Are You Worse at Maths than a Cicada?

A few days ago, I decided to catch up on my long-neglected manga reading-list. One particular manga¹ made a joke in passing about prime-number cicadas, leaving me stumped. Two days, and a lot of scrolling, later, here’s what I’ve learned.

Cicadas are loud, plant-sap-feeding insects famous for their buzzing songs, unfortunately not found in Ireland, and certainly not one I’d have expected to be mentioned in a sentence about number theory. Enter Magicicada², a genus of cicadas famous for their bizarre life cycles.

Unlike other types of cicadas, Magicicadas spend the majority of their exceptionally long lives underground, emerging only after 13 or 17 years to sing, mate, and die³. The beauty lies in these specific (prime, as you may have guessed from the title) numbers: why 13 or 17 years?

Prime numbers have an innate property: they don’t overlap easily with other numbers. For Magicicadas, this means fewer run-ins with peak (i.e. mature and most adept at feasting on these cicadas) predator populations, and thus more cicadas surviving to repeat the cycle. Many of the typical predators of cicadas, such as certain beetles or wasps, only emerge as adults (and pose a threat) during the final year of their life cycle.

For example, a 4-year predator and a 17-year cicada would only meet once every $\text{LCM}(4, 17) = 68$ years. Had the cicadas emerged every 16 years instead, the meeting would occur every $\text{LCM}(4, 16) = 16$ years - a relatively small change in emergence time leading to rendezvous four times more frequent—potentially devastating for the population.

This prime-number strategy, combined with the predator satiation strategy employed, where the cicadas emerge in overwhelming numbers to ‘flood’ predators, ensures the survival of enough individuals to reproduce even if many are eaten.

I never expected a casual manga binge to lead me to the real-world utilisations of number theory in nature, but here I was, marvelling at how a stupid one-liner on “prime-number watermelons” contained so much fascinating mathematical lore.