(i) For any fixed $B>0$, are there only finitely many triples $a,b,c$ of coprime positive integers, such that $a+b=c$ and all prime factors of $a,b,c$ are at most $B$?


(ii) For which $B$ all such triples are known?


A positive answer to (i) would follow from the abc conjecture. For (ii), we may assume $a\leq b$. There is one triple $1+1=2$ for $B=2$. For $B=3$, there are triples $1+1=2$, $1+2=3$, $1+3=4$, and $1+8=9$, and this list in complete. Indeed, the question reduces to equation $1+2^n=3^m$ or $1+3^m=2^n$. These equations has been solved by Gersonides in 1343. What about $B=5$?



a b c triples with bounded prime factors
Closely related is


* Jeffrey C. Lagarias, K. Soundararajan, *Smooth solutions to the abc equation: the xyz Conjecture*, J. Theor. Nombres Bordeaux 23 (2011), No. 1, 209–234, arXiv:[0911.4147](https://arxiv.org/abs/0911.4147):



> 
> This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor.
> 
> 
> 


Possible answer to your (ii) is on p. 4 of the pdf:



> 
> Example 2. de Weger [31, Theorem 5.4] found the complete set of primitive
> solutions to the xyz equation having S(X, Y, Z) ≤ 13; there are 545 such
> solutions. His table of large solutions ([31, Table IX]) yields the extremal
> values given in Table 1.
> 
> 
> 


