din. Their life above ground lasts only a few weeks. But before that, they spend years underground as larvae. Some North American cicadas emerge on a cycle of 13 or 17 years, both prime numbers. This odd choice of numbers is thought to have a shrewd survival reason behind it.
You have probably heard cicadas pour out of the trees at the height of summer. That noisy adult stage is actually only a tiny part of a cicada's life.
Most cicadas hatch from eggs, burrow into the soil as larvae, and spend years there sucking nutrients from tree roots. Only then do they climb out and moult into adults, who live for just a few weeks.
The North American group known as periodical cicadas spend exactly 13 or exactly 17 years underground, then emerge all at once. Why choose such a long time, and such an unusual number of years?
The larval stage lasts far longer than the adult stage.
These are prime numbers, divisible only by 1 and themselves. They are thought to be no accident but a survival strategy.
Let's look at the puzzle of "why primes" through the properties of numbers.
Most of a cicada's life is spent underground
Cicada larvae push their mouthparts into tree roots underground and grow by drinking the sap. Depending on the species, this stage often lasts several years. The dark soil, where enemies have trouble finding them, is thought to be a good place to take time and grow slowly.
Once above ground as adults, they spend their limited time finding a mate and leaving offspring. The short adult life makes sense if you think of it this way: the larval stage does almost all the "growing", while adults specialise in "passing life on".
North America has "prime-number cicadas" with 13- and 17-year cycles
North America is home to special cicadas called periodical cicadas. They are split into regional groups called "broods", and each brood emerges all at once on a strictly fixed cycle of 13 or 17 years.
13 and 17 are prime numbers, with no divisors other than 1 and themselves. Most animals' life cycles run to 1, 2, 4 or 6 years, numbers that divide up easily. Only periodical cicadas stick to these unusual prime numbers of years, and researchers have long been fascinated by that.
It is thought to be a strategy for using mathematics to reduce chance meetings with predators.
Why go to the trouble of a prime cycle?
One leading hypothesis is that "it reduces overlap with predators and with related cicadas on different cycles." If a cicada's cycle were divisible by many numbers, such as 2, 3, 4 or 6 years, its emergence would often coincide with that of predators or rivals on similar cycles.
But with a prime cycle, it rarely overlaps with anyone except those on that same cycle. It is thought to be extremely hard in practice for a predator with a long prime cycle of 13 or 17 years to evolve and become established. As a result, periodical cicadas manage to cut their encounters with many predators sharply.
Periodical cicadas emerge in enormous numbers in the same year. This is also thought to be a strategy called predator satiation: by emerging in numbers too great for predators to eat, the population as a whole raises its chance of survival. Combined with the prime cycle, this mass emergence is thought to greatly reduce the damage from natural enemies.
Something you can check yourself
- On a sheet of paper, write out the numbers from 1 to 60
- Circle the multiples of 4 (4, 8, 12…)
- On the same sheet, also mark the multiples of 12 (12, 24, 36…) and see where they land on a circle (they should overlap every time)
- Next, mark the multiples of 13 and look for places that fall on a circled multiple of 4 (within 60, they should hardly overlap at all)
12 and 4 overlap often, but 13 and 4 hardly ever do. This is the mathematical reason periodical cicadas choose prime numbers.
Summary
Cicadas spend years underground as a way to grow slowly and thoroughly as larvae. Periodical cicadas' choice of prime cycles of 13 and 17 years is thought to be a strategy for using mathematics to reduce how often they emerge at the same time as predators or related species on different cycles. The property of prime numbers, seemingly so plain, really does affect how living things evolve.
A cicada's long life underground is not just a time of waiting.
It is a shrewd survival strategy that puts the properties of numbers on its side.
The sound side of a related puzzle, why you can hear a voice but not see the person, is covered in Why can you hear a voice in hide-and-seek when you can't see the person?
For those who want to know more – terms, numbers and links to textbooksFrom middle-school science to ongoing research, each part is labelled with its level
- Middle schoolCovered in middle-school science and maths
- High schoolCovered in high-school "Mathematics A"
- High school+High-school "Biology", or textbook advanced material and sidebars
- UniversityUniversity-level specialist content (evolutionary ecology) not taught in high school
- ResearchTopics researchers are still working on, not yet taught even at university as settled fact
Middle schoolTerms: the vocabulary of periodical cicadas
- Prime number: an integer with no divisors other than 1 and itself (2, 3, 5, 7, 11, 13…).
