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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Small prime solutions of quadratic equations II


Authors: Kwok-Kwong Stephen Choi and Jianya Liu
Journal: Proc. Amer. Math. Soc. 133 (2005), 945-951
MSC (2000): Primary 11P32, 11P05, 11P55
Published electronically: November 19, 2004
MathSciNet review: 2117193
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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 $p_j\ll \sqrt{\vert n\vert}+ \max\{\vert b_j\vert\}^{25/2+\varepsilon};$ and (ii) if all the $b_j$ are positive and $n \gg \max\{\vert b_j\vert\}^{26+\varepsilon}$, then the quadratic equation $b_1p_1^2+\cdots +b_5p_5^2=n$ is soluble in primes $p_j.$ Our previous results are $\max\{\vert b_j\vert\}^{20+\varepsilon}$ and $\max\{\vert b_j\vert\}^{41+\varepsilon}$ in place of $\max\{\vert b_j\vert\}^{25/2+\varepsilon}$ and $\max\{\vert b_j\vert\}^{26+\varepsilon}$ 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

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07784-6
PII: S 0002-9939(04)07784-6
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
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.