Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums $a+b=c$ such that $D\mid abc$ for some integer~$D$. Mochizuki proved a theorem with an opposite restriction, that the full abc conjecture would follow if it were known for abc sums that are not highly divisible. We prove both theorems for general number fields.

中文翻译：

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两个受限的 ABC 猜想

Ellenberg 证明了 abc 猜想将遵循如果这个猜想是已知的和 $a+b=c$ 使得 $D\mid abc$ 对于某个整数~$D$。Mochizuki 证明了一个具有相反限制的定理，即如果已知 abc 和不是高度可分的，则完整的 abc 猜想将遵循。我们证明了一般数域的两个定理。