Two Restricted ABC Conjectures

TL;DR

This paper extends two conditional theorems related to the abc conjecture to general number fields, one for sums with divisibility constraints and one for sums with non-divisibility constraints.

Contribution

It generalizes Ellenberg's and Mochizuki's theorems to arbitrary number fields, broadening the scope of conditions under which the abc conjecture can be deduced.

Findings

Proved Ellenberg's divisibility-based theorem for number fields.

Proved Mochizuki's non-divisibility-based theorem for number fields.

Established theorems hold in the setting of general number fields.

Abstract

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.