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Peculiar pattern found in ‘random’ prime numbers: Last digits of nearby primes have ‘anti-sameness’ bias.

user-inactivated  ·  2642 days ago  ·  link  ·
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o11c  ·  2642 days ago  ·  link  ·

Wait, did people not know this? I thought it was obvious.

bioemerl  ·  2642 days ago  ·  link  ·

Why do you say this is obvious?

o11c  ·  2642 days ago  ·  link  ·

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.