# Mini-$abc$ conjecture

Define $$\text{rad}_{23}(2^m3^nr)=2^{\text{sign}(m)}3^{\text{sign}(n)}r$$, where $$m,n\ge0$$ and $$2,3\nmid r\in\mathbb{N}$$.

For a triple $$a+b=c$$ define the quality $$q_{23}(a,b,c)=\frac{\log(c)}{\log(\text{rad}_{23}(abc))}$$.

Has anyone attempted to specifically prove that only finitely many primitive triples have quality $$q_{23}>1+\epsilon$$ for any given $$\epsilon>0$$? Is there reason to hope this may be within reach of well established methods?

Based on this table of $$abc$$-triples, only 2 mini-$$abc$$-triples: are known with quality $$\ge1.4$$:

$$37 + 2^{15} = 3^8 \times 5,\ \ \ \ \ \ \ q_{23}=1.48291$$

$$5 + 3^{11} = 2^{10} \times 173,\ \ \ \ \ q_{23}=1.41268$$

• We are dealing with triples of the form $p+q2^m=r3^n$ or similar, with $p,q,r$ "small", which in particular gives $q2^m\approx r3^n$. Using Baker's theorem it should be possible to give some nontrivial bounds on the quality. – Wojowu May 5 at 13:13
• what is $\mathrm{sign}(m)$? – Fedor Petrov May 5 at 13:26
• @FedorPetrov: sign(0)=0, sign(a)=1 if a>0. – Yaakov Baruch May 5 at 13:27
• It's worth noting that the $2^{sign(m)}3^{sign(n)}$ is immaterial, since its log is obviously $o(\log c)$, so you might as well get rid of it. – Wojowu May 5 at 15:24