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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## Details

### 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* = *p*_{1} + *p*_{2} + *p*_{3}, where *p*_{1}, *p*_{2}, *p*_{3} are primes, such that *p*_{1} = *x*^{2} + *y*^{2} + 1, *p*_{2} = [*n ^{c}*].

### Keywords

- Goldbach problem
- Prime numbers
- Circle method

### 2010 Mathematics Subject Classification

- 11N36
- 11P32

### References

- Balog, A. & Friedlander, J. P.(1992) A hybrid of theorems of Vinogradov and Piatetski-Shapiro, Pacific J. Math., 156, 45–62.
- Dimitrov, S. I. (2017) Prime triples
*p*_{1},*p*_{2},*p*_{3}in arithmetic progressions such*p*_{1}=*x*^{2}+*y*^{2}+ 1,*p*_{2}= [*n*], Notes on Number Theory and Discrete Mathematics, 23(4), 22–33.^{c} - Halberstam, H. & Richert , H.-E. (1974) Sieve Methods, Academic Press.
- Hooley, C. (1976) Applications of sieve methods to the theory of numbers, Cambridge Univ. Press.
- Jia, C.-H. (1995) On the Piatetski-Shapiro-Vinogradov theorem, Acta Arith., 73, 1–28.
- Karatsuba, A. (1983) Principles of the Analytic Number Theory, Nauka, Moscow, (in Russian).
- Kumchev, A. (1997) On the Piatetski-Shapiro-Vinogradov Theorem, Journal de Theorie dés Nombres de Bordeaux, 9, 11–23.
- 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)
- Piatetski-Shapiro, I. I. (1953) On the distribution of prime numbers in sequences of the form [
*f(n)*], Mat. Sb., 33, 559–566. - Rivat, J. & Wu, J. (2001) Prime numbers of the form [
*n*], Glasg. Math. J, 43(2), 237–254.^{c} - Tenenbaum, G. (1995) Introduction to Analytic and Probabilistic Number Theory, Cambridge Univ. Press.
- Teräväinen, J. (2018) The Goldbach problem for primes that are sums of two squares plus one, Mathematika, 64(1), 20–70.
- 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.
- Tolev, D. (2010) The ternary Goldbach problem with arithmetic weights attached to one of the variables, J. Number Theory, 130, 439–457.
- 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

APADimitrov, 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.

ChicagoDimitrov, 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.

MLADimitrov, 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.