# Patterns of Primes

• The mathematicians made the discovery while performing a randomness check on the first hundred million primes. Within that set, a prime ending in 1 is followed by another ending in 1 just 18.5 percent of the time. That shouldn’t happen if they were truly random—we should expect to see this happen 25 percent of the time (keep in mind that primes can only end in 1, 3, 7, or 9). So while this isn’t a pattern—it’s also not totally perfectly random. In terms of the back-to-back distribution of the other numbers, primes ending in 3 and 7 appeared 30 percent of the time, and consecutive 9s appears about 22 of the time. Importantly, this observation has nothing to do with the base-10 numbering system, and is something inherent to primes themselves.

I find this very unpersuasive. They looked at an infinitesimal sample of primes and are trying to generalize an observation. Note that the article points this out a little later:

“As the numbers get larger, though, it sounds like this is less constraining, causing it to get closer and closer to an equal distribution of ending digits— which makes intuitive sense, since the primes get rarer and rarer,” he said.

I mean...it's an interesting pattern, but hardly surprising that a tiny sample ends up not being what you thought it would be. And yes, obviously the statement "keep in mind that primes can only end in 1, 3, 7, or 9" is wrong.

• "keep in mind that primes can only end in 1, 3, 7, or 9"

if you exclude 2, and 5 from that statement as being "trivial" then it is accurate. ay even number has two as a divisor, so the only even number prime is 2, and any number ending in 5 must be divisible by 5 and so 5 is the only prime with a ones digit of 5.

it gets a lot more complicated from there, but those two at least are trivial, leaving you with 4 other end digits for primes, which if they were truely random you would expect to see account for about 25% each of the primes.

i'll agree with you that it's not necessarily indicative of a larger pattern, though it is interesting and i would be interested in seeing more research on the matter.

all i can say for certain right now is.... Primes are WEIRD!

take Ulam Spirals for instance:

there is clearly some organization there, some diagonals have a much denser population of primes than others, yet no one has been able to prove a reason why, just to say that "hey, there's a lot of primes on this diagonal, so let's look along the diagonal for new primes!"

• "keep in mind that primes can only end in 1, 3, 7, or 9"

if you exclude 2, and 5 from that statement as being "trivial" then it is accurate. ay even number has two as a divisor, so the only even number prime is 2, and any number ending in 5 must be divisible by 5 and so 5 is the only prime with a ones digit of 5.

So you adjust your wording: "keep in mind that multi-digit primes can only end in 1, 3, 7, or 9".

• What about 11? or 10? Those are both primes. 101 is also a prime. And 111.

• 10

What? It's...

ohhh.

There are 10 kinds of primes in this world...

• there is clearly some organization there, some diagonals have a much denser population of primes than others,

The trouble with that argument is that the 'diagonals' are determined by how wide you draw the picture...

• So are B and D, but they always get ignored, poor sods.

• You can't feel too sorry for them though; they play starring roles in cryptography

• And checksum codes, they love them for that.

• The trouble with that argument is that the 'diagonals' are determined by how wide you draw the picture...

actually no, the diagonals are independant of the picture because of how ulam spirals are constructed

you build them this way:

so the diagonals are independant of the size of the spiral.

really the Ulam Spiral, is just a neat visualization of primes, a way to show that the primes are not randomly placed, but they are also not very predictably placed either. and the densely populated diagonals, which each have a specific mathematical formula for generating numbers along the spiral, are nothing more than a way of creating prime candidates that have a slightly better chance than average of being prime.

• a way to show that the primes are not randomly placed, but they are also not very predictably placed either.

"Not predictably (given some set of knowledge)" seems like the very definition of random, though.

• "Not predictably (given some set of knowledge)" seems like the very definition of random, though.

There are patterns in primes, far more than we would expect in a truly random set. but we don't know enough of the rules of that organization to predict with anything approaching certainty where primes will be.

Another famous example: Mersenne Primes. These are primes of the form . We know a lot of interesting things about these numbers. For instance we can fairly trivially prove that if is prime then `n` must be prime.*

As for the ulam spirals... I'll let James Grime, a PHD holding matematician try to explain it better while i go looking for more props.
Prime Spirals - Numberphile – 09:06
— Numberphile

*a full proof of this and other neat properties of mersenne primes is available on wikipedia

• There are patterns in primes, far more than we would expect in a truly random set.

Ah, the No True Random defense. Your still wrong. That the human brain, pattern detector par excellence, can detect patterns in randomness does not show that said randomness isn't random.

Another famous example: Mersenne Primes. These are primes of the form . We know a lot of interesting things about these numbers. For instance we can fairly trivially prove that if is prime then n must be prime.*

But not all Mersenne numbers are primes. Obviously, we know exactly how prime numbers "come about," but that's not the same thing as saying there's a predictable pattern.

• But not all Mersenne numbers are primes.

this is true, in fact mersenne primes are relatively rare compared to what one would expect from a random sample. we look for them because they are a lot easier for computers to work with.

Filed under: no meaningful input regarding primes

• Importantly, this observation has nothing to do with the base-10 numbering system, and is something inherent to primes themselves.

Yes it does. For example:

What about 11? or 10? Those are both primes. 101 is also a prime. And 111.

There's never going to be a prime ending in anything other than `1` (except `10`) in that base.

• Ah, the No True Random defense. Your still wrong. That the human brain, pattern detector par excellence, can detect patterns in randomness does not show that said randomness isn't random.

Why is there always someone who says this whenever the Ulam spiral is posted? There are real, clear patterns there, you can measure it if you want. Or just look at some actual random dots and see that they look completely different.

• Why is there always someone who says this whenever the Ulam spiral is posted?

Because I'm right?

I think you're substituting some other definition of random that the one I used in my argument.

• Primes aren't random, because they're not selected but a fixed set. Insert XKCD `return 4` meme here

A random selection can certainly have patterns If I have a random number generator that rejects even output, or doubles each number returned, then I have some random numbers with the pattern that they are all even.

• Primes aren't random, because they're not selected but a fixed set. Insert XKCD return 4 meme here

:sigh:

I think you're substituting some other definition of random that the one I used in my argument.

"Not predictably (given some set of knowledge)" seems like the very definition of random, though.

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