The article is too vague to assess how interesting the claims are, sadly.
It's too bad; I think it wouldn't have detracted from the article to put some more math in. It's not on the face of it at all surprising that sequential primes are more likely to be close to each other modulus any number (3, or 10, or what have you), than they are to be far apart.
By way of analogy, a train comes at 1:09pm. Trains come about every 5 minutes between 1 and 2 pm, and only on odd numbers. If you simulate a bunch of random 'next trains', 1 is much more likely than 9 because P(9) approx = !P(1,3,5,7). This is true for all bases.
I think what you'd need to be able to say to say something interesting is 1) calculate odds of finding the next prime. 2) Randomly generate numbers with a similar distribution to that of prime occurrence in that range using the Prime Number Theorem at the very least (1 / log(n) probability roughly). 3) check final digits and compare to actual distribution of final digits.
If those numbers are very different, then you have in fact found some underlying structure. But the article doesn't hit very hard on this angle, and its hard for (probably) any of us to say just thinking about it with minimal data whether or not there's structure.
> Lemke Oliver and Soundararajan’s first guess for why this bias occurs was a simple one: Maybe a prime ending in 3, say, is more likely to be followed by a prime ending in 7, 9 or 1 merely because it encounters numbers with those endings before it reaches another number ending in 3. For example, 43 is followed by 47, 49 and 51 before it hits 53, and one of those numbers, 47, is prime.
> But the pair of mathematicians soon realized that this potential explanation couldn’t account for the magnitude of the biases they found. Nor could it explain why, as the pair found, primes ending in 3 seem to like being followed by primes ending in 9 more than 1 or 7. To explain these and other preferences, Lemke Oliver and Soundararajan had to delve into the deepest model mathematicians have for random behavior in the primes.
They did mention this, but they didn't talk real numbers. And, my second point is (I think) slightly more subtle -- the probability distributions need to be considered, not just the counting upward angle.
As I'm writing this out, I'm a little less sure that this would matter, but I'll leave the comment out for the sake of discussion. :)
I don't know why you're nitpicking. The article's written for a more general audience that may be interested in the property, but not necessarily the nitty-gritty math behind it.
For your second point, I don't think there's anything wrong with a paper announcing they found something interesting, even if they haven't completely analyzed every aspect of it. Getting the info out early lets a wider audience look at it, and opens their current research up to scrutiny.
The article is too vague to assess how interesting the claims are, sadly.
By my reading, the article seems to state the key import of the finding quite clearly:
"This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random numbers."
However this statement:
If you simulate a bunch of random 'next trains', 1 is much more likely than 9 because P(9) approx = !P(1,3,5,7). This is true for all bases.
I'm afraid I don't follow at all. (Do you really mean we should expect that P(1|1) > P(1|9), for either random trains or for subsequent primes? Say wha?)
That said, perhaps you might want to skip straight to the arxiv article itself, or perhaps do some experiments on your own. It's definitely not hard to generate a non-"minimal" amount of data (out to the first few million primes or so) on one's laptop, these days.
It's too bad; I think it wouldn't have detracted from the article to put some more math in. It's not on the face of it at all surprising that sequential primes are more likely to be close to each other modulus any number (3, or 10, or what have you), than they are to be far apart.
By way of analogy, a train comes at 1:09pm. Trains come about every 5 minutes between 1 and 2 pm, and only on odd numbers. If you simulate a bunch of random 'next trains', 1 is much more likely than 9 because P(9) approx = !P(1,3,5,7). This is true for all bases.
I think what you'd need to be able to say to say something interesting is 1) calculate odds of finding the next prime. 2) Randomly generate numbers with a similar distribution to that of prime occurrence in that range using the Prime Number Theorem at the very least (1 / log(n) probability roughly). 3) check final digits and compare to actual distribution of final digits.
If those numbers are very different, then you have in fact found some underlying structure. But the article doesn't hit very hard on this angle, and its hard for (probably) any of us to say just thinking about it with minimal data whether or not there's structure.