According to [Lemke-Oliver][1], irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(n) \equiv 0\ (mod\ m)$) 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? [1]: http://math.tufts.edu/faculty/rlemkeoliver/papers/04-Quadratic.pdf

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?

According to Lemke-Oliver, irreducible quadratic polynomials (with positive leading coefficient and with $\rho(2)<2$) 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 $(p_1/p_2)=\pm1$, both options infinitely often (and with the same density). Is there any known result along these lines?

According to [Lemke-Oliver][1], irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(n) \equiv 0\ (mod\ m)$) 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? [1]: http://math.tufts.edu/faculty/rlemkeoliver/papers/04-Quadratic.pdf