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Cicadas In Their Prime

For years biologists have wondered why cicadas emerge from their underground habitats after a prime number (7, 13 or 17) of years. This week Dr Karl tells us how physicist Mario Markus unravelled the secret behind cicadas' understanding of number theory.

2,300 years ago, the Greek scientific writer, poet and astronomer, Eratosthenes of Cyrene, became the first person to calculate the circumference of the Earth. He also dabbled in a bit of maths, and invented the famous Sieve of Eratosthenes, which is an easy method for finding prime numbers. But now it seems that some recent research into cicadas gives us another way of finding prime numbers.

Prime numbers are numbers that can be divided only by themselves and one. So 2, 3, 5, 7, 11, 13 and 17 are all prime numbers - but 18 is not a prime number, because you can divide it by both 2 and 9. To work the Sieve of Eratosthenes you write down all the numbers, and then simply strike out every second number that comes after 2, every third number following the number 3, and so forth. All the numbers that you have left will be prime numbers. It's called a Sieve because all the numbers that are not prime just "fall through".

You also find prime numbers in the life cycles of cicadas. There are about 1,500 species of cicadas known. There are those that appear yearly in midsummer, and there are also the so-called "periodic" cicadas. They appear at prime number intervals - 7 years, 13 years and 17 years.

The cicadas are part of the insect order Homoptera. These are all sucking insects, which pierce plants with their pointy mouthparts and suck out the juices. The breeding cycle begins when huge numbers of adult cicadas emerge in the spring. They mate within a week, and a few days later, the female lays her eggs. She drills into the wood of trees, and inserts up to some 400-to-600 eggs. These eggs hatch up after two to six weeks. The little babies make their way down to the ground (by crawling down, or just dropping), dig their way into the soil with their claws and begin the next phase of their life, feeding on the roots of shrubs and trees for the next 6, 12 or 16 years. The 17-year cicadas are almost fully grown into nymphs by 8 years, but they continue to feed underground until the 17th year when they come out of the soil, and attach themselves to any nearby tree or post. Their shell splits open, the adults emerge and live only for a few weeks before dying.

Now biologists have asked for a long time whether it's just a coincidence that the emergence period of the three species of periodic cicadas (7, 13 and 17 years) are all prime numbers.

One previous theory was that if the cicadas are running on different cycles, and if these cycles are prime numbers, they'll cross over only very rarely. For example, a 13-year cycle and 17-year cycle will meet only every 221 years. That means that both species of cicadas would come out in huge numbers and all have to compete for the same amount of food only once every 221 years. The rest of the time, there would be enough food.

This is a nice theory, but Mario Markus, a physicist from the Max Planck Institute for Molecular Physiology in Germany has come up with a new theory. It's related to periodic predators. Suppose there are some predators (like birds, and the Cicada Killer Wasp) that attack cicadas, and that the cicadas emerge every 12 years. Then the predators that come out every two years will attack them, and so will the predators that come out every 3 years, 4 years and 6 years. But according Mario Markus, "if the cicadas mutate to 13-year cycles, they will survive."

So Markus and his colleagues created a mathematical model. In this mathematical model, if a prey happens to be met by a predator, then it loses. According to this mathematical model, as the years roll by, the length of the cycle increases until the cicadas hit a prime number, and then it stays there.

This model has an unexpected and delightful side effect. It turns out to be a machine, like the Sieve of Eratosthenes 2,300 years ago, that can generate prime numbers. Large prime numbers are rare, and they're difficult to find, but a biological mathematical model like this, based on cicadas, will click through the non-prime numbers, and land on the primes - and that will leave the mathematicians chirping.

A letter from Paul Norris

Hi Dr Karl

I just finished your article on Cicadas and prime numbers. I don't have anything to add, but just for your interest, here is a sequence of photographs I took of a Green Grocer making the change from nymph to adult.

I was surprised by the amount of volition shown by the cicada. I would have thought that bursting out of your skin would be something that just happens regardless of whether you're ready. I found the nymph above the ground early on a Sunday morning. I brought it home to show the kids and possibly treat them to the experience of it emerging. I put it into a tall jar with some moist soil in the bottom (to prevent dehydration), some leaves, and a stick for it to climb up.

Sunday night came and the nymph climbed the stick but just hung there.

Monday morning and no cicada in sight.

Monday night and nymph-on-a-stick again.

Tuesday morning and no cicada.

By this time I was very concerned that the nymph would die and I wondered if perhaps it wasn't ready to emerge. On Tuesday night I came home late from work (about 10pm) only to see the nymph-on-a-stick trick again. I decided to release it. I loosened some soil in the ground and emptied the nymph onto it. To my surprise, rather than digging, the nymph seemed more interested in climbing. I quickly planted the stick from the jar and put the nymph at the base of it. The nymph climbed to the top of the stick and started reaching upward. Clearly the nymph wanted to be higher.

I put the nymph at the base of our little lemon tree and rushed inside for the photographic equipment. By the time I had set up, the nymph was still climbing so I dashed inside, threw some food onto a plate and raced back out. I watched as I ate my dinner and in turn fed the local mosquitoes. From here, the photos tell the story.

I learned two things in particular from this experience:

Firstly the cicada seems to have absolute choice over whether it emerges or not. This one had waited at least three days and then hatched as soon as it was happy. I have in the past lifted paving slabs to discover the tunnels and bodies of cicada nymphs which were clearly ready to hatch. Clearly the nymph had chosen to die rather than hatch. If hatching were involuntary then I would have found the body of a dead cicada with the empty shell nearby. On one occasion I released a nymph from under the paving but it had waited too long and died during the day.

Secondly, it is clear that at some stage the nymph ceases to be a nymph and becomes a cicada wearing a nymph shell. Clearly this must cause complications for breathing, digging your way out of the ground, climbing a tree, feeding, and seeing

Hope you found this as interesting as I did.

Cheers,

Paul Norris

cicada images © Paul Norris

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