abc方程的光滑解：xyz猜想
Smooth solutions to the abc equation: the xyz Conjecture
摘要：
本文研究了ABC方程A+B+C=0的整数解，其中A，B，C都没有大素因子.设置H（A，B，C）= max（|A|，|B|，|C|），并将平滑度S（A，B，C）设置为ABC的最大素因子。我们考虑原始解（gcd（A，B，C）=1）的光滑性不大于log H的固定幂p.假设abc猜想，我们证明了当p<1时，有n个解。我们讨论了一个条件性结果，证明了当p>8时，广义黎曼假设（GRH）蕴涵有无穷多个原始解。我们勾勒出后一个结果的证明的一些细节。
英文摘要：
This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider primitive solutions (gcd(A, B, C)=1) having smoothness no larger than a fixed power p of log H. Assuming the abc Conjecture we show that there are finitely many solutions if p<1. We discuss a conditional result, showing that the Generalized Riemann Hypothesis (GRH) implies there are infinitely many primitive solutions when p>8. We sketch some details of the proof of the latter result.
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