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.



Weak form of the $abc$-conjecture?
The fact that the [S-unit equation](http://en.wikipedia.org/wiki/S-unit) 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.


