# Distribution of prime numbers and real life
**URL:** https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881
**Category:** Factual Questions
**Created:** [February 13, 2010, 5:29pm UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881 "2010-02-13T17:29:32Z")
**Posts on this page:** 18
**Page:** 1
### Author:  [@NineToTheSky](https://boards.straightdope.com/u/NineToTheSky)
#### Post date: [February 13, 2010, 5:29pm UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/1 "2010-02-13T17:29:32Z")
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?
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### Author:  [@John\_DiFool](https://boards.straightdope.com/u/John_DiFool)
#### Post date: [February 13, 2010, 7:27pm UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/2 "2010-02-13T19:27:44Z")
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.
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### Author:  [@NineToTheSky](https://boards.straightdope.com/u/NineToTheSky)
#### Post date: [February 13, 2010, 7:40pm UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/3 "2010-02-13T19:40:22Z")
> [@John\_DiFool](#):
>
> 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.
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### Author:  [@ZPhobiaZ](https://boards.straightdope.com/u/ZPhobiaZ)
#### Post date: [February 13, 2010, 7:57pm UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/4 "2010-02-13T19:57:37Z")
> [@NineToTheSky](#):
>
> 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(preda **tors** are alien trophy hunters from space) animals will not get them.
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### Author:  [@NineToTheSky](https://boards.straightdope.com/u/NineToTheSky)
#### Post date: [February 13, 2010, 8:02pm UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/5 "2010-02-13T20:02:32Z")
> [@ZPhobiaZ](#):
>
> 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(preda **tors** are alien trophy hunters from space) animals will not get them.
They may be aliens, but sometimes they’re not:
> [@](#):
>
> \*pred·a·tor (prd-tr, -tôr)
> n.
>
> 1. An organism that lives by preying on other organisms.
> 2. One that victimizes, plunders, or destroys, especially for one’s own gain.
> [Latin praedtor, pillager, from praedr, to plunder; see predatory.]
>
> The American Heritage® Dictionary of the English Language, Fourth Edition copyright ©2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.
> predator [ˈprɛdətə]
> n
>
> 1. (Life Sciences & Allied Applications / Zoology) any carnivorous animal
> 2. a predatory person or thing
>
> Collins English Dictionary – Complete and Unabridged 6th Edition 2003. © William Collins Sons & Co. Ltd 1979, 1986 © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003
> predator (prd-tr)
> An animal that lives by capturing and eating other animals.\*
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### Author:  [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)
#### Post date: [February 13, 2010, 10:28pm UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/6 "2010-02-13T22:28:14Z")
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.
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### Author:  [@beowulff](https://boards.straightdope.com/u/beowulff)
#### Post date: [February 13, 2010, 10:59pm UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/7 "2010-02-13T22:59:42Z")
> [@LSLGuy](#):
>
> 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.
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### Author:  [@hogarth](https://boards.straightdope.com/u/hogarth)
#### Post date: [February 13, 2010, 11:04pm UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/8 "2010-02-13T23:04:18Z")
Here’s a link to a description of the Prime Number Theorem (that the original poster is presumably referring to).
> **[Prime number theorem](https://en.wikipedia.org/wiki/Prime_number_theorem)**
>
> In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).
> The first such distribution foun...
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [February 13, 2010, 11:55pm UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/9 "2010-02-13T23:55:39Z")
> [@](#):
>
> 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.
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.
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### Author:  [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)
#### Post date: [February 14, 2010, 12:05am UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/10 "2010-02-14T00:05:27Z")
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.
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### Author:  [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)
#### Post date: [February 14, 2010, 12:20am UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/11 "2010-02-14T00:20:27Z")
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.
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### Author:  [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)
#### Post date: [February 14, 2010, 12:49am UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/12 "2010-02-14T00:49:36Z")
> [@Chronos](#):
>
> 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.
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.
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [February 14, 2010, 1:49am UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/13 "2010-02-14T01:49:09Z")
…Or possibly waste all their time and energy breeding with other cicadas with which they’re not interfertile. Yeah, I guess that makes sense.
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### Author:  [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)
#### Post date: [February 14, 2010, 1:52am UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/14 "2010-02-14T01:52:55Z")
> [@Chronos](#):
>
> …Or possibly waste all their time and energy breeding with other cicadas with which they’re not interfertile. Yeah, I guess that makes sense.
Again, it’s because their _predators_, with a different prime number cycle, are therefore less likely to emerge at the same time. [Cite (from Nature)](http://www.mariomarkus.com/save/nature_cicada_2.html)
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### Author:  [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)
#### Post date: [February 14, 2010, 3:34am UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/15 "2010-02-14T03:34:20Z")
> [@KarlGauss](#):
>
> Again, it’s because their _predators_, with a different prime number cycle, are therefore less likely to emerge at the same time. [Cite (from Nature)](http://www.mariomarkus.com/save/nature_cicada_2.html)
Here’s a cite for the alternative [negative consequence of hybridization](http://www.wired.com/wiredscience/2009/05/primecicadas/)\* 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](http://www.pnas.org/content/early/2009/05/15/0900215106.abstract)’s the paper it’s about.
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### Author:  [@Omphaloskeptic](https://boards.straightdope.com/u/Omphaloskeptic)
#### Post date: [February 14, 2010, 3:34am UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/16 "2010-02-14T03:34:50Z")
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](http://en.wikipedia.org/wiki/Riemann_zeta). 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](http://www.americanscientist.org/issues/pub/the-spectrum-of-riemannium): 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.
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### Author:  [@Lumpy](https://boards.straightdope.com/u/Lumpy)
#### Post date: [February 14, 2010, 3:41am UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/17 "2010-02-14T03:41:13Z")
Is this related to the [Ulam Spiral](http://en.wikipedia.org/wiki/Ulam_spiral), an indication that the distribution of primes has some sort of pattern?
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### Author:  [@Omphaloskeptic](https://boards.straightdope.com/u/Omphaloskeptic)
#### Post date: [February 14, 2010, 4:29am UTC](https://boards.straightdope.com/t/distribution-of-prime-numbers-and-real-life/528881/18 "2010-02-14T04:29:30Z")
> [@Lumpy](#):
>
> Is this related to the [Ulam Spiral](http://en.wikipedia.org/wiki/Ulam_spiral), an indication that the distribution of primes has some sort of pattern?
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.