AI 中文总结
本注记研究ABC猜想的例外集合问题，基于前人对例外三元组基数上界的结论，无条件证明该例外集合的基数为无穷大。
AI 中文摘要
取ε>0，设x>1为大实数，rad(n)=∏_{p|n}p为整数n≥1的根基，满足a+b=c且gcd(a,b,c)=1、c>(rad(abc))^{1+ε}的三元组(a,b,c)称为例外三元组。近期研究证明例外三元组集合ℰ的基数#ℰ(x)满足#ℰ(x)=O(x^{2/3})，本注记无条件证明该例外集合ℰ(x)的基数为无穷大。
英文摘要
Fix $\varepsilon>0$, let $x>1$ be a large real number and let $\text{rad}(n)=\prod_{p\mid n}p$ be the radical of an integer $n\geq1$. A triple $(a,b,c)$, with $a+b=c$ and $\gcd(a,b,c)=1$, such that $c>(\text{rad}(abc))^{1+\varepsilon}$, is called exceptional triple. Recent works have proved that the cardinality $\#\mathscr{E}(x)$ of set $\mathscr{E}$ of exceptional triples satisfies $\#\mathscr{E}(x)=O(x^{33/50})$. This note proves that the cardinality of the exceptional set $\mathscr{E}(x)$ of triples $(a,b,c)$ is an infinite set unconditionally for any $\varepsilon<2/35$.
CommentsSixteen Pages. Keywords: Diophantine equation; Integer inequality; Additive group; Multiplicative group; abc Conjecture
