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A quaternary Diophantine inequality with prime numbers of a special form

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Abstract

Let N be a sufficiently large real number. It is proved here that, for \(1<c<\frac{4803}{4040}\) and for any arbitrary large number \(E>0\), the Diophantine inequality

$$\begin{aligned} \big |p_1^c+p_2^c+p_3^c+p_4^c-N\big |<(\log N)^{-E} \end{aligned}$$

is solvable in prime variables \(p_1,p_2,p_3,p_4\) such that, for \(i=1,2,3,4\), each of the numbers \(p_i+2\) has at most \(\bigg [\frac{31540280}{12007500-10100000c}\bigg ]\) prime factors, counted according to multiplicity. When \(c\rightarrow 1\), each \(p_i+2\) is \({\mathcal {P}}_{16}\), which constitutes a large improvement upon the result of Dimitrov [14] who showed that each \(p_i+2\) is \({\mathcal {P}}_{32}\).

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The authors would like to appreciate the referee for his/her patience in refereeing this paper.

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Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology, Beijing, 100083, People’s Republic of China

    Jinjiang Li & Fei Xue

  2. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, People’s Republic of China

    Min Zhang

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  1. Jinjiang Li
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Correspondence to Min Zhang.

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This work is supported by the National Natural Science Foundation of China (Grant Nos. 11901566, 12001047, 11971476, 12071238), and the Fundamental Research Funds for the Central Universities (Grant No. 2022YQLX05).

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Li, J., Xue, F. & Zhang, M. A quaternary Diophantine inequality with prime numbers of a special form. Ramanujan J 63, 259–291 (2024). https://doi.org/10.1007/s11139-023-00700-w

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