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A PAIR OF EQUATIONS IN EIGHT PRIME CUBES AND POWERS OF 2

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  14 December 2022

XUE HAN  and
HUAFENG LIU
XUE HAN
Affiliation:
School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, Shandong, PR China e-mail: han_xue@stu.sdnu.edu.cn
HUAFENG LIU*
Affiliation:
School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, Shandong, PR China
*
e-mail: hfliu_sdu@hotmail.com
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Abstract

In this paper, we show that every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k powers of $2$. In particular, we prove that $k=335$ is admissible, which improves the previous result.

Keywords

circle methodGoldbach–Linnik problempowers of 2

MSC classification

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes
Secondary: 11P05: Waring's problem and variants 11P55: Applications of the Hardy-Littlewood method
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 244 - 253
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by the National Natural Science Foundation of China (Grant No. 12171286).

References

Elsholtz, C. and Schlage-Puchta, J. C., ‘The density of integers representable as the sum of four prime cubes’, Acta Arith. 192 (2020), 363–369.10.4064/aa180827-26-2CrossRefGoogle Scholar
Gallagher, P. X., ‘Primes and powers of $2$ ’, Invent. Math. 29 (1975), 125–142.10.1007/BF01390190CrossRefGoogle Scholar
Kong, Y. F., ‘On pairs of linear equations in four prime variables and powers of 2’, Bull. Aust. Math. Soc. 87 (2013), 55–67.10.1017/S0004972712000172CrossRefGoogle Scholar
Kong, Y. F. and Liu, Z. X., ‘On pairs of Goldbach–Linnik equations’, Bull. Aust. Math. Soc. 95 (2017), 199–208.10.1017/S000497271600071XCrossRefGoogle Scholar
Linnik, Y. V., ‘Prime numbers and powers of two’, Tr. Mat. Inst. Steklova 38 (1951), 151–169 (in Russian).Google Scholar
Linnik, Y. V., ‘Addition of prime numbers with powers of one and the same number’, Mat. Sb. (N.S.) 32 (1953), 3–60 (in Russian).Google Scholar
Liu, J. Y. and Liu, M. C., ‘Representation of even integers by cubes of primes and powers of 2’, Acta Math. Hungar. 91 (2001), 217–243.10.1023/A:1010671222944CrossRefGoogle Scholar
Liu, J. Y., Liu, M. C. and Wang, T. Z., ‘The number of powers of $2$ in a representation of large even integers (II)’, Sci. China Ser. A 41 (1998), 1255–1271.10.1007/BF02882266CrossRefGoogle Scholar
Liu, Y. H., ‘Two results on Goldbach–Linnik problems for cubes of primes’, Rocky Mountain J. Math. 52 (2022), 999–1007.10.1216/rmj.2022.52.999CrossRefGoogle Scholar
Liu, Z. X., ‘Density of the sums of four cubes of primes’, J. Number Theory 132 (2012), 735–747.10.1016/j.jnt.2011.12.003CrossRefGoogle Scholar
Liu, Z. X., ‘On pairs of quadratic equations in primes and powers of $2$ ’, J. Number Theory 133 (2013), 3339–3347.10.1016/j.jnt.2013.04.006CrossRefGoogle Scholar
Liu, Z. X. and Lü, G. S., ‘Eight cubes of primes and powers of $2$ ’, Acta Arith. 145 (2010), 171–192.10.4064/aa145-2-6CrossRefGoogle Scholar
Liu, Z. X. and Lü, G. S., ‘Two results on powers of 2 in Waring–Goldbach problem’, J. Number Theory 131 (2011), 716–736.10.1016/j.jnt.2010.11.007CrossRefGoogle Scholar
Pintz, J. and Ruzsa, I. Z., ‘On Linnik’s approximation to Goldbach’s problem II’, Acta Math. Hungar. 161 (2020), 569–582.10.1007/s10474-020-01077-8CrossRefGoogle Scholar
Platt, D. J. and Trudgian, T. S., ‘Linnik’s approximation to Goldbach’s conjecture, and other problems’, J. Number Theory 153 (2015), 54–62.10.1016/j.jnt.2015.01.008CrossRefGoogle Scholar
Ren, X. M., ‘Sums of four cubes of primes’, J. Number Theory 98 (2003), 156–171.10.1016/S0022-314X(02)00022-7CrossRefGoogle Scholar
Zhao, X. D., ‘Goldbach–Linnik type problems on cubes of primes’, Ramanujan J. 57 (2022), 239–251.10.1007/s11139-020-00303-9CrossRefGoogle Scholar
Zhao, X. D. and Ge, W. X., ‘Eight cubes of primes and $204$ powers of $2$ ’, Int. J. Number Theory 16 (2020), 1547–1555.10.1142/S1793042120500803CrossRefGoogle Scholar
Zhu, L., ‘Goldbach–Linnik type problems on eight cubes of primes’, Rocky Mountain J. Math. 52 (2022), 1127–1139.10.1216/rmj.2022.52.1127CrossRefGoogle Scholar