Quote:
Originally Posted by grandpascorpion
I have a 4th degree polynomial F(k) and I'm looking for a algorithm/heuristic to find solutions of the form: f(k) = r^2 where k, r, and F(x)'s coefficients are all integers.

We would ALL like such an algorithm. Unfortunately, no efficient ones
are known.
r^2 = F(k) is an elliptic (or hyperElliptic curve). While methods
are known for finding integer points, they are generally adhoc.
One general method is to find the Heegner points, but of course there
is no general method for doing that either.
Finding integer points on elliptic curves is a very very very DEEP subject.
And of course, there will only be finitely many. There may be none
if the rank of the curve is 0.