Diophantine approximation with four squares and one kth power of primes

Abstract

Let k be an integer with \(k\ge 3\) and \(\eta \) be any real number. Suppose that \(\lambda _1, \lambda _2, \lambda _3, \lambda _4, \mu \) are non-zero real numbers, not all of the same sign and \(\lambda _1/\lambda _2\) is irrational. It is proved that the inequality \(|\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^2+\lambda _4p_4^2+\mu p_5^k+\eta |<(\max \ p_j)^{-\sigma }\) has infinitely many solutions in prime variables \(p_1, p_2, \ldots , p_5\), where \(0<\sigma <\frac{1}{16}\) for \(k=3,\ 0<\sigma <\frac{5}{3k2^k}\) for \(4\le k\le 5\) and \(0<\sigma <\frac{40}{21k2^k}\) for \(k\ge 6\). This gives an improvement of an earlier result.