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 be a prime , and let be a primitive th root of unity. Let be the maximum number of consecutive units of the cyclotomic field . It is shown that , where is the maximum number of consecutive residues modulo , and the maximum number of consecutive non-residues modulo . This result implies that, for the primes under 100, is exactly 4 for (and possibly for the other primes as well). Another consequence is that .
-  Morris Newman, Units in arithmetic progression in an algebraic number field, Proc. Amer. Math. Soc. 43 (1974), 266–268. MR 330101, https://doi.org/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