[stromgo]

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.

[justin_vanw]

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

[learnstats2]

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.

[stromgo]

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".

[justin_vanw]

What makes you think that?

[stromgo]

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.

[justin_vanw]

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.

[stromgo]

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.