Number of integer solutions of $a^2+b^2=10c^2$ Find the number of integer solutions of the equation $a^2+b^2=10c^2$.
I can only get by inspection that $a=3m, b=m,c=m$ satisfies for any $m \in Z$.
Is there a formal logic to find all possible solutions? Any hint?
Also i tried taking $a=p^2-q^2$, $b=2pq$ and $10c^2=(p^2+q^2)^2$
which gives $$\frac{p^2+q^2}{c}=\sqrt{10}$$ which is invalid, since a rational can never be an irrational.

$$x^2+y^2=nz^2\tag{1}$$
In general, if equation $(1)$ has one integer solution $(x,y,z)=(x_0,y_0,z_0)$ then there exists an infinitely many integer solutions.
Substitute $x=t+x_0, y=t+y_0, z=at+z_0$ to equation $(1)$, then we get
$$t = \frac{2(x_0+y_0-nz_0a)}{-2+na^2}$$
Let $a=\frac{p}{q}$ with p,q are integers, hence we get a parametric solution $(x,y,z)=(x_0np^2-2qnz_0p+2q^2y_0, y_0np^2-2qnz_0p+2q^2x_0, nz_0p^2-2qx_0p-2qy_0p+2q^2z_0).$
Example of $x^2+y^2=10z^2.$
Let $n=10, (x_0,y_0,z_0)=(3,1,1).$
$(x,y,z)=(15p^2-10qp+q^2, 5p^2-10qp+3q^2, 5p^2-4qp+q^2).$
