Small prime solutions of quadratic equations II

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.