From Wikipedia for 'abc-conjecture':
"A third equivalent formulation of the conjecture involves the quality $q(a, b, c)$ of the triple $(a, b, c)$, defined as
$q(a,b,c)= \frac{\log(c)}{\log(\text{rad}(abc))}$
ABC conjecture III. For every positive real number ε, there exist only finitely many triples $(a, b, c)$ of coprime positive integers with $a + b = c$ such that $q(a, b, c) > 1 + ε$."
My doubt is if this definition really holds, because if we choose
$c = 3^n$ and $a = 2^k$ where $k$ is the maximum positive integer so that $2^k < 3^n$)
then
$q(a,b,c)= \log(3^n) / \log(\text{rad}(3^n . 2^k . b)) = n . \log(3) / (\log(6) + \log(\text{rad}(b)))$
As $\text{rad} \leq b$, let's choose the more restrictive case $\text{rad}(b) = b$ so that
$q(a,b,c) = n . \log(3) / (\log(6) + \log(b))$
By the way we define $a$ we have $b < c/2$ and $\log(b)$ will be much lower than $n$ (it will be equal or lower to the number of digits in $n$) so that the expression above will be clearly greater than $1+\epsilon$ for $\epsilon < 1$. This can be easily seen by the approximation $n = 10^p$ and $log(b) = p$ resulting in
$q(a,b,c) = 10^p . \log(3) / (p + \log(6))$
Any comments?
