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Finding primes in signals is seen as a sign of some kind of intelligence - see e.g. the role of primes in the search for extraterrestrial life (see e.g. here).
This is because there are relatively few examples of numbers that appear in nature because they are prime. One example of the use of prime numbers in nature is as an evolutionary strategy used by cicadas of the genus Magicicada (see e.g. here or here: [1])

My question:
Do you know of any other instances where prime numbers occur in nature? Could you please also give a source/link - and perhaps some background. Thank you.

[1] Goles, E., Schulz, O. and M. Markus (2001). "Prime number selection of cycles in a predator-prey model", Complexity 6(4): 33-38


Edit: Obviously many people misunderstood me. I didn't mean the occurrence of prime numbers just by coincidence - but because they are prime. The cicadas example - although being controversial - at least hints at some kind of evolutionary strategy.

Stefan Kohl
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vonjd
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    While this is a somewhat interesting question, I couldn't help but think that mathematicians perhaps ought to be the last bunch you want to ask about natural phenomena. Good luck on your research. – Willie Wong Oct 24 '10 at 16:43
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    does stuff like "there are (usually) two genders" count as an example of prime-numbers in nature? Or that there are several situations in nature where there is a dichotomy. Or am I thinking in the wrong direction? – Suvrit Oct 24 '10 at 16:44
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    @Suvrit: Interesting comment. The question is: Are there two genders because "2" is prime? I don't think so... (but who knows).
    I would think more into the direction of some sequence of primes or at least bigger primes in nature.     – vonjd Oct 24 '10 at 16:47
  • Gopal Prasad once remarked in class that his daughter had found in her research that ants "use" the Chinese remainder theorem. I can't recall any more than that or her name, unfortunately. – Dylan Moreland Oct 24 '10 at 16:52
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    I think a good way to interpret this question is as a special case of the more general interesting question: "What are examples of mathematical principles that get used by Nature to accomplish some purpose?" – Dick Palais Oct 24 '10 at 16:56
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    I waited a while to see if a genuine applied mathematics question would arise from the discussion, but none seems to. I have thus voted to close. – Andy Putman Oct 25 '10 at 03:06
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    Andy, I'd vote with you to close this. Even edited, the question is not mathematical and the answers are all anecdotal and coincidental. How does primeness or indivisibility by any other integer than $1$ or itself play into the occurence of these incidents of prime numbers in nature? Even with the cicadas, I don't see how the indivisibility of the number of years plays a role. And aren't some families/swarms of cicadas phase-shifted relative to others? – sleepless in beantown Oct 25 '10 at 22:26
  • From http://www.cicadamania.com/faq.html#a30 :There is a cicada in India called Chremistica that emerges in synch with the World Cup (that is known as the World Cup cicada). Last I heard 4 was not a prime number. –  Dec 26 '11 at 14:10

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Cicadas spend most of their lives underground, emerging to mate every $k$ years where the integer $k$ varies from species to species. Biologists have observed that $k$ tends to be a prime number -- for example, there are "13 year" cicadas and "17 year" cicadas. To the best of my knowledge, it is still controversial whether there is an evolutionary reason for this or whether it is just a coincidence.

David E Speyer
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I think this question is better asked as "are there examples of prime SEQUENCES in nature?"

The fact that a starfish has 5 points, or that a sunflower has 23 petals or whatever, proves very little about primes in nature. It could be complete coincidence that the number happens to be prime.

What if starfish are found to have either 3, 5, or 7 points? Does that imply a prime sequence? Maybe its just something about odd numbers > 1.

The Cicada example definitely implies a prime sequence, not just a single number which happens to be prime.

John
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Somewhat longish to be a comment, so here goes:

How about examples like:

  • Polygons in nature? Unfortunately, the most famous one of them is a hexagon ;-)

But I am certain there are several chemical compounds, physical structures such as crystals, and so on that exhibit polygonal structures with prime number of edges / facets --- perhaps because the "primeness" there leads to a physically / chemically more stable configuration.

Is this an acceptable kind of primeness? Perhaps not. It seems to me that the question you pose is almost at a "meta" level!

