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ON SUMS OF TWO PRIME SQUARES, FOUR PRIME CUBES AND POWERS OF TWO

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  08 January 2020

YUHUI LIU Open the ORCID record for YUHUI LIU [Opens in a new window]
YUHUI LIU*
Affiliation:
School of Mathematical Sciences,Tongji University, Shanghai, 200092, PR China email [email protected]
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Abstract

We prove that every sufficiently large even integer can be represented as the sum of two squares of primes, four cubes of primes and 28 powers of two. This improves the result obtained by Liu and Lü [‘Two results on powers of 2 in Waring–Goldbach problem’, J. Number Theory 131(4) (2011), 716–736].

Keywords

Waring–Goldbach problemHardy–Littlewood methodpowers of 2

MSC classification

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes
Secondary: 11P55: Applications of the Hardy-Littlewood method
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 2 , October 2020 , pp. 207 - 216
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Project supported by the National Natural Science Foundation of China (Grant No. 11771333).

References

Brüdern, J. and Kawada, K., ‘Ternary problems in additive prime number theory’, in: Analytic Number Theory, Developments in Mathematics, 6 (eds. Jia, C. and Matsumoto, K.) (Springer, Boston, MA, 2002), 39–91.CrossRefGoogle Scholar
Heath-Brown, D. R. and Puchta, J. C., ‘Integers represented as a sum of primes and powers of two’, Asian J. Math. 6 (2002), 535–565.CrossRefGoogle Scholar
Linnik, Y. V., ‘Prime numbers and powers of two’, Tr. Mat. Inst. Steklov. 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., ‘Enlarged major arcs in additive problems II’, Proc. Steklov Inst. Math. 276 (2012), 176–192.CrossRefGoogle Scholar
Liu, Z. X., ‘Goldbach–Linnik type problems with unequal powers of primes’, J. Number Theory 176 (2017), 439–448.CrossRefGoogle Scholar
Liu, J. Y. and Liu, M. C., ‘Representation of even integers by cubes of primes and powers of 2’, Acta Math. Hungar. 91(3) (2001), 217–243.CrossRefGoogle Scholar
Liu, J. Y., Liu, M. C. and Zhan, T., ‘Squares of primes and powers of 2’, Monatsh. Math. 128(4) (1999), 283–313.CrossRefGoogle Scholar
Liu, Z. X. and Lü, G. S., ‘Two results on powers of 2 in Waring–Goldbach problem’, J. Number Theory 131(4) (2011), 716–736.CrossRefGoogle Scholar
Platt, D. and Trudgian, T., ‘Linnik’s approximation to Goldbach’s conjecture, and other problems’, J. Number Theory 153 (2015), 54–62.CrossRefGoogle Scholar
Ren, X. M., ‘Density of integers that are the sum of four cubes of primes’, Chin. Ann. Math. Ser. B 22 (2001), 233–242.CrossRefGoogle Scholar
Zhao, L. L., Some Results on Waring–Goldbach Type Problems, PhD Thesis, University of Hong Kong, Hong Kong, 2012.Google Scholar
Zhao, L. L., ‘On the Waring–Goldbach problem for fourth and sixth powers’, Proc. Lond. Math. Soc. (3) 108(6) (2014), 1593–1622.CrossRefGoogle Scholar
Zhao, L. L., ‘On unequal powers of primes and powers of 2’, Acta Math. Hungar. 146 (2015), 405–420.CrossRefGoogle Scholar