Here is my attempt to work through the math and figure out how "surprising" this result is. Clearly, we should expect that for small primes (< 100e6) it is les

Here is my attempt to work through the math and figure out how "surprising" this result is. Clearly, we should expect that for small primes (< 100e6) it is less likely that a prime ending in K (in base B) will be followed by another prime ending in K - because for that to happen, none of the B-1 numbers in between can be prime. A (very naive) model of the distribution of primes says that every number n has probability p(n) = 1/log(n) of being prime. Assume that a number n ends with a k in base b. Define p = 1/log(n). Then the probability that the next prime ends in k+j is, roughly, q(j) = p * (1-p)^(j-1) * sum_{i=0}^{infinity} (1-p)^(i*b) = p * (1-p)^(j-1) / (1 - (1-p)^b) In this formula, j takes values 1 to b (where j = b represents another prime ending in k). For n ~ 1,000,000 and working in base b, under this model we would expect to see around 6.97% of primes ending in k followed by another prime ending in k, whereas we expect to see 13.7% of primes ending in k+1 (it is apparent how naive the model is, since in fact we never see a prime ending in k followed by a prime ending in k+1, except for 2,3). It would not be hard to extend the model to rule out even primes, or multiples of 3 and 5, but I have not done this. Around n ~ 10^60 the distribution starts to look more equal, as the primes are "spread out" enough that you expect to have long sequences of non-primes between the primes, which blurs out the distribution to be roughly constant. I think this is what the article is getting at when it quotes James Maynard as saying "“It’s the rate at which they even out which is surprising to me". With a naive model of 'randomness' in the primes, you expect to see this phenomenon at low numbers (less then 10^60) and for it to slowly disappear at higher numbers. And indeed, you do see that, but the rate at which the phenomenon disappears is much slower than the random model predicts. I think that is why it is surprising.