# A Diophantine inequality with four squares and one kth power of primes

• Quanwu Mu
• Minhui Zhu
• Ping Li
Article

## Abstract

Let k ⩾ 5 be an odd integer and η be any given real number. We prove that if λ1, λ2, λ3, λ4, μ are nonzero real numbers, not all of the same sign, and λ12 is irrational, then for any real number σ with 0 < σ < 1/(8ϑ(k)), the inequality
$$|{\lambda _1}p_1^2 + {\lambda _2}p_2^2 + {\lambda _3}p_3^2 + {\lambda _4}p_4^2 + \mu{p^k_5}+\eta | < {(\mathop {max}\limits_{1 \leqslant j \leqslant 5} {p_j})^{ - \sigma }}$$
has infinitely many solutions in prime variables p1, p2,...,p5, where ϑ(k) = 3 × 2(k−5)/2 for k = 5, 7, 9 and ϑ(k) = [(k2 + 2k + 5)/8] for odd integer k with k ⩾ 11. This improves a recent result in W.Ge, T.Wang (2018).

## Keywords

Diophantine inequalities Davenport-Heilbronn method prime

11D75 11P55

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