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?
Best Answer
That’s not true. From $b\lt\frac c2=\frac{3^n}2$ it follows only that $\log b\lt n\log3-\log2$, so we merely have
$$ q(a,b,c)=\frac{n\log 3}{\log6+\log b}\gt\frac{n\log3}{\log6+n\log3-\log2}=\frac n{n+1}\lt1\;. $$
In fact, this third form of the conjecture is directly equivalent to the first form via taking logarithms.
What this calculation does show, though, is that if the conjecture is true, then $3^n$ and $2^k$ usually don’t get very close and their difference is usually not highly factorizable.
Here’s Java code that calculates the quality up to $n=48$. The results are in the table below. As expected, the values are mostly very close to $1$. About every second value is greater than $1$, but the distance from $1$ doesn’t seem to grow; if so, this would be be compatible with the conjecture.
\begin{array}{r|r} n&k&b&\operatorname{rad}b&\text{quality}\\\hline 1&1&1&1&0.613147\\ 2&3&1&1&1.226294\\ 3&4&11&11&0.786661\\ 4&6&17&17&0.950157\\ 5&7&115&115&0.840343\\ 6&9&217&217&0.919128\\ 7&11&139&139&1.143327\\ 8&12&2465&2465&0.915348\\ 9&14&3299&3299&0.999432\\ 10&15&26281&26281&0.917930\\ 11&17&46075&9215&1.106626\\ 12&19&7153&7153&1.235895\\ 13&20&545747&545747&0.952025\\ 14&22&588665&53515&1.213029\\ 15&23&5960299&5960299&0.947494\\ 16&25&9492289&9492289&0.984323\\ 17&26&62031299&62031299&0.946364\\ 18&28&118985033&118985033&0.970017\\ 19&30&88519643&88519643&1.038981\\ 20&31&1339300753&1339300753&0.963392\\ 21&33&1870418611&1870418611&0.996961\\ 22&34&14201190425&2840238085&1.025916\\ 23&36&25423702091&25423702091&0.981258\\ 24&38&7551629537&7551629537&1.074578\\ 25&39&297532795555&297532795555&0.973583\\ 26&41&342842572777&342842572777&1.007464\\ 27&42&3227550973883&3227550973883&0.969538\\ 28&44&5284606410545&480418764595&1.072202\\ 29&45&33446005276051&33446005276051&0.967420\\ 30&47&65153643739321&65153643739321&0.980918\\ 31&49&54723442862635&54723442862635&1.018905\\ 32&50&727120282009217&727120282009217&0.976222\\ 33&52&1055460939185027&1055460939185027&0.996419\\ 34&53&7669982444925577&1095711777846511&1.025558\\ 35&55&14002748080035739&14002748080035739&0.986698\\ 36&57&5979447221143249&5979447221143249&1.037545\\ 37&58&162053529739285619&162053529739285619&0.981414\\ 38&60&197930213066145113&197930213066145113&1.003095\\ 39&61&1746712143805282315&1746712143805282315&0.978305\\ 40&63&2934293422202152993&2934293422202152993&0.991645\\ 41&64&18026252303461234787&783750100150488469&1.047645\\ 42&66&35632012836674152745&3239273894243104795&1.038909\\ 43&68&33109062215184251771&33109062215184251771&1.010746\\ 44&69&394475091824905581169&394475091824905581169&0.982182\\ 45&71&593129465116011091795&593129465116011091795&0.996249\\ 46&72&4140571636782855882233&4140571636782855882233&0.980012\\ 47&74&7699348427478922433003&7699348427478922433003&0.989415\\ 48&76&4208579350958186444225&841715870191637288845&1.055222\\ \end{array}