Small prime solutions of quadratic equations II

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.