Assume that $a$, $b$, $c$, and $d$ are [positive integers](https://artofproblemsolving.com/wiki/index.php/Positive_integer) such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$.
It follows from the givens that $a$ is a [ perfect fourth power](https://artofproblemsolving.com/wiki/index.php/Perfect_power), $b$ is a perfect fifth power, $c$ is a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square) and $d$ is a [perfect cube](https://artofproblemsolving.com/wiki/index.php/Perfect_cube).  Thus, there exist [integers](https://artofproblemsolving.com/wiki/index.php/Integer) $s$ and $t$ such that $a = t^4$, $b = t^5$, $c = s^2$ and $d = s^3$.  So $s^2 - t^4 = 19$.  We can factor the left-hand side of this [equation](https://artofproblemsolving.com/wiki/index.php/Equation) as a difference of two squares, $(s - t^2)(s + t^2) = 19$.  19 is a [prime number](https://artofproblemsolving.com/wiki/index.php/Prime_number) and $s + t^2 > s - t^2$ so we must have $s + t^2 = 19$ and $s - t^2 = 1$.  Then $s = 10, t = 3$ and so $d = s^3 = 1000$, $b = t^5 = 243$ and $d-b=\boxed{757}$.