Much less information?

I think you're actually missing a sign: $757=b-d$ , not $d-b$ .

Since $a,d$ are integers and $a^5=d^4$ , $a=n^4$ for some integer $n$ . Similarly, since $c,b$ are integers and $c^3=b^2$ , $c=m^2$ for some integer $m$ . That is, $a+19=m^2$ . Checking the first three numbers that are powers of four to find $a$ , we get:

$a=81, d=243, c=100,b=1000,$ so $b-d=\boxed{757}$ .