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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


An extension of the theorem on primitive divisors in algebraic number fields

Author: A. Schinzel
Journal: Math. Comp. 61 (1993), 441-444
MSC: Primary 11R47; Secondary 11R04
MathSciNet review: 1189523
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Abstract: The theorem about primitive divisors in algebraic number fields is generalized in the following manner. Let A, B be algebraic integers, $ (A,B) = 1, AB \ne 0$, A/B not a root of unity, and $ {\zeta _k}$ a primitive root of unity of order k. For all sufficiently large n, the number $ {A^n} - {\zeta _k}{B^n}$ has a prime ideal factor that does not divide $ {A^m} - \zeta _k^j{B^m}$ for arbitrary $ m < n$ and $ j < k$.

References [Enhancements On Off] (What's this?)

  • [1] J. Browkin, 𝐾-theory, cyclotomic equations, and Clausen’s function, Structural properties of polylogarithms, Math. Surveys Monogr., vol. 37, Amer. Math. Soc., Providence, RI, 1991, pp. 233–273. MR 1148382,
  • [2] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), no. 4, 391–401. MR 543210 (80i:10040)
  • [3] A. Schinzel, Primitive divisors of the expression 𝐴ⁿ-𝐵ⁿ in algebraic number fields, J. Reine Angew. Math. 268/269 (1974), 27–33. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II. MR 0344221 (49 #8961)

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Additional Information

PII: S 0025-5718(1993)1189523-2
Article copyright: © Copyright 1993 American Mathematical Society

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