Cicadas in their primes

I somehow developed an interest in prime numbers. But I know when I’m beat. I understand enough though to enjoy this passage from Marcus du Sautoy’s The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics (New York: HarperCollins, 2003). It’s about two species of cicada, Magicicada septendecim and Magicicada tredecim. The first emerges from the ground to mate and die every seventeen years; the second, every thirteen years. “Why,” du Sautoy asks, “has each species chosen a prime number of years as the length of their life cycle?”

There are several possible explanations. Since both species have evolved prime number life cycles, they will be synchronised to emerge in the same year very rarely. In fact they will have to share the forest only every 221 = 17 × 13 years. Imagine if they had chosen cycles which weren’t prime, for example 18 and 12. Over the same period they would have been in sync 6 times, namely in years 36, 72, 108, 144, 180 and 216. These are the years which share the prime building blocks of both 18 and 12. The prime numbers 13 and 17, on the other hand, allow the two species of cicada to avoid too much competition.

Another explanation is that a fungus developed which emerged simultaneously with the cicadas. The fungus was deadly for the cicadas, so they evolved a life cycle which would avoid the fungus. By changing to a prime number cycle of 17 or 13 years, the cicadas ensured that they emerged in the same years as the fungus less frequently than if they had a non-prime life cycle. For the cicadas, the primes weren’t just some abstract curiosity but the key to their survival.

I remember the Great Confluence of 1998 well. An endless din. Shells everywhere. Every moment outside was aural agony. The cicadas were in their primes.

[From a review of the book: “just enough maths to befuddle the layman.”]