Comment by learnstats2 | Hacker News Reader

Comment by learnstats2 | Hacker News Reader

Perhaps I have missed something, but the introductory example seems to follow from simple probability and therefore I do not find it mathematically remarkable.

Say, there is a fixed and equal probability that each number ending with 9 and 1 is prime. I could go along with that assumption, although the fact that primes get less likely as you go higher is potentially relevant.

What the authors consider here is starting with a prime ending in 9. So the next potential prime ends in 1. If only because 1 is the next number to be checked, a 1-prime is more likely to appear next than a 9-prime. The probability of that can be calculated, depending on your assumptions, as a geometric sequence. In any case, P(next prime is 1) > P(next prime is 9).

"Most mathematicians would have assumed, Granville and Ono agreed, that a [known] prime should have an equal chance of being followed by a prime ending in 1, 3, 7 or 9" So - I'm a definite nope on that.

This result appears to be exactly what I would have assumed was the case.

This is explicitly ruled out in the article -

> 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.

Ok so the random model "1,3,7,9 mod 10" doesn't fully work, but let's look at what happens mod 30. Large primes have the following possible remainders mod 30: 1, 7, 11, 13, 17, 19, 23, 29. We see that when a prime ends with a 3 then p + 6 (ending in 9) is always an option, but p + 4 (ending in 7) is an option only half of the time. I think that this fully explains why a prime ending with 3 is more likely to be followed by a prime ending in 9. So basically the OP is on the right track, and his random model just needed to be refined a bit.

So you figure two respected mathematicians, and of course everyone they have told about this result so far, missed a completely trivial observation?

No I don't. The article continues:

> The primes' preferences about the final digits of the primes that follow them can be explained, Soundararajan and Lemke Oliver found, using a much more refined model of randomness in primes, something called the prime k-tuples conjecture.

So I guess that my observation is just a special case of this "prime k-tuples conjecture".

What makes you think that?

Are you contesting it, or just curious? You already know that my observation explains the "3 followed by 9" bias. You already know that the mathematicians call the conjecture "a much more refined model of randomness in primes" which is similar to how I described what I was doing. In addition, MathWorld's article on the k-Tuple Conjecture talks about residues mod q, which is similar to what I'm doing when I look at primes mod 10*3. All these elements point at some connection between the k-tuple conjecture and my observation.

Well 'points to a connection' is not the same as 'is a special case of', I'm not an expert on this subject, but looking at the conjecture is about the asymptotic distribution of certain patterns in prime numbers. I don't think that an example like the one you are giving is 'related' except in a hand-wavy vague way that anything dealing with prime numbers and patterns is related to everything else dealing with prime numbers and patterns of primes.

All I meant by "special case" was that "mod 30" isn't the whole story -- more like the most significant correction on top of what the OP said, with other smaller corrections possible, and the entire set of corrections being described by the k-tuple conjecture.

It's amazing how people can be picky and negative on HN. Someone positive would instead congratulate me for making the gist of what the prime k-tuple conjecture says about the biases easily understandable. Oh well.

So you are making comments about pure mathematics. If you want to use imprecise language and not be corrected, you should probably go write a book review or something. In math, precise language and correcting someone or forcing someone to give justification for something is expected and completely usual. It would be bizarre when talking to a mathematician about mathematics if they didn't immediately correct or demand clarification and justification when you say something vague or incorrect or unjustified.

I think the article and quotes have over-sensationalised a piece of maths.

The top comment here on HN does not recognise the 'trivial' point that I have made, hence the need to point it out.

And, the "most mathematicians would have assumed..." quote is probably false.

>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.

That line of thinking is specifically addressed as their first assumption in the article. It's not indicated why but they apparently 'quickly realised this couldn't account for such a bias'.

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