I posted this question over at Stackexchange, where a user informed me that it was probably more appropriate for Mathoverflow. Here's to hoping that the answer is out there:
The ABC conjecture states that there are a finite number of integer triples (a,b,c) such that $\frac {\log \left( c \right)}{\log \left( \text{rad} \left( abc \right) \right)}>1+\varepsilon $, where $a+b=c$ and $\varepsilon > 0$.
I am however more interested in a weaker version of the ABC conjecture where the following inequality holds true: $\frac {\log \left( c \right)}{\log \left( a \: \text{rad} \left( bc \right) \right)}>1+\varepsilon $. This weaker conjecture has a number of applications in music theory — specifically concerning temperament theory. For instance, it establishes a type of intuitive complexity metric on various temperaments, and then lets us bound a finite number of these temperaments underneath a given complexity. (if you are not familiar with temperament theory, you can think of these "temperaments" as z-module homomorphisms from one free abelian group to another of lower rank)
It is easy to see that this conjecture is implied by the ABC conjecture. However, I was wondering if this weaker version is already proven? And if not, what is the best approach to a proof that does not rely on ABC? I'm not very familiar with number theory so I don't know where to start.
Best Answer
According to Rockytheflyingsquirrel, this is still an open problem. I made this answer community wiki so as not to benefit from a squirrel's hard work.