The primes are equidistributed in the residue classes $1(\!\!\!\mod{4})$ and $3(\!\!\!\mod 4)$. We also know (for example, by Rubinstein-Sarnak) that the patterns cannot be eventually alternating, i.e. there exists no $n_0\in \mathbb N$ such that for all $n>n_0$ one has $$p_{n+1}\equiv -p_n \mod 4,$$ where $p_n$ is the $n$-th smallest prime. My question is whether there exists an easy proof of this, (maybe unconditional?)
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4See here for a simple proof, by MathOverflow's lucia: https://mathoverflow.net/questions/168378/the-prime-numbers-modulo-k-are-not-periodic – so-called friend Don Feb 20 '22 at 14:47
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it is great he foresaw my question! – Dr. Pi Feb 21 '22 at 11:34