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.

`$S=\{p_j\}$`

with`$p_{j+1}>f(p_j)$`

. All the counterexamples below fail dramatically even for $f(x)=x^2$. – fedja Apr 29 '10 at 3:53