Article
| Title: | A Diophantine equation involving one Linnik prime (English) |
| Author: | Liu, Yuhui |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 2 |
| Year: | 2025 |
| Pages: | 655-668 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | Let $[\theta ]$ denote the integral part of the real number $\theta .$ We prove that for $1<c<\frac {25 908}{18 905}$, the Diophantine equation $$ [p_{1}^{c}]+[p_{2}^{c}]+[p_{3}^{c}]+[p_{4}^{c}]+[p_{5}^{c}]=N $$ is solvable in prime variables $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ such that $p_1=x^2+y^2+1$ with integers $x$ and $y$ for sufficiently large integer $N$, and we also establish the corresponding asymptotic formula. This result constitutes a refinement upon that of S. Dimitrov (2023). (English) |
| Keyword: | Diophantine equation |
| Keyword: | exponential sum |
| Keyword: | prime |
| MSC: | 11L07 |
| MSC: | 11L20 |
| MSC: | 11P32 |
| DOI: | 10.21136/CMJ.2024.0372-24 |
| . | |
| Date available: | 2025-05-20T11:50:30Z |
| Last updated: | 2025-05-26 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152964 |
| . | |
| Reference: | [1] Baker, R.: Some Diophantine equations and inequalities with primes.Funct. Approximatio, Comment. Math. 64 (2021), 203-250. Zbl 1484.11195, MR 4278752, 10.7169/facm/1912 |
| Reference: | [2] Buriev, K.: Additive Problems with Prime Numbers: Thesis.Moscow State University, Moscow (1989), Russian. MR 0946769 |
| Reference: | [3] Dimitrov, S.: A ternary Diophantine inequality by primes with one of the form $p=x^2+y^2+1$.Ramanujan J. 59 (2022), 571-607. Zbl 1497.11088, MR 4480301, 10.1007/s11139-021-00545-1 |
| Reference: | [4] Dimitrov, S.: On an equation by primes with one Linnik prime.Georgian Math. J. 29 (2022), 455-470. Zbl 1487.11079, MR 4431316, 10.1515/gmj-2022-2144 |
| Reference: | [5] Dimitrov, S.: A Diophantine equation involving special prime numbers.Czech. Math. J. 73 (2023), 151-176. Zbl 1538.11172, MR 4541094, 10.21136/CMJ.2022.0469-21 |
| Reference: | [6] Dimitrov, S.: A quinary Diophantine inequality by primes with one of the form $p=x^2+y^2+1$.Indian J. Pure Appl. Math. 55 (2024), 168-188. Zbl 1535.11137, MR 4703140, 10.1007/s13226-022-00354-2 |
| Reference: | [7] Graham, S. W., Kolesnik, G.: Van Der Corput's Method of Exponential Sums.London Mathematical Society Lecture Note Series 126. Cambridge University Press, Cambridge (1991). Zbl 0713.11001, MR 1145488, 10.1017/CBO9780511661976 |
| Reference: | [8] