Equivalent formulation of the $abc$-conjecture

Question

In a 2015 paper titled Lecture on the abc conjecture and some of its consequences by Michel Waldschmidt it is stated that:

It can be easily seen that the $abc$-conjecture is equivalent to the following statement:

For each $\epsilon > 0$, there exists $\kappa(\epsilon)$ such that, for any abc triple $(a, b, c)$,

$$c < \kappa(\epsilon) \cdot \operatorname{Rad}(abc)^{1+\epsilon}$$

Another paper, Modular Forms, Elliptic curves and the ABC-Conjecture, written by Dorian Goldfeld (apparently as dedication to Alan Baker), states that:

Let $A$, $B$, $C$ be non–zero, pairwise relatively prime, rational integers satisfying $A + B + C = 0$. Define $N$ to be the square free part of $ABC$. Then for every $\epsilon > 0$, there exists $\kappa(\epsilon) > 0$ such that:

$$\max(|A|, |B|, |C|) < \kappa(\epsilon) \cdot N ^{1+\epsilon}$$

I have the following questions:

1. Are both of these equivalent formulations of the $abc$-conjecture correct?
2. What does $\kappa(\epsilon)$ denote here? I know it's a constant but why it's $\kappa(\epsilon)$ and not simply $\kappa$? Is it like a function which varies for changing values of $\epsilon$?
3. Assuming the answer to the above question is a yes, does the constant $\kappa(\epsilon)$ strictly depend only on the value of $\epsilon$ or can it also depend on the values of say $abc$?
4. Does the constant $\kappa(\epsilon)$ have any additional conditions?
5. What does $\max(|A|, |B|, |C|)$ mean in the $2$nd formulation? I am assuming that it means to select/consider the maximum/largest value among the absolute values of $A$, $B$, and $C$, to ensure that the comparison is made based on the magnitude of the numbers, regardless of their sign.

Sources:

[1] https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf

[2] https://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf

I think this question will help me (as well as others interested about this conjecture) to better understand the equivalent formulations of the $abc$-conjecture in question so I am posting it. Thanks!

Answer 1

I think this question will help me as well as the public to better understand the equivalent formulations of the $abc$-conjecture

Well, the "public" doesn't really care about the $abc$ conjecture or any other higher math.

The number $\kappa(\varepsilon)$ depends on $\varepsilon$. It absolutely does not depend on $a$, $b$, or $c$.

Any equation $A + B + C = 0$ in nonzero integers can have its terms negated and/or moved to the other side in order rewrite it as $a+b = c$ where $a$, $b$, and $c$ are all positive: $\{a, b, c\} = \{|A|, |B|, |C|\}$. Then $\max(|A|,|B|,|C|) = \max(a,b,c) = c$ since $c = a + b$ with $a$, $b$, and $c$ all positive.