According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(n) \equiv 0\ (\mathrm{mod}\ m)$) represent $p_1p_2$ infinitely often, with $p_1,p_2$ two distinct primes.

Is there anything known about the distribution of $(p_1,p_2)$, with $(\cdot,\cdot)$ the Legendre symbol? I am working with a specific family of polynomials which appear to have $(p_1,p_2)=\pm1$, with each case occurring infinitely often (and with the same density). Is there any known results along these lines?

Update: if this is unknown, can we at least argue that the density of $(p_1,p_2)=\pm1$ are both strictly positive? That is, that there is no polynomial where *only one* of the options is realised. Or is this out of reach too?

at mosttwo prime factors, not exactly two, so if you really need semiprimes, you might be out of luck. OTOH, if you want something weaker (e.g., that there's some factorization of G(n) = d_1*d_2, not necessarily into primes, with (d_1,d_2) of each sign), then there might be some games you could play, but this would have very a different feel. $\endgroup$ – rlo Nov 13 '18 at 18:52exactlytwo prime factors, notup to; my bad). In my case, prime or semi-prime are both fine, but other composites do not really work. In fact, the existence of a single prime, or a single pair with $(p_1,p_2)=-1$ would suffice, but I guess that's too much to ask... Anyway, thank you again! $\endgroup$ – Delmastro Nov 13 '18 at 22:28