When ab = x² or ab = px²

$a$ and $b$ are positive integers.

78.1

If $ab=x^2$ and $\gcd(a,b)=1$, then $a=y^2,\ b=z^2$.

78.2

If $ab=x^2$ and $\gcd(a,b)=d$, then $a=dy^2,\ b=dz^2$.

$p$ is a prime number.

78.3

If $ab=px^2$ and $\gcd(a,b)=1$, then $a=py^2,\ b=z^2$ or $a=y^2,\ b=pz^2$.

78.4

If $ab=px^2$ and $\gcd(a,b)=d$, then $a=dpy^2,\ b=dz^2$ or $a=dy^2,\ b=dpz^2$.