December 19, 2005

Why not 17?

Sailesh Ganesh reminded me that I haven't posted a solution to my Cicada question, posed here ten days ago. Why does Magicicada septendecim have a life-cycle as long as 17 years, and what's the significance of the 17?

As I indicated there, I read about this in Simon Singh's "Fermat's Last Theorem." Here's a paraphrase of his explanation:

It's likely that the cicada has a parasite that also has a long life-cycle. Now let's say the parasite's life-cycle is two years long. Any cicada life-cycle that is an even number of years long means the parasite and cicada will "meet" regularly, with disastrous consequences for the cicada. What if the parasite had a three-year life-cycle? Then the cicada must evolve to avoid life-cycles that are multiples of three years long.

Continue with this reasoning, and you see that the cicada's best option is a life-cycle that's long, and prime. Like 17 years. Now, if the parasite has a two-year cycle, cicada and parasite will meet every 34 years. If the parasite has a longer cycle, they will meet even more rarely: if it has a 16-year cycle, onle every 272 years (16 x 17 = 272).

Only two cycle-lengths will make these meets more frequent: a one-year cycle and a 17-year cycle. But if the parasite has a one-year cycle, it will have to live out its life 17 successive times without the cicada to feed on. Not a likely option. If it has to evolve to a 17-year cycle, it would first have to evolve to and through the 16-year cycle, and that means there will necessarily be one of those 272-year gaps.

Both ways, the cicada's long prime life-cycle is its protection.

I tell you, primes are a delight.


Sailesh Ganesh said...

I am going to take a palmful of water and try and drown in it.

Kartik said...

That's just cruel :-) But definitely, primes are one of the most interesting branches of all science.

Dilip D'Souza said...

Sailesh, in that case, may I suggest using 17 cc of water?

wise donkey said...

the parasite's name and its life cycle?

Dilip D'Souza said...

WiseD, Singh suggests that it is because of this long cycle and the odds against fighting its power that the cicada parasite hasn't been found yet. Perhaps it evolved through those lifecycles, hoping every time to find cicadas to feed on, and then went extinct.

Or maybe we're in the 272-year period now, where the parasite is showing up every 16 years and the cicada every 17, and we just don't know that that's the parasite.

wise donkey said...