3
$\begingroup$

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?

Improve this question
$\endgroup$
7
  • $\begingroup$ 58% is not far from $\gamma$. $\endgroup$
    – Sylvain JULIEN
    Commented Apr 25, 2020 at 12:53
  • $\begingroup$ @SylvainJULIEN I strongly doubt this holds in general, we are examining very small set. $\endgroup$
    – joro
    Commented Apr 25, 2020 at 13:08
  • $\begingroup$ "good" = $\operatorname{rad}(abc)<c$? $\endgroup$
    – Will Sawin
    Commented Apr 25, 2020 at 14:00
  • $\begingroup$ @WillSawin Yes. Or in other words quality >1. $\endgroup$
    – joro
    Commented Apr 25, 2020 at 14:07
  • 1
    $\begingroup$ If one assumes $\operatorname{rad}(a), \operatorname{rad}(b), \operatorname{rad}(a+b)$ behave as independent random variables, the key quantity is understanding the number of $n \approx N$ with $\operatorname{rad}(n) \approx n^\alpha$. Surely this must be known/studied. I think the highest-order terms in the asymptotic are $n^\alpha e^{ \sqrt{\log n}}$ but I don't know what the next ones are. $\endgroup$
    – Will Sawin
    Commented Apr 25, 2020 at 14:42

0

Reset to default

You must log in to answer this question.