For any prime $p\equiv\pm1\pmod5$, we can write $p$ uniquely in the form $x_p^2+3x_py_p+y_p^2$ with $x_p,y_p\in\mathbb Z$ and $x_p>y_p>0$.

I have the following conjecture.

**Conjecture.** We have
$$\lim_{N\to+\infty}\frac{\sum_{p\le N\atop p\equiv\pm1\pmod5}(3x_p^2+2x_py_p)}
{\sum_{p\le N\atop p\equiv\pm1\pmod5}(3y_p^2+2x_py_p)}=4.\tag{$*$}$$

I have checked this conjecture via computation. For example, \begin{align}\sum_{p\le 2\times10^9\atop p\equiv\pm1\pmod5}(3x_p^2+2x_py_p)=&88916125007243531, \\\sum_{p\le 2\times10^9\atop p\equiv\pm1\pmod5}(3y_p^2+2x_py_p)=&22228898519387861, \\\frac{88916125007243531}{22228898519387861}\approx&4.000023885.\end{align}

Similarly, any prime $p\equiv1\pmod3$ can be written uniquely as $u_p^2+u_pv_p+v_p^2$ with $u_p,v_p\in\mathbb Z$ and $u_p>v_p>0$, and I observe that $$\lim_{N\to+\infty}\frac{\sum_{p\le N\atop p\equiv1\pmod3}(u_p^2+2u_pv_p)} {\sum_{p\le N\atop p\equiv1\pmod3}(v_p^2+2u_pv_p)}=2.$$ This is essentially provable in view of Hecke's equidistribution theorem.

**QUESTION.** How to prove the equality $(*)$?

notan imaginary quadratic field. $\endgroup$ – Zhi-Wei Sun Dec 23 '19 at 4:08