Say we have three infinite sequences $\{a\_i\},\{b\_i\},\{c\_i\}$ of natural numbers, satisfying the equations $$a\_1+b\_1=c\_1,\dots, a\_n+b\_n=c\_n,\dots $$
Assume further that $gcd(a\_i,b\_i,c\_i)=1$ for each $i$ and that $(a\_i,b\_i,c\_i)\neq (a\_j,b\_j,c\_j)$ for all $i,j$. Now let's define $S$ as the set of primes $p$ which divide $a\_ib\_ic\_i$ for at least one $i$. From the S-unit theorem we know that $S$ has to be infinite. 


Now the question is: Can $S$ be sparse? This can be taken to mean Dirichlet density zero, for example.


I haven't thought much about this but there are reasons to believe the answer is yes, indeed if there are infinitely many Mersenne primes $q=2^p-1$ then the equations $1+q=2^p$ give such a sparse $S$. However, I am looking for an unconditional result.



An S-unit equation, with S an infinite and sparse set of primes.
Yes, in fact, you can make $S$ grow as slowly as you like. This follows, for example, from the fact that there exist 3-term arithmetic progressions of primes $(p,q,r)$ with $\min(p,q,r)$ arbitrarily large. For each such arithmetic progression, you can take $(a\_i,b\_i,c\_i)=(p,r,2q)$. Now just choose these arithmetic progressions one at a time, making sure that the primes involved each time are much larger than any primes appearing earlier.


