Prime Spirals

Five Guys
Five Guys
Dec 20, 2016 · 7 min read

12–19–16, Lefko

My goal for this FF is to give y’all as much of a math hard-on for prime numbers as Squilldaddy Vic gets any time someone mentions topology.

Let’s talk primes.

What do we know about them? — Because of the nature of primes, mathematicians often refer to them as the “atoms” of numbers, since they can’t really be “broken down” into other numbers.

What don’t we know about them? We have no idea what their distribution is within the set of prime numbers — there’s no pattern for predicting which numbers are prime or for how far apart successive primes are.

This seeming randomness of prime numbers makes them very appealing and mysterious. There have been a few interesting proposals of the value of prime numbers or for how they appear in nature.

With regard to useful man-made ventures, the major use for prime numbers is in cryptography, where they are important because of this “building block” nature of the prime number. Prime numbers are also pretty useful in pseudo-random number generation and fourier transforms, apparently.

On a more whimsical note, in his 1980 documentary series, Cosmos Carl Sagan proposed a use for primes in alien communication. He proposed that we use a signal like:


with prime numbers length strings at the start of messages we send to aliens. His rationale is that this is likely to catch their attention, since there are no (?) natural processes that generate such a sequence, even though it is a sequence we expect alien mathematicians to immediately understand, and thus recognize as a sign of intelligence. (here, the lack of physical application of primes leads to their usefulness!)

Intriguingly, Sagan was able to put primes into practice with alien communication on other fronts too. In 1974, he worked with Dr. Frank Drake of Cornell, and many other physicists, to craft the Arecibo message. This messafe was an interstellar radio message carrying basic information about humanity and Earth and was sent to globular star cluster M13 in the hope that extraterrestrial intelligence might receive and decipher it. The message consisted of 1,679 binary digits arranged rectangularly as 73 rows by 23 columns. 1,679 was chosen because it is a semiprime number (the product of two primes), and 73 and 23 were chosen because they are prime. Apparently, this leaves only two interpretations of the rectangle, and somehow 73 columns by 23 rows creates jumbled nonsense. If the aliens somehow manage to translate this code into graphics, characters, and spaces, they would get this (without color):

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What does it mean? The message consists of seven parts that encode the following (from the top down):[4]

  1. The numbers one (1) to ten (10) (white)
  2. The atomic numbers of the elements hydrogen, carbon, nitrogen, oxygen, and phosphorus, which make up deoxyribonucleic acid (DNA) (purple)
  3. The formulas for the sugars and bases in the nucleotides of DNA (green)
  4. The number of nucleotides in DNA, and a graphic of the double helix structure of DNA (white & blue)
  5. A graphic figure of a human, the dimension (physical height) of an average man, and the human population of Earth (red, blue/white, & white respectively)
  6. A graphic of the Solar System indicating which of the planets the message is coming from (yellow)
  7. A graphic of the Arecibo radio telescope and the dimension (the physical diameter) of the transmitting antenna dish (purple, white, & blue)

“Because it will take 25,000 years for the message to reach its intended destination (and an additional 25,000 years for any reply), the Arecibo message was more a demonstration of human technological achievement than a real attempt to enter into a conversation with extraterrestrials. In fact, the core of M13, to which the message was aimed, will no longer be in that location when the message arrives.[1] However, as the proper motion of M13 is small, the message will still arrive near the center of the cluster.[5]”

There has also been at least one somewhat validated proposal for prime numbers in nature:

For instance, the Cicada-prime connection. Cicadas spent most of their life in “hibernation,” they emerge every 13- or 17-years for one brief shot at sex — then they die. Why these two prime numbers? There’s one interesting (and very debated) hypothesis:

“by emerging every 13 and 17 years, [the famous evolutionary biologist Stephen Jay] Gould argues in his 1977 book, cicadas minimize the chance that their infrequent invasions will sync with the life cycles of birds and other creatures that dine on them.

For example, imagine bird species that wax and wane on a five-year cycle. If cicadas emerged every 10 years, their arrival might coincide with the peak of this avian predator, setting up a pattern that could drive the cicadas to extinction.

By cycling at a large prime number, cicadas minimize the chance that some bird or other predator can make a living off them. The emergence of a 17-year cicada species, for example, would sync with its five-year predator only every (5 multiplied by 17) 85 years.” — the math here has been checked, and it seems to work out!

Now for my favorite interesting prime fact: Ulam Spirals.

In 1963, the mathematician Stanislaw Ulam noticed an odd pattern while doodling in his notebook during a presentation: When integers are written in a spiral, prime numbers always seem to fall along diagonal lines, like so:

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This in itself wasn’t so surprising, because all prime numbers except for the number 2 are odd, and diagonal lines in integer spirals are alternately odd and even. Much more startling was the tendency of prime numbers to lie on some diagonals more than others — and this happens regardless of whether you start with 1 in the middle, or any other number.

Here’s what happens if you extend this spiral out to a 200x200 square:

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Look at how many diagonal line segments you get, and look how many diagonal line segments occur along the same lines…Why do the prime numbers tend to occur in clusters along the diagonals of this spiral? I don’t have a clue. Nor, to my knowledge, does anyone else!

We can take this one step further and look at Sack’s spiral:

To make one, we just write the non-negative integers on a ribbon and roll it up with zero at the center. The trick is to arrange the spiral so all the perfect squares (1, 4, 9, 16, etc.) line up in a row on the right side:

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If we continue winding for a while and zoom out a bit, the result looks like this:

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If we zoom out even further and remove everything except the dots that indicate the locations of integers, we get the next illustration. It shows 2026 dots:

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Bold the primes:

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The primes seem to cluster along certain curves. Let’s zoom out a bit. The following number spiral shows all the primes that occur within the first 46,656 non-negative integers. (For clarity, non-primes have been left out.)

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It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow.

I thought this was cool in itself, but there are entire careers spent decoding this shit. Check out this site for some examples of the crazy is people have discovered on this circle:

Alright I’m out boyos, hang on while I go deal with this prime number boner.


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