Peculiar pattern found in ‘random’ prime numbers: Last digits of nearby primes have ‘anti-sameness’ bias. · 4
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At first, I thought this was crazy, because I thought they only meant base 10, but it looks like the authors are conjecturing that it's every base. That seems less interesting. Would another way to say this be: "Consecutive primes are unlikely to have the same value modulo n, where n is every integer"? Because that makes it sound boring and obvious. Whatever integer(s) k that they do have the same value (mod k) for are bound to outnumbered by those integers that they do not share the same value for.
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I think this is the k-tuple conjecture that the article talks about, but: Look at the odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ... every third one is a multiple of 3, every fifth one is a multiple of 5, every seventh one is a multiple of 7, etc. The same exact pattern holds with any stride: 1, 11, 21, 31, 41, 51, 61, 71, 81, 91 ... because if α is a multiple of β, then α+βγ is obviously also a multiple of β. I originally thought of this in the context of twin primes. Since every third odd number is a multiple of 3, twin primes can only happen in the gaps, but the gaps are more likely to get hit by being a multiple of some other odd prime the further you go. Now, with a stride of 10 (but still considering multiples of 3), we know that adjacent primes are much more likely if they don't have to cross the multiple of 3 gap (or rather, don't have to cross it very often). But, at the same time, there are many other smaller strides which might generate a prime even closer. While each of them is also vulnerable to the 3, their strides are smaller so they are more likely to have something* escape.