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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Consecutive units


Author: Morris Newman
Journal: Proc. Amer. Math. Soc. 108 (1990), 303-306
MSC: Primary 11R27; Secondary 11R18
MathSciNet review: 994782
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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}}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-0994782-1
PII: S 0002-9939(1990)0994782-1
Article copyright: © Copyright 1990 American Mathematical Society