Suvrit
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Here you go: The Holy Trinity, the Kremlin 5-pointed stars, the Magnificent Seven (and the Seven Samurai), the 7-11 convenience stores, the Devil's Dozen.

Edit 1 Of course, I forgot the 101 Dalmatians.

Edit 2 Also if $\alpha$ is the fine structure constant, then $\alpha^{-1}\approx 137$. But dalmatians are much cuter.

Edit 3 2011 is the year when this question is going to be deleted, I hope. Note that 2011 is prime.

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    I didn't realize the 5-pointed stars on the Kremlin or 7-11 stores are appearing "in nature". :) – KConrad Oct 25 '10 at 09:07
  • This boils down to the question: "What is nature"? For a member of some Amazon tribe, Kremlin stars and 7-11s are not in nature while the fine-structure constant is. –  Oct 25 '10 at 18:15
  • The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is the fifth Mersenne prime. – Stopple Oct 25 '10 at 20:05
  • Right! So 8191 also appears in nature now. By the way a typical soccer team (on a field) consists of 11 players. This is much more natural than some obscure cicadas. On the other hand a hockey team consists of 6 players, which is not prime. That is why soccer is much more popular than hockey. –  Oct 25 '10 at 20:49
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    non-answer, to a non-question, non-mathematical, n'est-ce pas? Or is that the point you were trying to make, Mark? :) And "7-11" is a construct by humans, as much as a hexagonal-walled cell in the honey-comb is a construct by honey-bees, so one is as validly an occurence in nature as is the other. It's numerology, like ascribing importance to the recurrence of digits in a date (with an artificial zero-point in the calendar), or to the relative phase of one object next to another, or marking the passage of $10x$ years, $x \gt 0, x\in \mathbb{Z}$ for birthdays or anniversaries. – sleepless in beantown Oct 25 '10 at 22:32
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    @sleepless in beantown: Humans are part of nature. Just like cicadas, only bigger. Hence every prime number they (we) produce is a product of nature. Yes, the question does not make any mathematical sense. It is amusing to see what kind of questions do not get closed on MO. –  Oct 25 '10 at 23:16
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Nontrival primes: human chromosome number is 23, Sunflower chromosome number is 17, both of which are prime. Most flowers have an odd and often prime number of petals: five is a common number.

I assume you really meant non-trivial primes, where $p>2$ or $p>3$, but you didn't specify that, so let me point out some basic regions where 2 and 3 are prominent. And $2$ and $3$ are very prominent through-out nature, so these are not very special concepts.

Diploid genomes come in copies of 2, and 2 is prime. DNA chains come in duplicate copies, with one side reading in one direction as the sense and the other side being the "inverted carbon copy" and called the nonsense side. When DNA chains are copied, they split apart like a zipper and complementary copies are made on both, yielding two identical doubled-DNA chains.

Three is also a prime number:

The human chromosomal set consists of 23 pairs of DNA chromosomes: $46$ in total, with $44$ of those being non-sex chromosomes, and $2$ of them being sex chromosomes: $XX$ for females and $XY$ for males.

Also, when you convert from 2 copies of these chromosomes to three, for example Trisomy 21, you can end up with Down's syndrome.

If, instead of having two sex chromosomes of the usual type, $XX$ or $XY$, you can have XYY syndrome or XXY syndrome also known as Klinefelter's syndrome. Technically, you could say that these types of sex-chromosome sets are not in the set of the usual two genders of male ($XY$) and female ($XX$), but are actually outside of the two genders. So to answer @vonjd's comment to the question, there are not just 2 genders.

Extra random point: the sunflower's spiral follows Fermat's spiral which at various points, is a prime number. So there must be some sunflowers which have a prime number of sunflower seeds.

Many flowers have five petals, five is prime.

chris
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    Since chimpanzees have 24 pairs of chromosomes, and one of our pairs is very similar to 2 fused chimpanzee chromosomes (one of the predictions of common ancestry which was verified recently), I see no importance in the fact that the number of pairs of human chromosomes is prime. – Douglas Zare Oct 24 '10 at 21:51
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    2 is a highly non-trivial prime. – Cam McLeman Oct 25 '10 at 00:31
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Do you have any evidence of the higher incidence or prevalence of prime numbers in nature, other than happen-stance or co-incidence?

