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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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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