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 $\#\{a :a \le c^\alpha\}$ 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?

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?