Proceedings of the Steklov Institute of Mathematics

, Volume 299, Issue 1, pp 246–267

# On a Diophantine Inequality with Prime Numbers of a Special Type

Article

## Abstract

We consider the Diophantine inequality |p 1 c + p 2 c + p 3 c N| < (logN)E, where 1 < c < 15/14, N is a sufficiently large real number and E > 0 is an arbitrarily large constant. We prove that the above inequality has a solution in primes p1, p2, p3 such that each of the numbers p1 + 2, p2 + 2 and p3 + 2 has at most [369/(180 − 168c)] prime factors, counted with multiplicity.

## Preview

### References

1. 1.
R. Baker and A. Weingartner, “A ternary Diophantine inequality over primes,” Acta Arith. 162 (2), 159–196 (2014).
2. 2.
J. Brüdern and E. Fouvry, “Lagrange’s Four Squares Theorem with almost prime variables,” J. Reine Angew. Math. 454, 59–96 (1994).
3. 3.
J. R. Chen, “On the representation of a larger even integer as the sum of a prime and the product of at most two primes,” Sci. Sin. 16, 157–176 (1973).
4. 4.
H. Davenport, Multiplicative Number Theory, 2nd ed. (Springer, New York, 1980).
5. 5.
S. I. Dimitrov, “Investigation of Diophantine inequalities and arithmetic progressions with methods of number theory,” PhD Thesis (Tech. Univ., Sofia, 2016).Google Scholar
6. 6.
S. I. Dimitrov, “A ternary Diophantine inequality over special primes,” JP J. Algebra Number Theory Appl. 39 (3), 335–368 (2017).
7. 7.
S. I. Dimitrov and T. L. Todorova, “Diophantine approximation by prime numbers of a special form,” Annu. Univ. Sofia, Fac. Math. Inform. 102, 71–90 (2015).
8. 8.
G. Greaves, Sieves in Number Theory (Springer, Berlin, 2001).
9. 9.
H. Iwaniec and E. Kowalski, Analytic Number Theory (Am. Math. Soc., Providence, RI, 2004).
10. 10.
A. A. Karatsuba, Basic Analytic Number Theory (Nauka, Moscow, 1983; Springer, Berlin, 1993).Google Scholar
11. 11.
K. Matomäki, “A Bombieri–Vinogradov type exponential sum result with applications,” J. Number Theory 129 (9), 2214–2225 (2009).
12. 12.
K. Matomäki and X. Shao, “Vinogradov’s three primes theorems with almost twin primes,” Compos. Math. 153 (6), 1220–1256 (2017); arXiv: 1512.03213v1 [math.NT].
13. 13.
H. L. Montgomery, Topics in Multiplicative Number Theory (Springer, Berlin, 1971).
14. 14.
B. I. Segal, “On a theorem analogous to Waring’s theorem,” Dokl. Akad. Nauk SSSR, No. 2, 47–49 (1933).
15. 15.
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, rev. by D. R. Heath-Brown (Clarendon Press, Oxford, 1986).
16. 16.
D. I. Tolev, “On a diophantine inequality involving prime numbers,” Acta Arith. 61 (3), 289–306 (1992).
17. 17.
D. I. Tolev, “Arithmetic progressions of prime-almost-prime twins,” Acta Arith. 88 (1), 67–98 (1999).
18. 18.
D. I. Tolev, “Representations of large integers as sums of two primes of special type,” in Algebraic Number Theory and Diophantine Analysis: Proc. Int. Conf., Graz, 1998 (W. de Gruyter, Berlin, 2000), pp. 485–495.Google Scholar
19. 19.
D. I. Tolev, “Additive problems with prime numbers of special type,” Acta Arith. 96 (11), 53–88 (2000); “Corrigendum,” Acta Arith. 105 (2), 205 (2002).
20. 20.
R. C. Vaughan, “An elementary method in prime number theory,” Acta Arith. 37 (1), 111–115 (1980).
21. 21.
I. M. Vinogradov, “Representation of an odd number as a sum of three primes,” Dokl. Akad. Nauk SSSR 15 (6–7), 291–294 (1937).Google Scholar

## Authors and Affiliations

1. 1.Faculty of Mathematics and InformaticsSofia University “St. Kliment Ohridski,”SofiaBulgaria