Consecutive units
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- by Morris Newman PDF
- Proc. Amer. Math. Soc. 108 (1990), 303-306 Request permission
Abstract:
Let $p$ be a prime $> 3$ , and let $\zeta$ be a primitive $p$th root of unity. Let $k$ be the maximum number of consecutive units of the cyclotomic field ${\mathbf {Q}}\left ( \zeta \right )$. It is shown that $k \leq \max \left ( {4,R,N} \right )$, where $R$ is the maximum number of consecutive residues modulo $p$ , and $N$ the maximum number of consecutive non-residues modulo $p$. This result implies that, for the primes $p > 3$ under 100,$k$ is exactly 4 for $p = 5,7,11,13,17,19,23,29,31,37,47,73$ (and possibly for the other primes as well). Another consequence is that $k < 2{p^{1/2}}$.References
- Morris Newman, Units in arithmetic progression in an algebraic number field, Proc. Amer. Math. Soc. 43 (1974), 266–268. MR 330101, DOI 10.1090/S0002-9939-1974-0330101-2
- I. M. Vinogradov, Elements of number theory, Dover Publications, Inc., New York, 1954. Translated by S. Kravetz. MR 0062138
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 303-306
- MSC: Primary 11R27; Secondary 11R18
- DOI: https://doi.org/10.1090/S0002-9939-1990-0994782-1
- MathSciNet review: 994782