Recently, I have asked a question about the balance of abc triples. Since then I have come up with a different idea of a new criterion that somewhat combines balance and magnitude and has two advantages:

- it does not imply a (somewhat arbitrarily chosen) threshold, like e.g. the idea of "good abc triples" or the question when to consider a triple "sufficiently balanced".
- it is symmetric in $a,b,c$, meaning that it can be possibly modelized in terms of (hyper-?)elliptic curves, with things happening in $\mathbb Z$ rather than in $\mathbb N$.

If we denote the usual abc triples by "*c-abc triples*", my idea would be to introduce a subset called "*a-abc triples*" or for short, "*a-triples*" (maintaining the hyphen to avoid grammatical ambiguities), defined as follows:

A triple $(a,b,c)$ with $a<b$ and $a+b=c$ is an

iff $a>\text{rad}(abc)$.a-triple

It is natural to define the ** a-quality** of such a triple as $\frac{\log a}{\log\text{rad}(abc)}>1$ .

Since we have automatically $b,c>\text{rad}(abc)$ as well, we could consider equivalently

triples $(a,b,c)\in\mathbb Z^3$ with $a+b+c=0$ and $|a|,|b|,|c|>\text{rad}|abc|$.

It turns out that $95$ of the $241$ known "good" abc triples (i.e. with quality $\geqslant1.4$) are a-triples. The 10 ones with best a-quality are the following:

```
rk quality size merit a/b a-quality
66 1.4420 15.51 15.53 0.6363 1.4038
95 1.4316 13.28 12.18 0.8366 1.3948
151 1.4158 23.92 24.63 0.5997 1.3906
173 1.4121 29.38 31.48 0.3006 1.3815
9 1.5270 9.78 11.02 0.1139 1.3723
105 1.4290 10.44 8.74 0.6055 1.3710
240 1.4003 16.79 14.68 0.6427 1.3662
43 1.4526 9.43 8.28 0.3550 1.3629
28 1.4646 21.58 25.80 0.0302 1.3605
72 1.4403 16.98 17.38 0.1058 1.3538
```

Note that the penultimate one of those is quite imbalanced, but still has a good a-quality. As the size grows, the contribution of the imbalance is mitigated by taking the logs. Or look at the third one in the list (rank 151): "big" in size, very balanced, thus the a-quality is "hardly" smaller than the (c-)quality.

Looking at a-triples might shed some new light on the abc conjecture. My first question:

Are there still infinitely many a-triples?