TITLE: Weak form of the $abc$-conjecture?
QUESTION [6 upvotes]: The $abc$-conjecture implies that the equation $a+b=c$ has only finitely many primitive solutions in the multiplicative semigroup generated by any particular finite set of primes.
I would appreciate any information about the status of this a priori weaker statement,
and citations to the literature if any exist.

REPLY [10 votes]: The fact that the S-unit equation has finitely many solutions is due to Siegel and Mahler. The statement has been generalized quite a bit (for example to the fact that $u_1+\cdots +u_n=1$ has finitely many solutions, which uses some generalization of Schmidt's subspace theorem). You can find a proof and proper references in "Diophantine Geometry: An introduction" by M. Hindri, J.H. Silverman, among other places.