Thue theorem ($F(x,y)=m$)

An irreducible integral binary form of degree at least three has only finitely many integral solutions at a fixed nonzero value. Factor the form in a splitting field. Each linear factor generates an ideal dividing a fixed ideal, so it is one of finitely many chosen factors times a unit. The linear relation among any three distinct factors gives a unit equation over a number field, hence finitely many ratios and finitely many rational directions. Homogeneity and the fixed nonzero value then bound the scale on each direction. The same argument works for a binary form with at least three distinct projective roots, even without irreducibility.