On a binary Diophantine inequality involving prime numbers

Jing Huang; Qian Wang; Rui Zhang; Jing Huang; Qian Wang; Rui Zhang

doi:10.3934/math.2024407

AIMS Mathematics

2024, Volume 9, Issue 4: 8371-8385. doi: 10.3934/math.2024407

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On a binary Diophantine inequality involving prime numbers

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China

Received: 15 January 2024 Revised: 21 February 2024 Accepted: 23 February 2024 Published: 28 February 2024
MSC : 11J25

Let $ N $ denote a sufficiently large real number. In this paper, we prove that for $ 1 < c < \frac{104349}{77419} $, $ c\neq\frac{4}{3} $, for almost all real numbers $ T\in(N, 2N] $ (in the sense of Lebesgue measure), the Diophantine inequality $ |p_1^c+p_2^c-T| < T^{-\frac{9}{10c}\left(\frac{104349}{77419}-c\right)} $ is solvable in primes $ p_1, p_2 $. In addition, it is proved that the Diophantine inequality $ |p_1^c+p_2^c+p_3^c+p_4^c-N| < N^{-\frac{9}{10c}\left(\frac{104349}{77419}-c\right)} $ is solvable in primes $ p_1, p_2, p_3, p_4 $. This result constitutes a refinement upon that of Li and Cai.
- Diophantine inequality,
- prime,
- exponential sum
Citation: Jing Huang, Qian Wang, Rui Zhang. On a binary Diophantine inequality involving prime numbers[J]. AIMS Mathematics, 2024, 9(4): 8371-8385. doi: 10.3934/math.2024407

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Abstract

Let $ N $ denote a sufficiently large real number. In this paper, we prove that for $ 1 < c < \frac{104349}{77419} $, $ c\neq\frac{4}{3} $, for almost all real numbers $ T\in(N, 2N] $ (in the sense of Lebesgue measure), the Diophantine inequality $ |p_1^c+p_2^c-T| < T^{-\frac{9}{10c}\left(\frac{104349}{77419}-c\right)} $ is solvable in primes $ p_1, p_2 $. In addition, it is proved that the Diophantine inequality $ |p_1^c+p_2^c+p_3^c+p_4^c-N| < N^{-\frac{9}{10c}\left(\frac{104349}{77419}-c\right)} $ is solvable in primes $ p_1, p_2, p_3, p_4 $. This result constitutes a refinement upon that of Li and Cai.

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