It's been a long time since I've posted on here, but a friend of mine recently observed something in number theory and wants to know if anyone can help prove or disprove it, the conjecture is as follows. He's foreign, I apologize for the awkward formatting. I would format it myself, but I'm in a bit of a rush and cannot re-learn how to do so.
For reference, he has defined his "$\text{rad}(x)$" function as "the minimum of the unique prime factors of $x$." $a$ and $b$ are assumed to be positive integers.
$$\begin{array}{c}\large \mathbf{ \text{ Papava's conjecture}} \\ \\ \text{If }a+b=c>2 \text{ and }\gcd(a,b)=1\\ \text{then}\\ c<\gcd(abc, (\text{rad}(abc))^3)\\ \\ \end{array}$$
EDIT:
The OP's definition of $\text{rad}(x)$, as given above, is surely not the one used by the actual proposer of the conjecture (else there are lots of instant counterexamples). The definition which is both standard and also makes sense is this:
For a positive integer $x$, define $\text{rad}(x)$ to be the product of the distinct prime factors of $x$.
With that definition, the conjecture at least survives testing with small numbers.
Assuming that $\text{rad}(16x)=2$, it looks like $a=5, b=11$ is a counterexample:
$c=16$
$\text{rad}(abc)=2$
$\text{rad}(abc)^3=8$
$\gcd(abc, 8) = 8 < 16$
If instead $\text{rad}(x)$ indicates the largest square-free factor of $x$ - the product of all its prime factors - giving eg. $\text{rad}(144)=6$ we could investigate the sum and difference of prime cubes (or higher powers), looking for something with lower prime factors.