# Mathematicians explore cicada's mysterious link with primes

Sun Staff

It's probably no surprise that periodical cicadas like the clan invading Maryland this month are a big draw for biologists. Less obvious is why someone like Glenn Webb would care.

Webb is a mathematician. Working at Nashville's Vanderbilt University, he spends his days immersed in formulas, not fieldwork. But a backyard encounter with periodical cicadas several years ago led him to a mystery that has seduced more than a few members of his profession over the years: the cicada-prime connection.

Periodical cicadas crawl from their subterranean hideouts en masse every 13 or 17 years, depending on the kind.

For Webb and others, it's a pattern that immediately raises eyebrows: 13 and 17 are prime numbers, integers divisible by only themselves and 1. Primes, like cicadas, have been a source of fascination for centuries. So it didn't take long before scientists wondered: Is it mere coincidence that cicada emergences are timed to primes, or is some deeper mechanism at work?

"To me it's a little puzzle from evolution," says Webb, who has devised a mathematical model of cicada behavior and in 2001 published a tentative conclusion: The prime-number lifecycle is no coincidence but evolved as an effort to avoid predators.

The prime-number conundrum gets to the heart of what most people find beguiling about the bugs in the first place: why an insect would spend most of its life suckling on tree roots and a few short weeks singing and mating in the sunshine.

Adding to the intrigue is that it's difficult to find other examples of cicadalike behavior in nature. In his 1977 essay collection Ever Since Darwin, paleontologist Stephen Jay Gould notes one: Phyllostachys bambusoides, a bamboo native to Japan and China. The plant, writes Gould, flowers and sets seed every 120 years or so.

But as a grass, bamboos can propagate asexually, and typically they do many times before they flower. The 13- and 17-year cicadas have just one brief shot at sex - then they die.

Gould, a polymath who died in 2002, was among the first to propose that the cicada's unusual lifestyle is a strategy it evolved to avoid its predators.

"Some individuals hide, others taste bad, others grow spines or thick shells, still others evolve to look conspicuously like a noxious relative," Gould wrote. Periodical cicadas, he argued, did it by evolving a highly unusual reproductive cycle.

By springing forth from the ground by the millions, cicadas help ensure that no single predator can devour them, a tactic that evolutionary biologists now call the "predator satiation" strategy.

And by emerging every 13 and 17 years, Gould argues in his 1977 book, cicadas minimize the chance that their infrequent invasions will sync with the life cycles of birds and other creatures that dine on them.

For example, imagine bird species that wax and wane on a five-year cycle. If cicadas emerged every 10 years, their arrival might coincide with the peak of this avian predator, setting up a pattern that could drive the cicadas to extinction.

By cycling at a large prime number, cicadas minimize the chance that some bird or other predator can make a living off them. The emergence of a 17-year cicada species, for example, would sync with its five-year predator only every (5 multiplied by 17) 85 years.

That's the theory, anyway.

Intrigued by Gould's explanation, Glenn Webb at Vanderbilt spent several years, off and on, creating a mathematical model of periodical cicadas and hypothetical predators with two- and three-year life cycles. He found that Gould's argument held up: By emerging only every 13 or 17 years, periodical cicadas better ensured their survival.

But other scientists have done the math and concluded it's only coincidence that the insect's lifecycles also happen to be prime numbers.

Still others have argued it's not predators but weather that helped shape the cicada's behavior. The insects are thought to have evolved 1.8 million years ago during the Pleistocene epoch, when glaciers advanced and retreated across North America.

Scientists have created models showing that the more years cicadas remained nestled underground, the less likely they would emerge during a killing summer cold spell.

But that still doesn't explain why periodical cicadas settled on 13 and 17 and not 11, 19 or some other prime number.

"It's pretty controversial," Webb says of the whole cicada/prime business. "I don't know if there will ever be a satisfying scientific resolution."

• #### Cicada boundaries

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