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Old 2009-09-16, 15:12   #2
R.D. Silverman
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Nov 2003

22·5·373 Posts

Originally Posted by grandpascorpion View Post
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 hyper-Elliptic curve). While methods
are known for finding integer points, they are generally ad-hoc.
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.
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