Cicadas and Prime Numbers

You may have heard that this year, for the first time since 1803, two different broods of cicadas will emerge at the same time.

This year, though, will be a rare event. Two groups, or “broods,” are waking up during the same season. There will likely be billions, if not trillions, of the insects.

There’s the 17-year-group called Brood XIII, which is concentrated in northern Illinois (brown on the map below), and the 13-year clutch, Brood XIX, which will emerge in southern Illinois, Missouri, Arkansas, and throughout the Southeast.

You may have noticed the lengths of both periodicities (13, 17) are prime numbers — and that does not appear to be a coincidence. Scientists haven’t nailed down an exact cause, but one hypothesis has to do with predator cycles:

According to the paleontologist Stephen J. Gould, in his essay “Of Bamboo, Cicadas, and the Economy of Adam Smith,” these kind of boom-and-bust population cycles can be devastating to creatures with a long development phase. Since most predators have a two-to-ten-year population cycle, the twelve-year cicadas would be a feast for any predator with a two-, three-, four-, or six-year cycle. By this reasoning, any cicada with a development span that is easily divisible by the smaller numbers of a predator’s population cycle is vulnerable.

Prime numbers, however, can only be divided by themselves and one; they cannot be evenly divided into smaller integers. Cicadas that emerge at prime-numbered year intervals, like the seventeen-year Brood II set to swarm the East Coast, would find themselves relatively immune to predator population cycles, since it is mathematically unlikely for a short-cycled predator to exist on the same cycle. In Gould’s example, a cicada that emerges every seventeen years and has a predator with a five-year life cycle will only face a peak predator population once every eighty-five (5 x 17) years, giving it an enormous advantage over less well-adapted cicadas.

See also Long-lived insects raise prime riddle.

• mathematics
• science