A pair of equations in four prime squares and powers of 2
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A pair of equations in four prime squares and powers of 2 Liqun Hu1
· Yafang Kong2 · Zhixin Liu3
Received: 3 February 2019 / Accepted: 2 April 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, we obtain when k = 98, every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of four prime squares and k powers of 2. This improves the previous result k = 142 of Hu and Liu. Keywords Circle method · Linnik problem · Powers of 2 Mathematics Subject Classification 11P32 · 11P05 · 11P55
1 Introduction As an approximation to Goldbach’s problem, Linnik proved in 1951 [5] under the assumption of the Generalized Riemann Hypothesis (GRH), and later in 1953 [6] unconditionally, that each large even integer N is a sum of two primes p1 , p2 and a bounded number of powers of 2, namely N = p1 + p2 + 2ν1 + · · · + 2νk .
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11761048 and 11871367).
B
Liqun Hu [email protected] Yafang Kong [email protected] Zhixin Liu [email protected]
1
Department of Mathematics, Nanchang University, Nanchang 330031, Jiangxi, People’s Republic of China
2
College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, People’s Republic of China
3
School of Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China
123
L. Hu et al.
In 2002, Heath-Brown and Puchta [1] applied a rather different approach to this problem and showed that k = 13 and, on the GRH, k = 7. In 2003, Pintz and Ruzsa [12] established this latter result and announced that k = 8 is acceptable unconditionally. This paper is yet to appear in print. Elsholtz, in an unpublished manuscript, showed that k = 12; this was proved independently by Liu and Lü [10]. In 1999, Liu, Liu and Zhan [11] proved that every large even integer N can be written as a sum of four squares of primes and a bounded number of powers of 2, namely N = p12 + p22 + p32 + p42 + 2v1 + · · · + 2vk . Subsequently Liu and Liu [8] got that k = 8330 suffices. Later Liu and Lü [9] improved the value of k of (1.2) to 165, Li [4] improved it to 151 and Zhao [15] improved it to 46. Finally Platt and Trudgian [13] revised it to 45. As a comparison, Liu [7] considered the equations
N1 = p12 + p22 + p32 + p42 + 2v1 + · · · + 2vk , N2 = p52 + p62 + p72 + p82 + 2v1 + · · · + 2vk ,
(1.1)
and proved that this equations are solvable for k = 584. Later, Hu and Liu [2] improved it to k = 142. In this paper, we sharpen this result considerably by establishing the following theorem. Theorem 1.1 For k = 98, the Eq. (1.1) are solvable for every pair of sufficiently large positive even integers N1 and N2 satisfying N2 N1 > N2 and N1 ≡ N2 (mod 24). Our proof of Theorem 1.1 uses the Hardy–Littlewood circle method. We make a new estimate of minor arcs and draw on some strategies adopted in the works of Zhao [15] and Kong and Liu [3]. Throughout this paper, the letter denotes a positive constan
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