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Small Prime Solutions of Quadratic Equations


Published:20020201
Printed: Feb 2002
KwokKwong Stephen Choi
Jianya Liu
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Abstract
Let $b_1,\dots,b_5$ be nonzero 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}