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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Covers of the integers with odd moduli and their applications to the forms $x^{m}-2^{n}$ and $x^{2}-F_{3n}/2$
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by Ke-Jian Wu and Zhi-Wei Sun PDF
Math. Comp. 78 (2009), 1853-1866 Request permission


In this paper we construct a cover $\{a_{s}(\operatorname {mod} \ n_{s})\}_{s=1}^{k}$ of $\mathbb {Z}$ with odd moduli such that there are distinct primes $p_{1},\ldots ,p_{k}$ dividing $2^{n_{1}}-1,\ldots ,2^{n_{k}}-1$ respectively. Using this cover we show that for any positive integer $m$ divisible by none of $3, 5, 7, 11, 13$ there exists an infinite arithmetic progression of positive odd integers the $m$th powers of whose terms are never of the form $2^{n}\pm p^{a}$ with $a,n\in \{0,1,2,\ldots \}$ and $p$ a prime. We also construct another cover of $\mathbb {Z}$ with odd moduli and use it to prove that $x^{2}-F_{3n}/2$ has at least two distinct prime factors whenever $n\in \{0,1,2,\ldots \}$ and $x\equiv a (\operatorname {mod} M)$, where $\{F_{i}\}_{i\geqslant 0}$ is the Fibonacci sequence, and $a$ and $M$ are suitable positive integers having 80 decimal digits.
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Additional Information
  • Ke-Jian Wu
  • Affiliation: Department of Mathematics, Zhanjiang Normal University, Zhanjiang 524048, People’s Republic of China
  • Email:
  • Zhi-Wei Sun
  • Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China and State Key Laboratory of Novel Software Technology, Nanjing University, Nanjing 210093, People’s Republic of China
  • MR Author ID: 254588
  • Email:
  • Received by editor(s): February 15, 2007
  • Received by editor(s) in revised form: July 4, 2008
  • Published electronically: January 30, 2009
  • Additional Notes: The second author is responsible for communications, and supported by the National Natural Science Foundation (grant 10871087) of People’s Republic of China.
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 78 (2009), 1853-1866
  • MSC (2000): Primary 11B25; Secondary 11A07, 11A41, 11B39, 11D61, 11Y99
  • DOI:
  • MathSciNet review: 2501080