Let $\{p\_i\},\{q\_i\}$ be disjoint sets of primes. For natural $e\_i,f\_i$
define $A=\prod p\_i^{e\_i},B=\prod q\_i^{f\_i}$.


Is it true that for all real $d < 1$, $|A-B| < \max(A,B)^d$
has finitely many solutions $e\_i,f\_i$?


What about $p\_i=(2,3),q\_i=(5,7)$?


abc implies this.



Small $|2^x 3^y - 5^z 7^t|$ and generalization
[Baker's theorem](https://en.wikipedia.org/wiki/Baker%27s_theorem) implies that


$$ |\log (A/B)| = |\sum e\_i \log p\_i - \sum f\_j \log q\_j| > \max (e\_i, f\_j)^{-C} $$


where $C$ depends only on $\{p\_i\}, \{q\_j\}$. Since $A \leq \left( \prod p\_i \right) ^{\max (e\_i)}$ and $B \leq \left( \prod q\_j \right) ^{\max (f\_j)}$ it follows that $\max (e\_i, f\_j) \sim \log (\max (A, B))$. Assume for convenience that $A > B$. Then


$$ A/B - 1 \geq \log (A/B) \gtrsim \log (A)^{-C} $$


It follows that $|A - B| \gtrsim \log (A)^{-C} A \gtrsim A^{1-\epsilon}$ for all $\epsilon > 0$. Picking $\epsilon < 1 - d$ we get $|A - B| > A^d$ for sufficiently large $A$.


