I watched a (serious, science) TV programme some time ago, in which it was said that a German mathematician worked out the distribution of prime numbers, and that recently it has been discovered that the distribution is echoed, very accurately, in quite a few examples in real life. From vibrations of a quartz ball, to the distribution of parked cars in London (?!) and the distribution of particles in an atom.
This was mentioned almost in passing. So, is this right? Has it been discovered why this correlation exists?
Well, cicadas and other long-lived insects often have life cycles in terms of prime numbers (13 or 17 years) so that their cycles will more rarely synch up with their predators.
Sorry - I don’t get this. I understand life cycles to be stages of development, so are you saying that they take 13/17 years to mature? If so, I’d say that they’re at risk for a large number of years. I thought insect predators are often able and willing to eat their prey at all sorts of stages of development.
Many people are confused when they hear 17 year life cycle, but there is no need to wonder. Most of those years are spent in hibernation where the predatory(predators are alien trophy hunters from space) animals will not get them.
Read up on periodiocal cicadas and all will be clear.
The species uses a survival technique called predator flooding. In any given area where the live, once every few years they hatch en masse, live a month, breed, then die en masse, leaving behind eggs buried in trees. The same number of years later those eggs all hatch at once. Lather rinse repeat.
When they hatch, the local explosion of cicadas is so vast that even though 80% are eaten, they still have plenty to produce the next generation. And because they appear in any one location so rarely, no predators are explicitly evolved to eat them since those predators can’t wait years for their next meal.
Prime numbers have nothing to do with this. But …
Different populations of cicadas have different dormancy periods. If two populations overlap geographically, there would be a surival advantage to them not appearing at the same time. And sure enough, the two populations common in the US use a 13 year and a 17 year cycle. As these numbers are prime, this produces a minimum liklihood of both appearing in any given year in any given location.
Not exactly.
The eggs hatch soon after being laid, and the young Cicadas (nymphs) fall to the ground, where they bury themselves, and spend the next 17 (or 13) years feeding on tree roots. This makes Cicadas one of the longest-lived insects.
Why would this be? Given that the tactic is based around overwhelming the predators in the first place, wouldn’t it be even better to overwhelm the predators with large numbers of some other species? It seems to me that the same evolutionary pressures that cause one cicada to emerge at the same time as all its siblings would also cause one cicada species to emerge at the same time as all the other species.
Isn’t the prime number number of years in the cicada cycle, “because” (pardon the teleology), there is a natural predator with a life cycle of a different prime number of years. So, it’s not that two populations of cicadas don’t overlap, so much as they and their predators don’t.
hogarth, if what the OP was referring to was the Prime Number Theorem, could you explain to us what it has to do with the following in the OP:
> . . . a German mathematician worked out the distribution of prime numbers, and
> that recently it has been discovered that the distribution is echoed, very
> accurately, in quite a few examples in real life. From vibrations of a quartz ball,
> to the distribution of parked cars in London (?!) and the distribution of particles
> in an atom . . .
I’m not convinced that this is what the OP was talking about. I’m really not sure what he’s talking about, to be honest. I’m pretty sure, on the other hand, that it has nothing to do with the supposed tendency for long-lived insects to have life cycles of prime numbers of years.
IIRC, when the populations appear in the same year, they interbreed, giving not just 13 and 17 year cicadas, but years in between. This dilutes their numbers, so it’s best that that not happen often.
Here’s a cite for the alternative negative consequence of hybridization* theory. It sounds like they don’t really know the cause. Both your cite and mine are for numerical simulations showing they can explain the prime numbers. Neither proves that’s what happened.
That’s a Wired article. Here’s the paper it’s about.
The TV program the OP saw was probably talking about Riemann. Riemann showed that the distribution of primes is related to the locations of the zeroes of what’s now called the Riemann zeta function. This is already kind of surprising, since the zeta function as commonly defined doesn’t involve primes in any obvious way (there is a way of rewriting it due to Euler which at least makes this a little more clear). The Riemann hypothesis is an unsolved conjecture about the location of these zeroes which in turn would imply bounds on the distribution of primes; in particular, it would imply the Prime Number Theorem and give nice bounds on the error term. (The Prime Number Theorem is not originally due to Riemann; Chebyshev is the one most commonly mentioned in connection with PNT, though he only proved a weaker version.)
The relation to other distributions is probably the sort of thing described here: the distribution of zeroes of the zeta function looks similar to distributions of random matrix eigenvalues. For example, this paper mentions the spectrum of energy levels of atomic nuclei. This is the sort of thing that’s easy to hype, but I don’t know how significant it really is.
An explanation of the Riemann hypothesis and the relation of zeta to the distribution of primes is probably beyond the scope of a single post. I’d recommend a recent book (Derbyshire, Prime Obsession) makes a pretty good attempt at explaining these results to a lay reader. This book has lots of equations, but Derbyshire tries to explain the notation and meaning starting from a pretty basic level, and the book is organized pretty well so that you can probably understand at least the first two-thirds of the book, including the Euler-product formula for the zeta function, as long as you have a reasonable high-school algebra background. It’d probably require a lot of patience to make it to the end with that background, but I don’t really know. There’s at least one other recent popular book on the subject (Stalking the Riemann Hypothesis). I haven’t read it, but I have the impression that it is less technical than the Derbyshire.
If this is a reply to my post: The Prime Number Theorem and Ulam spiral are not very obviously related; one is concerned with the average, asymptotic behavior of the primes while the other is concerned with local, detailed behavior. The lines appearing in the Ulam spiral are related to the existence of particular quadratic polynomials which produce substantially more, or fewer, primes than average. The Prime Number Theorem is concerned with the asymptotic density of primes, not with the exact locations of particular primes. (Plot an Ulam spiral; notice that the prime dots get more sparse the farther you get from the center. The rate at which these dots get more sparse, on average, is what the Prime Number Theorem is talking about.)
On the other hand, I don’t think the lines in the Ulam spiral are well understood, so it can’t really be said that there isn’t a connection, either.