We did some statistics about the 14 million good abc triples below 10^18
taken from Bart de Smith site.

This was examining just the top of the iceberg, since the
interesting triples grow very likely well above 10^18.

We were interested in $\#\{(a,b,c) :a \le c^\alpha\}$ for real $\alpha$,
as $a,b,c$ are taken from the good triples, assuming $a < b$.


Here are the numbers:

    alpha , #a<=c^alpha in percents
    0 (a=1) 0.31
    1/4     2.38
    1/2    11.75
    3/4    38.59
    9/10   70.34

The arithmetic mean of $\log{a}/\log{c}$ is 75.894%,
which is very close to $3/4$.

We did the same computation for the 234 (give or take few)
high quality triples (quality > 1.4) and the data is:

    0    4.27
    1/4 16.24
    1/2 35.47
    3/4 68.38
    9/10 91.03


The arithmetic mean of $\log{a}/\log{c}$ is 58%.
 
Can we get some heuristic arguments about the distribution
of good abc triples?