- Periodical cicada: a group of cicadas that emerge above ground on a fixed cycle of 13 or 17 years.
- Brood: a group of periodical cicadas that emerge together in the same year and the same region.
- Lowest common multiple: the smallest of the multiples shared by two or more numbers.
High schoolChecking with a formula: how rarely do prime cycles overlap?
Let's use the lowest common multiple (LCM) to actually calculate how often cycles overlap.
LCM = (product of the two numbers) ÷ greatest common divisor
| Greatest common divisor | The largest divisor shared by the two numbers |
| Lowest common multiple | The interval at which the two cycles line up exactly |
As an example predator, imagine a creature with a 4-year cycle.
| Greatest common divisor of 12 and 4 | 4 |
| LCM of 12 and 4 | 12 × 4 ÷ 4 ＝ 12 |
| Interval at which the 12-year cicada overlaps | Once every 12 years |
| Greatest common divisor of 13 and 4 | 1 |
| LCM of 13 and 4 | 13 × 4 ÷ 1 ＝ 52 |
| Interval at which the 13-year cicada overlaps | Once every 52 years |
Let's compare these two intervals.
| Ratio of overlap intervals | 52 ÷ 12 ≒ 4.33 |
| Ratio of overlap intervals | About 4.33 times |
The calculation shows that a 13-year cicada meets the same 4-year rival only about a quarter as often as a 12-year cicada does.
The same reasoning gives the interval at which the 13-year and 17-year broods meet each other. Both are prime, so their greatest common divisor is 1.
| LCM of 13 and 17 | 13 × 17 ＝ 221 |
| Interval at which the two broods overlap | Once every 221 years |
We can see that 13-year and 17-year cicadas emerge together only once every 221 years, a very rare event.
* The 4-year predator is just an illustration to help explain the mechanism. The cycles of animals that could actually overlap with periodical cicadas are said to vary by species.
High school+This hypothesis is supported from several angles
Besides the idea of reducing encounters with predators, another explanation for prime cycles has been proposed: preventing related cicadas on different cycles from interbreeding. The thinking is that when cicadas on different cycles rarely emerge in the same year, each cycle is more easily kept as a separate population. Both explanations share the point that "cycles that rarely overlap help survival and breeding".
UniversityA link to the ice ages has also been suggested
In evolutionary ecology, there is also a discussion of the idea that the long cycle of periodical cicadas itself may come from slow growth during the harsh climate of past ice ages. It has been suggested that in a harsh climate where growth could only be slow, groups that happened to have similar cycles gradually converged on prime numbers of years, but no single factor is thought to explain it completely.
ResearchWhat is still unclear
- How larvae underground count "how many years have passed" so accurately is said to be not yet fully understood. It has been suggested that they may record seasonal changes in the nutrients from tree roots as physiological changes in the body, but the detailed molecular mechanism is still under study.
- Reports of "early emergence", where some periodical cicadas come out earlier than their proper cycle because of climate change, are said to be increasing, and how temperature changes affect the maintenance of the cycle itself is a research question.
- Even the mathematical models that support the evolution of prime cycles differ in their positions, and debate is said to continue over which factors mattered and by how much.
Behind the familiar sound of summer cicadas lies a topic where mathematics and biology meet, and research continues today.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Middle school | Maths: primes and divisors / Science: insect growth | Basic terms: prime numbers, periodical cicadas, lowest common multiple |
| High school | Mathematics A: properties of integers | Using the LCM to work out how rarely cycles overlap |
| High school+ | Biology: evolution and adaptation | Predator-avoidance and interbreeding-prevention hypotheses |
| University | Evolutionary ecology | Ice-age climate and the evolution of cycles |
| Research | Insect physiology and climate ecology (ongoing research) | How the body clock works; climate change and early emergence |
- Explanations of the prime-cycle hypothesis for periodical cicadas (the predator-avoidance hypothesis) in evolutionary ecology literature.
- Explanations of the greatest common divisor and lowest common multiple in high-school maths textbooks.
- Explanations of cicada life history (larval and adult stages) in entomology textbooks.
- Research reviews on the evolution of periodical cicada cycles and the ice ages, from palaeoclimatology and evolutionary ecology.
- Research reviews on the body clock and early emergence of periodical cicadas, from insect physiology.
* Numbers such as the predator's cycle are illustrations to help explain the mechanism. Real ecosystems are said to be more complicated.
* This article is a general-audience science explainer. For detailed information on cicada and periodical cicada ecology, please consult entomological institutions and academic literature.