# Heuristic on the percentage of "lesser" twin primes congruent to 1 modulo 4

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 $$p \equiv 3 \mod 4$$.

• 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 ? Commented Jan 18, 2022 at 8:35
• 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. Commented Jan 26, 2022 at 19:37