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Goldbach–Linnik type problems with mixed powers of primes

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Abstract

In this paper, we prove that every sufficiently large even integer can be represented as a sum of two squares of primes, four cubes of primes, and 21 powers of 2.

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References

  1. Hua, L.K.: Additive Theory of Prime Numbers. Science Press. Amer. Math. Soc, Providence (1965)

    MATH  Google Scholar 

  2. Kumchev, A.V.: On Weyl sums over primes and almost primes. Michigan Math. J. 54, 243–268 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Languasco, A., Zaccagnini, A.: On a Diophantine problem with two primes and \(s\) powers of two. Acta Arith. 145, 193–208 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Linnik, Y.V.: Prime numbers and powers of two. Trudy Mat. Inst. Steklov 38, 152–169 (1951)

    MathSciNet  Google Scholar 

  5. Linnik, Y.V.: Addition of prime numbers with powers of one and the same number. Mat. Sbornik N. S. 32, 3–60 (1953)

    MathSciNet  Google Scholar 

  6. Liu, Z.X.: Goldbach–Linnik type problems with unequal powers of primes. J. Number Theory 176, 439–448 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Liu, Z.X., Lü, G.S.: Two results on powers of 2 in Waring–Goldbach problem. J. Number Theory 131, 716–736 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lü, X.D.: On unequal powers of primes and powers of 2. Ramanujan J. 50, 111–121 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhao, L.L.: On the Waring–Goldbach problem for fourth and sixth powers. Proc. Lond. Math. Soc. 108, 1593–1622 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhao, L.L.: The additive problem with one cube and three cubes of primes. Michigan Math. J. 63, 763–779 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhao, L.L.: On unequal powers of primes and powers of 2. Acta Math. Hung. 146, 405–420 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhao, X.D.: Two prime squares, four prime cubes and powers of 2. Acta Arith. 187, 143–150 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhao, X.D.: Goldbach–Linnik type problems on cubes of primes. Ramanujan J. 57, 239–251 (2022)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Rui Zhang.

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This work is supported by the National Natural Science Foundation of China (Grant No. 11871367, Grant No. 11871307, and Grant No. 12031008) and Natural Science Foundation of Tianjin City (Grant No. 19JCQNJC14200).

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Zhang, R. Goldbach–Linnik type problems with mixed powers of primes. Ramanujan J 60, 1069–1080 (2023). https://doi.org/10.1007/s11139-022-00662-5

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  • DOI: https://doi.org/10.1007/s11139-022-00662-5

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