In Question 337879, I conjectured that for any prime $p\equiv3\pmod4$ the equation $$3x^2+4\left(\frac p3\right)=py^2\tag{1}$$ always has integer solutions, where $(\frac p3)$ is the Legendre symbol. Motivated by this, here I pose the following conjectures.

**Conjecture 1.** For any prime $p\equiv13\pmod{24}$, the equation
$$3x^2+1=py^2\tag{2}$$
always has integer solutions.

For example, when $p=829$ the least positive integer solution of $(2)$ is $$(x,\,y)=(1778674,\,106999).$$

**Conjecture 2.** For any prime $p\equiv3\pmod4$, the equation
$$2x^2-py^2=\left(\frac 2p\right)\tag{3}$$
always has integer solutions, where $(-)$ is the Legendre symbol.

For example, when $p=167$ the smallest positive integer solution of $(3)$ is $$(x,\,y)=(3993882,\,437071).$$

**Conjecture 3.** For any prime $p\equiv3\pmod4$ and $q\in\{7,11,19,43,67,163\}$, the equation
$$qx^2+4\left(\frac pq\right)=py^2\tag{4}$$
always has integer solutions.

**QUESTION.** How to solve the conjectures?

Your comments are welcome!