Quadratic equations with five prime unknowns
type: Item
instance of: scholarly article
title: Quadratic equations with five prime unknowns
zbMATH Open document ID: 1080.11069
author: Stephen Choi
author: Angel V. Kumchev
published in: Journal of Number Theory
publication date: 2004-08-16T00:00:00Z
review text: Let \(b_1, \cdots , b_5\) be non-zero integers such that \(\gcd (b_i,\, b_j,\, b_k) = 1\) for \(1\leq i < j < k \leq 5\) and \(| b_j| \leq | b_5| \) for \(1\leq j\leq 5\). The authors consider the quadratic equation \[ b_1 p^2_1 + \cdots + b_5 p^2_5 = n\tag{E} \] in the prime variables \(p_1, \, p_2,\, \cdots ,\, p_5\). The main results are: (i) if the \(b_j\)'s are all positive, then the equation (E) is soluble whenever \(n\gg b_5 (b_1 \cdots b_4)^{4+\varepsilon}\); (ii) if the \(b_j\)'s are not all of the same sign, then the equation (E) has solutions in which all \(p_j \ll \sqrt n + | b_1 \cdots b_4| ^{2+\varepsilon}\). Here \(\varepsilon >0\) is arbitrarily small and the implied constants depend only on \(\varepsilon\). These estimates on \(n\) and \(p_j\) improve on earlier works of \textit{M. C. Liu} and the reviewer [Monatsh. Math. 111, No. 2, 147--169 (1991; Zbl 0719.11064)] as well as those of \textit{K. K. Choi} and \textit{J. Y. Liu} [Can. J. Math. 54, No. 1, 71--91 (2002; Zbl 1040.11068)]. The present work also uses the circle method and the improvement arises from sharper estimates for Weyl sums over primes obtained by the second author. More recently the second author and \textit{G. Harman} obtained further improvements in the above results. The exponent 4 in case (i) is reduced to \(7/2\) and the exponent 2 in case (ii) is reduced to \(7/4\) [to appear in Math. Proc. Camb. Philos. Soc.].
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 2091717
zbMATH Keywords: quadratic equations
zbMATH Keywords: small prime solutions
zbMATH Keywords: Hardy-Littlewood method
zbMATH Keywords: circle method
zbMATH Keywords: Weyl sums over primes
reviewed by: Kai-Man Tsang
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.03.007
OpenAlex ID: W2119476796
cites work: Small Prime Solutions of Quadratic Equations
cites work: Small prime solutions of quadratic equations II
cites work: On Weyl sums over primes and almost primes
cites work: On the Waring–Goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers
cites work: Small prime solutions of some additive equations
cites work: The exceptional set of Goldbach's problem
DOI: 10.1016/J.JNT.2004.03.007
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description: scientific article; zbMATH DE number 2091717