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.

  • $\begingroup$ Benne de Weger surely knows... $\endgroup$ – Luis H Gallardo Jan 14 '11 at 7:14

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.

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    $\begingroup$ Gjergji, for the equation $u_1+\cdots + u_n=1$ one needs to assume that no subsum is zero (if I recall correctly). Otherwise one has (trivially) infinitely many solutions. $\endgroup$ – GH from MO Jan 14 '11 at 14:20

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