Question: Do you have any idea how to solve this problem?

In a french magazine written by High Schools teachers I found this problem:

let a, b, p, q four strictly positive integers such that a > b^2 and p > q+1;
find 4-tuples (a, b, p, q) such that 

(a^2 - b^4) = p!/q!

Given the source of this problem I suspect that there is a trick to answering this question.
After some hours spent, I have found no general method to solve it, only a few solutions (first one and second one are almost obvious), for instance

rel := a^2 - b^4 = p!/q!:

eval(rel, [a= 5, b=1, q=1, p=4]);   
eval(rel, [a=11, b=1, q=1, p=5]);
eval(rel, [a=71, b=1, q=1, p=7]);
eval(rel, [a= 2, b=1, q=2, p=3]);
eval(rel, [a=19, b=1, q=2, p=6]);
eval(rel, [a=21, b=3, q=2, p=6]);

Do you have any idea how to solve this problem?
Could it be handled by Maple (without a systematic exploration of a part of N^4)?

Thanks in advance