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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Small prime solutions of quadratic equations II
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by Kwok-Kwong Stephen Choi and Jianya Liu PDF
Proc. Amer. Math. Soc. 133 (2005), 945-951 Request permission

Abstract:

Let $b_1, \ldots , b_5$ be non-zero integers and $n$ any integer. Suppose that $b_1+\cdots +b_5 \equiv n \pmod {24}$ and $(b_i,b_j)=1$ for $1 \leq i < j \leq 5$. In this paper we prove that (i) if the $b_j$ are not all of the same sign, then the above quadratic equation has prime solutions satisfying $p_j\ll \sqrt {|n|}+ \max \{|b_j|\}^{25/2+\varepsilon };$ and (ii) if all the $b_j$ are positive and $n \gg \max \{|b_j|\}^{26+\varepsilon }$, then the quadratic equation $b_1p_1^2+\cdots +b_5p_5^2=n$ is soluble in primes $p_j.$ Our previous results are $\max \{|b_j|\}^{20+\varepsilon }$ and $\max \{|b_j|\}^{41+\varepsilon }$ in place of $\max \{|b_j|\}^{25/2+\varepsilon }$ and $\max \{|b_j|\}^{26+\varepsilon }$ above, respectively.
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Additional Information
  • Kwok-Kwong Stephen Choi
  • Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
  • Email: kkchoi@cecm.sfu.ca
  • Jianya Liu
  • Affiliation: Department of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China
  • Email: jyliu@sdu.edu.cn
  • Received by editor(s): February 3, 2003
  • Published electronically: November 19, 2004
  • Additional Notes: The first and second authors were supported by the NSERC and the NSF of China (Grant #10125101), respectively
  • Communicated by: David E. Rohrlich
  • © Copyright 2004 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 945-951
  • MSC (2000): Primary 11P32, 11P05, 11P55
  • DOI: https://doi.org/10.1090/S0002-9939-04-07784-6
  • MathSciNet review: 2117193