Let $p$ be a prime with $p\equiv1\pmod3$. It is well known that we can write $p$ uniquely as $a_p^2+a_pb_p+b_p^2$ with $a_p,b_p\in\mathbb Z$ and $a_p>b_p>0$. Note that $a_b\not \equiv b_p\pmod3$. Thus $p$ can be written uniquely as $c_p^2+c_pd_p+d_p^2$ with $c_p,d_p\in\{1,2,3,\ldots\}$ and $c_p-d_p\equiv1\pmod3$.
Motivated by Question 348562 on MathOverflow, via computation I have formulated the following conjectures concerning primes $p\equiv1\pmod3$.
Conjecture 1. We have $$\sum_{p\le N\atop p\equiv1\pmod3}\left(\frac{a_p-b_p}3\right)=O(\sqrt N)\ \ \quad \text{for}\ N\ge1,$$ where $(-)$ is the Jacobi symbol.
Conjecture 2. $$\frac{\sum_{p\le N\atop p\equiv1\pmod3}c_p}{\sum_{p\le N\atop p\equiv1\pmod3}d_p}=1+O\left(\frac1{\sqrt N}\right)\quad\text{for}\ N\ge7.$$
Conjecture 3. $$\lim_{N\to+\infty}\frac{\sum_{p\le N\atop p\equiv1\pmod3}c_p}{\sum_{p\le N\atop p\equiv1\pmod3}b_p}=1+\frac{\sqrt3}2.$$
It is well known that any prime $p\equiv1\pmod4$ can be written uniquely as $x_p^2+y_p^2$ with $x_p,y_p\in\mathbb Z$ and $x_p>y_p>0$. Also, $p$ can be written uniquely as $u_p^2+v_p^2$ with $u_p,v_p\in\{1,2,3,\ldots\}$ and $u_p-v_p\equiv1\pmod4$. For primes $p\equiv1\pmod4$, I also have conjectures similar to Conjectures 1-3 for primes $p\equiv1\pmod3$.
Conjecture 4. We have $$\sum_{p\le N\atop p\equiv1\pmod4}\left(\frac{-1}{x_p-y_p}\right)=O(\sqrt N)\ \ \quad \text{for}\ N\ge1,$$ where $(-)$ is the Jacobi symbol.
Conjecture 5. For any $\varepsilon>0$, we have $$\frac{\sum_{p\le N\atop p\equiv1\pmod4}u_p}{\sum_{p\le N\atop p\equiv1\pmod4}v_p}=1+O(N^{-1/2+\varepsilon})\quad\text{for}\ N\ge5.$$
Conjecture 6. $$\lim_{N\to+\infty}\frac{\sum_{p\le N\atop p\equiv1\pmod4}u_p}{\sum_{p\le N\atop p\equiv1\pmod4}y_p}=1+\frac{\sqrt2}2.$$
QUESTION. How to solve the above conjectures?
Your comments are welcome!
