Finiteness proofs for S-integral points on P^1−{0, 1, ∞}

Abstract: We present 4 different finiteness proofs for S-integral points on \(\mathbb{P}^1−\lbrace 0, 1, \infty\rbrace\) in a number field K, or equivalently for S-unit equations in K, using Diophantine approximation, logarithmic forms, Faltings’ finiteness theorems, or modularity (K=Q). We also discuss effectivity of these proofs. In particular we explain for K=Q how one can use modularity to get effective height bounds, obtaining a new effective finiteness proof.