S. I. Dimitrov

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 23, 2017, Number 4, Pages 22–33

**Full paper (PDF, 225 Kb)**

## 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 there exist infinitely many arithmetic progressions of three different primes *p*_{1}, *p*_{2}, *p*_{3} = 2*p*_{2} − *p*_{1} such that *p*_{1} = *x*^{2} + *y*^{2} + 1, *p*_{3} = [*n ^{c}*].

### Keywords

- Arithmetic progression
- Prime numbers
- Circle method

### AMS Classification

- 11N36
- 11P32

### References

- Dimitrov, S. I. (2016) Arithmetic progressions of three prime numbers with two of the form
*p*=*x*^{2}+*y*^{2}+ 1, Proc. Techn. Univ.-Sofia, 66, 3, 55–64. - Halberstam, H., & Richert, H.-E. (1974) Sieve Methods, Academic Press, London Mathematical Society.
- Hooley, C. (1976) Applications of sieve methods to the theory of numbers, Cambridge Univ. Press.
- Karatsuba, A. (1983) Principles of the Analytic Number Theory, Nauka, Moscow (in Russian).
- Kumchev, A. (1997) On the Piatetski-Shapiro-Vinogradov Theorem, Journal de Th´eorie des Nombres de Bordeaux, 9, 11–23.
- Linnik, J. (1960) An asymptotic formula in an additive problem of Hardy and Littlewood, Izv. Akad. Nauk SSSR, Ser.Mat., 24, 629–706 (in Russian).
- Mirek, M. (2014) Roth’s theorem in the Piatetski-Shapiro primes, arXiv:1305.0043v2 [math.CA] 10 Apr 2014.
- Peneva, T. & Tolev, D. (1998) An additive problems with primes and almost-primes, Acta Arith., 83, 155–169.
- 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. (2016) The Goldbach problem for primes that are sums of two squares plus one, arXiv:1611.08585v1 [math.NT] 25 Nov 2016.

## Related papers

## Cite this paper

Dimitrov, S. I. (2017). Prime Triples *p*_{1}, *p*_{2}, *p*_{3} in Arithmetic Progressions such that *p*_{1} = *x*^{2} + *y*^{2} + 1, *p*_{3} = [*n ^{c}*].

*Notes on Number Theory and Discrete Mathematics*, 23(4), 22-33.