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Nov 11, 2021 at 21:49 comment added Lucia See arxiv.org/abs/1603.03720 which discusses such problems for consecutive primes $\pmod q$ for any $q$.
Nov 11, 2021 at 18:58 comment added John Omielan There was a few years ago a series of articles about a shocking "pattern" in primes found, I believe somewhat similar to what you're discussing but in base $10$ instead. The Math SE Mathematicians shocked(?) to find pattern in prime numbers post discusses this.
Nov 11, 2021 at 18:43 history asked Vincent Granville CC BY-SA 4.0