On the magnitudes of some small cyclotomic integers

TL;DR

This paper proves a longstanding conjecture about the magnitudes of conjugates of small cyclotomic integers, classifying their possible sizes and extending the result to slightly larger magnitudes.

Contribution

It completes the proof of Robinson's conjecture for cyclotomic integers with conjugates up to magnitude 5 and extends the classification to magnitudes up to 5+1/25.

Findings

Classifies conjugates of cyclotomic integers with magnitude at most 5

Identifies four explicit types of maximum conjugate magnitudes

Extends classification to conjugates with magnitude up to 5+1/25

Abstract

We prove the last of five outstanding conjectures made by R.M. Robinson from 1965 concerning small cyclotomic integers. In particular, given any cyclotomic integer $\beta$ all of whose conjugates have absolute value at most 5, we prove that the largest such conjugate has absolute value one of four explicit types given by two infinite classes and two exceptional cases. We also extend this result by showing that with the addition of one form, the conjecture is true for $\beta$ with magnitudes up to 5+1/25.