Quadratic equations with five prime unknowns
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.].
scientific article; zbMATH DE number 2091717