On the magnitudes of some small cyclotomic integers

Volume 160 / 2013

Frederick Robinson, Michael Wurtz Acta Arithmetica 160 (2013), 317-332 MSC: Primary 11R18. DOI: 10.4064/aa160-4-2


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 of 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 $5+ {1/25}$.


  • Frederick RobinsonDepartment of Mathematics, UCLA
    Box 951555 Los Angeles, CA, U.S.A.
  • Michael WurtzMicrosoft Corporation
    1 Microsoft Way miwurtz 36/3511
    Redmond, WA 98052, U.S.A.

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