A Diophantine inequality with mixed powers of four primes of a specific type
DOI:
https://doi.org/10.55630/mem.2026.55.060-077Keywords:
Diophantine inequality, Davenport-Heilbronn method, Rosser’s weights, vector sieve, almost primesAbstract
In this paper we study a Diophantine inequality with one prime and three prime squares. We prove that it has infinitely many solutions in primes p_i such that each p_i + 2 (i = 1, . . . , 4) is an almost-prime of prescribed order.
References
Baker, A. On some Diophantine inequalities involving primes. J. Reine Angew.Math., 228, 1967, 166–181.
Baker, R. Harman, G. Diophantine approximation by prime numbers. J. Lond. Math. Soc., 25:2, 1982, 201–215.
Brüdern, J.; Fouvry, E. Lagrange’s Four Squares Theorem with almost prime variables. J. Reine Angew. Math., 454, 1994, 59–96.
Cai Y. On Chen’s theorem (II). Journal of Number Theory, 128:5, 2008, 1336–1357.
Chen J. R. On the representation of a large even integer as the sum of a prime and the product of at most two primes. Sci. Sinica, 16, 1973, 157–176.
Dimitrov, St. Diophantine approximation by special primes. AIP Conf. Proc., 2048, 2018.
Dimitrov, St.; Todorova, T. L. Diophantine approximation by prime numbers of a special form. Annuaire Univ. Sofia, Fac. Math. Inform., 102, 2015, 71–90.
Gambini, A. Diophantine approximation with one prime two squares of primes and one k th power of a prime. Open Mathematics, 19.1, 2021, 373-387.
Ge, Wenxu, Feng Zhao, and Tianqin Wang. On Diophantine approximation with one prime and three squares of primes. Frontiers of Mathematics in China, 14.4, 2019, 761-779.
Greaves, G. Sieves in number theory. Springer-Verlag New York, 2001.
Languasco, A. , Zaccagnini A. A Diophantine problem with a prime and three squares of primes. Journal of Number Theory, 132.12, 2012, 3016-3028.
Matomäki, K. Diophantine approximation by primes. Glasgow Math. J., 52, 2010, 87–106.
Segal, B. I. On a theorem analogous to Waring’s theorem. Dokl. Akad. Nauk SSSR (N. S.), 2, 1933, 47–49.
Shmidt W. M. Diophantine Approximation, Mir, Moscow, 1983.
Todorova, T. L. Diophantine approximation by prime numbers of a special form. Serdica Mathematical Journal, 47:3, 2021, 255–272.
Todorova, T.L. On the distribution of αp2 +β modulo one for primes p such that p + 2 has no more two prime divisors. Mathematics and Education in Mathematics, 53, 2024, 39—56.
Todorova, T.L.; Georgieva, A. A Diophantine Inequality Involving Mixed Powers of Primes with a Specific Type. Mathematics, 13(19), 2025, 3065.
Todorova, T. L.; Tolev, D. I. On the distribution of αp modulo one for primes p of a special form. Math. Slovaca, 60, 2010, 771–786.
Wang, Y., Yao, W. Diophantine approximation with one prime and three squares of primes Journal of Number Theory, 180, 2017, 234–250.
Vaughan, R. C. The Hardy–Littlewood method. Cambridge Univ. Press, 2nd ed.; 1997.
