## Prime numbers and a couple of mysteries

If you are mathematically challenged, please don’t panic. This piece is indeed about prime numbers, but my intent here is not to give a proof of some theorem, but only to discuss a couple of interesting points, one each from Physics and Biology, that greatly increased by admiration for prime numbers.

Prime numbers are those numbers that cannot be divided by any number other than 1 and itself. Thus, examples of prime numbers are 7, 11, 13, 17 and so on as they cannot be divided by other numbers, whereas numbers like 12, 14, 15, 16 are all composite (non-prime) numbers.

Suppose you are asked to list out all the even numbers, you will immediately reel them off, 2, 4, 6, 8 and so on till you are asked to stop. If you are asked for the list of squares, you will again start listing out squares 1 (1^2), 4(2^2), 9(3^2), 16(4^2), 25(5^2) and so on. It won’t be as easy as listing out even numbers, but you clearly know how to proceed, and given a calculator/computer, you can go on producing perfect squares, theoretically, forever.

Can you do the same thing with prime numbers? The curious thing about prime numbers is that you cannot keep producing them the way you produce even numbers or perfect squares. They don’t follow a pattern. This might not sound very strange, but it is indeed so. Think of it. Can you think of any property of numbers, that you cannot express as a pattern? It is tough. In fact, generating genuine random numbers (numbers without any pattern) is one of the toughest problems in computer science. To be sure, you can definitely look at each number, try dividing it by all numbers less than it and if it is divisible by none, then you can conclude it is a prime number and not if otherwise. This is the simplest way to do it, though you can find many more efficient algorithms on the Internet. But this process cannot go on for long, since with big numbers, it becomes increasingly tough.

Now if you wonder why this inability to easily generate prime numbers is important, how would you react if I said prime numbers could be a means of communication with extra terrestrial intelligence (ETI)? This idea is that of the great science communicator Carl Sagan. Prime numbers are a candidate for alien-human signaling because no naturally occurring phenomenon can be complex enough to generate prime numbers. Imagine yourself receiving some radio waves from somewhere in space and that they follow a pattern of increasing frequency of consecutive even numbers. Could this be a sign from some intelligence in space? It could be. But the fact that generating even numbers is a simple task greatly increases its probability of occurring naturally, without the need of an intelligent being. Thus the chances of some intelligent brain being the origin of those waves are reduced. But imagine if these waves were of frequencies that are in sync with prime numbers. Because there is no obvious pattern that could generate prime numbers, it would be very tough for nature on its own to produce prime numbers one after another. Such a signal has a very high likelihood of being able to be produced only by an intelligent source. Carl Sagan used this idea in his science fiction novel Contact, where ETI contacts Earth by listing out the series of first 261 prime numbers.

Fascinated as I was by Carl Sagan’s idea, I found Cicadas even more wonderful. Cicadas are plant eating insects. If anything can claim to have generated prime numbers naturally without a thought process, Cicadas would be it. There are numerous species of cicadas, most of them having 2 to 8 year life cycles. But there are some species of Cicadas which have life cycles of 13 and 17 years. That is, these cicadas lie underground as larvae for 12 years (or 16 years), and in the 13th year (or the 17th year) come out as adults, mate, lay their eggs, and disappear. They are so well synchronised, that scientists can predict beforehand when their outbreak will happen and give a warning, so that farmers, among others, can take suitable preventive measures.

Considering the topic in hand, you will see that these two species are special not because their life cycles are longer than usual, but because the lengths of their life cycles are prime numbers. Many biologists have suggested that the life cycles being prime numbers helps the Cicadas, since any predators having a shorter life cycle than these will have lesser likelihood to prey on them. To make that clear, imagine a 15 year life cycle cicada. If its predator has a 3 or 5 year life cycle then the outbreaks of predator and prey may coincide and the prey (cicadas) might go extinct soon. But if it is a 17 year cicada the outbreaks of the shorter life cycle predators, say of length 3 years, has a lesser chance of coinciding with the outbreak of the prey (17 is not divisible by 3, or for that matter any number lesser than 17).

There is no conclusive proof in favour of this hypothesis. In fact this hypothesis leaves many questions unanswered, but the fact that there are 3 species with 13 year life cycles, and 3 more with 17 year life cycles, seems to suggest that there is indeed some advantage in prime numbered life cycle lengths. Some Mathematical models of insects have also suggested that there could indeed be an advantage for prey species to have prime number life cycles. By the way, just to be clear, I am not suggesting that Cicadas do number division in their heads, like we do, and find out prime numbers. If true, it would only mean that evolution favoured Cicadas which have prime numbered life cycles over those with non-prime number lifecycles. The life cycle length is part of the Cicadas’ genetic makeup. But the fact that prime numbers can come up naturally seems to contradict Carl Sagan’s conjecture, though it is still tough to imagine some natural process producing a long list of the consecutive prime numbers.

Nobody, least of all I, knows if ETI, if and when it contacts us, will indeed use prime numbers to say “Are you there?” (In all probability they won’t, since there are many other values they can use, like the digits of pi, which is, 3.1415925654…. Moreover, for all we know, they might just decide to land here directly). Neither do we know, yet, if Cicadas do indeed have an advantage in having a life-cycle length of prime numbers. But the very possibility that these questions could be connected, and that too in such fascinating ways by the simple concept of prime numbers that we learnt in school, is exciting. Is it not?