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.


2 Answers 2


Baker's 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 $A$.


This is a supplement to Bar-Natan's answer. Let $n_1<n_2<\dots$ be the sequence of numbers composed of a given finite set of primes $S$. Tijdeman published in two papers (Compositio Math. 26 (1973), Compositio Math. 28 (1974)) the result that for $i$ sufficiently large (in terms of $S$), $$ \frac{n_i}{(\log n_i)^{c_1}}<n_{i+1}-n_i<\frac{n_i}{(\log n_i)^{c_2}}, $$ where $c_1,c_2>0$ are two constants (depending only on $S$). Moreover, all constants here are effectively computable. This implies the affirmative answer to your question in sharper form.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.