The ternary Goldbach problem with prime numbers of a mixed type

S. I. Dimitrov
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 2, Pages 6—20
DOI: 10.7546/nntdm.2018.24.2.6-20
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Authors and affiliations

S. I. Dimitrov
Faculty of Applied Mathematics and Informatics, Technical University of Sofia
8 St. Kliment Ohridski Blvd., 1756 Sofia, Bulgaria

Abstract

In the present paper we prove that every sufficiently large odd integer N can be represented in the form N = p1 + p2 + p3, where p1, p2, p3 are primes, such that p1 = x2 + y2 + 1, p2 = [nc].

Keywords

  • Goldbach problem
  • Prime numbers
  • Circle method

2010 Mathematics Subject Classification

  • 11N36
  • 11P32

References

  1. Balog, A. & Friedlander, J. P.(1992) A hybrid of theorems of Vinogradov and Piatetski-Shapiro, Pacific J. Math., 156, 45–62.
  2. Dimitrov, S. I. (2017) Prime triples p1, p2, p3 in arithmetic progressions such p1 = x2 + y2 + 1, p2 = [nc], Notes on Number Theory and Discrete Mathematics, 23(4), 22–33.
  3. Halberstam, H. & Richert , H.-E. (1974) Sieve Methods, Academic Press.
  4. Hooley, C. (1976) Applications of sieve methods to the theory of numbers, Cambridge Univ. Press.
  5. Jia, C.-H. (1995) On the Piatetski-Shapiro-Vinogradov theorem, Acta Arith., 73, 1–28.
  6. Karatsuba, A. (1983) Principles of the Analytic Number Theory, Nauka, Moscow, (in Russian).
  7. Kumchev, A. (1997) On the Piatetski-Shapiro-Vinogradov Theorem, Journal de Theorie dés Nombres de Bordeaux, 9, 11–23.
  8. Linnik, Ju. (1960) An asymptotic formula in an additive problem of Hardy and Littlewood, Izv. Akad. Nauk SSSR, Ser.Mat., 24, 629–706 (in Russian)
  9. Piatetski-Shapiro, I. I. (1953) On the distribution of prime numbers in sequences of the form [f(n)], Mat. Sb., 33, 559–566.
  10. Rivat, J. & Wu, J. (2001) Prime numbers of the form [nc], Glasg. Math. J, 43(2), 237–254.
  11. Tenenbaum, G. (1995) Introduction to Analytic and Probabilistic Number Theory, Cambridge Univ. Press.
  12. Teräväinen, J. (2018) The Goldbach problem for primes that are sums of two squares plus one, Mathematika, 64(1), 20–70.
  13. Tolev, D. (1997) On the number of representations of an odd integer as a sum of three primes, one of which belongs to an arithmetic progression, Proc. Steklov Math. Inst., 218, 415–432.
  14. Tolev, D. (2010) The ternary Goldbach problem with arithmetic weights attached to one of the variables, J. Number Theory, 130, 439–457.
  15. Vinogradov, I. M. (1937) Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk. SSSR, 15, 291–294 (in Russian).

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Cite this paper

APA

Dimitrov, S. I. (2018). The ternary Goldbach problem with prime numbers of a mixed type. Notes on Number Theory and Discrete Mathematics, 24(2), 6-20, doi: 10.7546/nntdm.2018.24.2.6-20.

Chicago

Dimitrov, S. I. “The Ternary Goldbach Problem with Prime Numbers of a Mixed Type.” Notes on Number Theory and Discrete Mathematics 24, no. 2 (2018): 6-20, doi: 10.7546/nntdm.2018.24.2.6-20.

MLA

Dimitrov, S. I. “The Ternary Goldbach Problem with Prime Numbers of a Mixed Type.” Notes on Number Theory and Discrete Mathematics 24.2 (2018): 6-20. Print, doi: 10.7546/nntdm.2018.24.2.6-20.

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