On pairs of equations in eight prime cubes and powers of 2

Gen Li; Liqun Hu; Xianjiu Huang; Gen Li; Liqun Hu; Xianjiu Huang

doi:10.3934/math.2023197

AIMS Mathematics

2023, Volume 8, Issue 2: 3940-3948. doi: 10.3934/math.2023197

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Research article

On pairs of equations in eight prime cubes and powers of 2

1.
Ji luan Academy, Nanchang University, Nanchang, Jiangxi 330031, China
2.
Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China

Received: 23 September 2022 Revised: 17 November 2022 Accepted: 23 November 2022 Published: 30 November 2022
MSC : 11P32, 11P05, 11P55

In this paper, it is proved that every pair of large positive even integers satisfying some necessary conditions can be represented in the form of a pair of eight cubes of primes and 287 powers of $ 2 $. This improves the previous result.
- circle method,
- linnik problem,
- powers of 2
Citation: Gen Li, Liqun Hu, Xianjiu Huang. On pairs of equations in eight prime cubes and powers of 2[J]. AIMS Mathematics, 2023, 8(2): 3940-3948. doi: 10.3934/math.2023197

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Abstract

In this paper, it is proved that every pair of large positive even integers satisfying some necessary conditions can be represented in the form of a pair of eight cubes of primes and 287 powers of $ 2 $. This improves the previous result.

References

[1]	D. R. Heath-Brown, J. C. Puchta, Integers represented as a sum of primes and powers of two, Asian J. Math., 6 (2002), 535–566. https://doi.org/10.4310/AJM.2002.v6.n3.a7 doi: 10.4310/AJM.2002.v6.n3.a7
[2]	L. Q. Hu, Y. F. Kong, Z. X. Liu, A pair of equations in four prime squares and powers of 2, Ramanujan J., 54 (2021), 79–92. https://doi.org/10.1007/s11139-019-00171-y doi: 10.1007/s11139-019-00171-y
[3]	L. Q. Hu, L. Yang, On pairs of equations in unlike powers of primes and powers of 2, Open Math., 15 (2017), 1487–1494. https://doi.org/10.1515/math-2017-0125 doi: 10.1515/math-2017-0125
[4]	Y. V. Linnik, Prime numbers and powers of two (in Russian), Trudy Mat. Inst. Steklova, 38 (1951), 152–169.
[5]	Y. V. Linnik, Addition of prime numbers and powers of one and the same number (in Russian), Mat. Sb., 74 (1953), 3–60.
[6]	J. Y. Liu, M. C. Liu, Representation of even integers by cubes of primes and powers of 2, Acta Math. Hung., 91 (2001), 217–243. https://doi.org/10.1023/A:1010671222944 doi: 10.1023/A:1010671222944
[7]	Y. H. Liu, Two results on Goldbach–Linnik problems for cubes of primes, Rocky Mountain J. Math., 52 (2022), 999–1007. https://doi.org/10.1216/rmj.2022.52.999 doi: 10.1216/rmj.2022.52.999
[8]	Z. X. Liu, G. S. Lü, Eight cubes of primes and powers of 2, Acta Arith., 145 (2010), 171–192.
[9]	Z. X. Liu, Density of the sums of four cubes of primes, J. Number Theory, 132 (2012), 735–747. https://doi.org/10.1016/j.jnt.2011.12.003 doi: 10.1016/j.jnt.2011.12.003
[10]	Z. X. Liu, On pairs of quadratic equations in primes and powers of 2, J. Number Theory, 133 (2013), 3339–3347. https://doi.org/10.1016/j.jnt.2013.04.006 doi: 10.1016/j.jnt.2013.04.006
[11]	Z. X. Liu, G. S. Lü, Density of two squares of primes and powers of 2, Int. J. Number Theory, 7 (2011), 1317–1329. https://doi.org/10.1142/S1793042111004605 doi: 10.1142/S1793042111004605
[12]	J. Pintz, I. Z. Ruzsa, On Linnik's approximation to Goldbach's problem. I, Acta Arith., 109 (2003), 169–194.
[13]	D. J. Platt, T. S. Trudgian, Linnik's approximation to Goldbach's conjecture, and other problems, J. Number Theory, 153 (2015), 54–62. https://doi.org/10.1016/j.jnt.2015.01.008 doi: 10.1016/j.jnt.2015.01.008
[14]	L. L. Zhao, On the Waring-Goldbach problem for fourth and sixth powers, Proc. London Math. Soc., 108 (2014), 1593–1622. https://doi.org/10.1112/plms/pdt072 doi: 10.1112/plms/pdt072
[15]	X. D. Zhao, Goldbach-Linnik type problems on cubes of primes, Ramanujan J., 57 (2022), 239–251. https://doi.org/10.1007/s11139-020-00303-9 doi: 10.1007/s11139-020-00303-9
[16]	X. D. Zhao, W. X. Ge, Eight cubes of primes and 204 powers of 2, Int. J. Number Theory, 16 (2020), 1547–1555. https://doi.org/10.1142/S1793042120500803 doi: 10.1142/S1793042120500803

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