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Refer to the smaller prime in a twin prime pair as a lesser twin prime. As an odd number, a lesser twin prime is congruent to either 1 or 3 mod 4: is anyone aware of existing heuristics which predict the percentage of lessser twin primes congruent to 1 mod 4? I checked using Python that 50,013 of the first 100,000 lesser twin primes are congruent to 1 modulo 4, which hints at the rather satisfying answer of 50%.

In case it is helpful, I'll say a bit about what inspired this question. I am currently reading Washington's Introduction to Cyclotomic Fields which involves a lot of expressions involving $(p-1)/2$ and $(p+1)/2$ for an odd prime $p$ (for example, as subscripts of Bernoulli numbers). I started thinking about how a twin prime pair $(p,q)$ has $(p+1)/2 = (q-1)/2$ and thus that it could be interesting to compare expressions involving $(p+1)/2$ for $p \equiv 1 \mod 4$ with expressions involving $(p-1)/2$ for $p \equiv 3 \mod 4$.

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  • $\begingroup$ I only know that the lead in the "prime-race" between the primes of the form $4k-1$ and the primes of the form $4k+1$ switches infinite many often despite Chebychev's bias (in numerical analysis the primes of the form $4k-1$ usually seem to be preferred). I do not know whether similar results are known in the case of twin primes, but I see no reason that the frequency should not be $1/2$ in the long run. Are you interested in the count in larger ranges ? $\endgroup$
    – Peter
    Commented Jan 18, 2022 at 8:35
  • $\begingroup$ Ah hmm I'd forgotten about Chebychev's bias; I would be interested to know if something similar holds for lesser twin primes. And I have no specific interest in a particular direction. My question was an idle observation I wanted to pose just to see if there existed something known. I do think that something like counts in short/long ranges could also be interesting. $\endgroup$
    – xion3582
    Commented Jan 26, 2022 at 19:37

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