WHY A PRIME NUMBER IS SO IMPORTANT

We have probably heard the cicadas sing during the nights in the countryside. But we may not be aware that cicadas spend most of their life underground, coming out only for about two weeks, during which they sing, reproduce and die. Most surprising in this respect are two species of cicadas that live in the Eastern and Southern United States. These cicadas spend respectively seventeen and thirteen years underground. Then they emerge all at once, in areas hundreds of miles wide, within a couple of days. They lay eggs in trees; these eggs fall to the ground and, when new cicadas are born in the form of nymphs, they enter the ground and attach themselves to roots of the trees. There they slowly develop over the next thirteen or seventeen years (thirteen in the South, seventeen in the Eastern United States), until within a couple of days they are all out again to repeat the cycle. They climb up the trees and begin to 'sing'. Thus there are “cicada years”, rare events that take place every so many years that they come each time as a surprise: suddenly there are myriads of holes in the ground, and cicadas all around singing and mating.

The strange behavior of these periodic cicadas requires an explanation. Why do they wait so long to come out? And why do they do it all at once? A plausible answer involves a mechanism called predator satiation. The idea is the following: When the cicadas are out, predators –particularly birds– feed off them. If the cicadas emerged frequently, these birds would prosper; so much in fact that they could drive the cicadas to extinction. By emerging at very long intervals, the cicadas make sure that no birds will make a living off them; almost every third or fourth generation of birds will have a formidable banquet once in their life. Moreover, when the cicadas come out, they do it by the millions. Thus even if the birds manage to eat a few, there will always be many left to perpetuate the species; hence the name predator satiation.

This mechanism also explains the need for such precise synchronization: if a cicada emerges too early or too late, it will make a sure prey for the waiting birds. By coming at the same time as everybody else, it reduces significantly the probability of ending up completely devoured.

For the lay person, this is probably the end of the mystery of the peculiar way of life of the cicadas. But the mathematically aware person may still be worried: he or she has probably noticed that the life-spans of the cicadas, thirteen and seventeen years, are not just any large numbers: they are also Prime Numbers. One may wonder: is this a coincidence? Or is there any reason why prime numbers should be evolutionary preferable to non-prime ones? This is in fact a matter of debate for biologists: many think that 13 and 17 just came out of chance. However, Stephen Jay Gould provides a beautiful explanation of why prime numbers may have been selected. His argument goes as follows: Many potential predators have 2–5–year life cycles. Such cycles are not set by the availability of cicadas (for they peak too often in year of non-emergence), but cicadas might be eagerly harvested when the cycles coincide. Consider a predator with a life-cycle of five years: if cicadas emerged every 15 years, each bloom would be hit by the predator. By cycling at a large prime number, cicadas minimize the number of coincidences (every 5×17, or 85 years, in this case). Thirteen –and 17–year cycles cannot be tracked by any smaller number.

The cicadas have studied mathematics, especially Prime Numbers, a favourite subject of our genius Srinivasan Ramanujan.