solve $b^2c-a^2=d^3$ with some conditions.

Question

Solve $b^2 c-a^2=d^3$

Conditions $b^2c>a^2$,　 $b$>0, $c$>0,　 $a$, $b$, $c$, $d$ are rational number.

Example Solution $a=108$, $b=12$, $c=849$, $d=48$

Is Solving this equation impossible?

Answer 1

Note that any positive integer can be written in the form $b^2c$ - some in many ways. For example $7=1^2\times 7$; $36=1^2\times 36=2^2\times 9= 3^2\times 4=6^2\times 1$

So you can take any positive integers $a, d$ and write $a^2+d^3$ in this form. And that is without considering rational numbers (which you ought to be able to sort out by clearing denominators).

Answer 2

$c = \frac {d^3 + a^2}{b^2}$ solves your equation. Note that if $a, b$ and $d$ are rational, so is $c$.