LLL \(\and\) ABC (Q1827525)

Of course we do not know whether the ABC-Conjecture, or the closely related conjecture of Szpiro, is true. Thus it's a serious sport to construct examples \(A+B=C\) with \(\gcd(A,B,C)=1\) so that variously \[ P=P(A,B,C)=\log\max(| A| , | B| , | C| )/\log\text{rad}(A,B,C), \] \[ \rho=\rho(A,B,C)=\log| ABC| /\log\text{rad}(A,B,C), \] is large. In this context, \(P>1.4\) is a `good' ABC-triple and \(\rho>4\) a good Szpiro-triple. The author makes the very useful observation that selecting three `powerful' numbers \(a\), \(b\), \(c\) of comparable size and using LLL lattice reduction to find integers \(\alpha\), \(\beta\), \(\gamma\) so that \(\alpha a+\beta b+\gamma c=0\) heuristically has business providing good triples. Indeed, the author's relatively straightforward implementation of this idea leads him to rediscover 145 of 154 known and 41 new good ABC-triples, and 44 of 47 known and 48 new good Szpiro-triples. The idea also applies in the algebraic case of the ABC-conjecture, as is illustrated by a new record example, with \(P=2.029\ldots\,\).