**QUESTION.** Are my following conjectures true? How to prove them?

**Conjecture 1.** For each prime $p>100$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that 
$$\left(\frac ap\right)=\left(\frac bp\right)=\left(\frac cp\right)=1\ \mbox{and}\ a^2+b^2=c^2,$$
where $(-)$ denotes the Legendre symbol.

**Conjecture 2.** For each prime $p>50$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that 
$$\left(\frac ap\right)=\left(\frac bp\right)=\left(\frac cp\right)=-1\ \mbox{and}\ a^2+b^2=c^2.$$

**Conjecture 3.** For each prime $p>32$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that 
$$\left(\frac ap\right)=\left(\frac bp\right)=1,\ \left(\frac cp\right)=-1\ \mbox{and}\ a^2+b^2=c^2.$$

**Conjecture 4.** For each prime $p>72$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that 
$$\left(\frac ap\right)=\left(\frac bp\right)=-1,\ \left(\frac cp\right)=1\ \mbox{and}\ a^2+b^2=c^2.$$

*Remark.* I formulated these conjectures in 2015, see http://oeis.org/A260911 . Perhaps, it is practical to prove them.