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# Small Prime Solutions of Quadratic Equations

Published:2002-02-01
Printed: Feb 2002
• Kwok-Kwong Stephen Choi
• Jianya Liu
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## Abstract

Let $b_1,\dots,b_5$ 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 \begin{enumerate}[(ii)] \item[(i)] if $b_j$ are not all of the same sign, then the above quadratic equation has prime solutions satisfying $p_j \ll \sqrt{|n|} + \max \{|b_j|\}^{20+\ve}$; and \item[(ii)] if all $b_j$ are positive and $n \gg \max \{|b_j|\}^{41+ \ve}$, then the quadratic equation $b_1 p_1^2 + \cdots + b_5 p_5^2 = n$ is soluble in primes $p_j$. \end{enumerate}
 MSC Classifications: 11P32 - Goldbach-type theorems; other additive questions involving primes 11P05 - Waring's problem and variants 11P55 - Applications of the Hardy-Littlewood method [See also 11D85]

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