I believe you're mistaken in claiming that prime numbers are used in nature. I would simply state that prime numbers occur in nature; numbers occur in nature and must be used to describe natural phenomena and it is a mere coincidence that some numbers which occur are primes.

While it may be true that the cicada emergence cycle happens to be a prime number, is it always exactly so, or in some cycles is it an even number? What about other insects or varieties of cicadas that may use a non-prime number yearly cycle? As in the examples stated above, humans have twenty-three pairs of chromosomes, chimps have twenty-four. That chimps have 24 pairs does not make the human's 23 pairs less prime, just more likely to be co-incidental. It would appear that the majority of the comments and answers to this question are joke answers, and perhaps this question should be closed if joke answers are the only answers which mathematicians can provide.

If you could come up with a mechanism to collate all of the occurences of numbers in natural phenomena as a multi-set $S$, and then show somehow that the prevalence of prime numbers in this multiset $S$ is greater than would be expected by chance for another randomly selected multiset over the integers $\mathbb{Z}$, then perhaps I could buy the argument that primes are a sign of intelligence in nature.

The argument about SETI searching for prime number like signals from beyond is about the expectation that intelligent creatures would transmit a signal that would appear out of the ordinary and as a non-natural phenomenon. I think it's ridiculous to mix that up with looking for prime numbers in nature (and then jumping to the conclusion that some sort of intelligence or design might be behind it). Natural phenomena, even emergent evolutionary phenomena, have the statistical properties which they have.

Do you have any evidence of the higher incidence or prevalence of prime numbers in nature, other than happen-stance or co-incidence?

PamNDRome
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  • I overstated my point when I said "perhaps I could buy the argument that primes are a sign of intelligence in nature..." --- If it could be shown that prime numbers occured more frequently than would be expected by random draws in that multiset $S$ of numbers occuring in natural phenomena, I would only be willing to buy the possibility that there is some possible phenomenon that underlies the selection of those particular prime numbers in those particular cases. I would not buy there presence as an indication of intelligence in nature or underlying natural phenomena. – PamNDRome Oct 25 '10 at 04:03
  • to andy putnam, I can't comment at the top, but thanks for voting to close. I don't see how mathematicians takes on the coincidental presence of prime numbers in natural phenomena rises to the level of research which ought to be investigated here... – PamNDRome Oct 25 '10 at 04:05
  • @PamNDRome: I think you didn't get the point. I was not implying that the occurrence of prime numbers in nature hinted at some kind of intelligent design. The only thing that you could possibly deduce is the presence of some kind of information processing (and I think the prevalent example of the cicadas could also fall under this category). And as we all know information processing takes place all the time and is esp. accelerated by evolution. – vonjd Oct 25 '10 at 06:54
  • -1 for insinuating that the OP is promoting intelligent design: from an intellectual standpoint, I can hardly think of a more insulting charge to level against someone. In this case, the evidence doesn't seem to back it up. – Pete L. Clark Oct 25 '10 at 14:49
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    The question itself still states "...of the use of prime numbers in nature is as an evolutionary strategy...", and look at my $2$nd paragraph, where I say that prime numbers are not used in nature, they occur in nature. I don't charge anything but point out the short walk from "the use of prime numbers", and it's silly to gang up on me with negative points. The first statement the OP makes is "Finding primes in signals is seen as a sign of some kind of intelligence"... It's bizarre group/gang behaviour; the people at night got my point. What would the relative frequency of primes show? – PamNDRome Oct 25 '10 at 16:27
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    How can you (the OP) make the comment that prime numbers are relatively infrequent in nature without some sort of collating mechanism to calculate the incidence of integer values in general. Group-think is leading people to vote up a question that has not got a sound mathematical basis, or any sound science underneath it which I can discern. Please argue with my point about the relative incidence of primes vs. nonprimes in nature; note that the OP did not have anything factual or scientific to say about it. Pete, look at my comment carefully, I just point out the short jump to ID from this Q. – PamNDRome Oct 25 '10 at 16:32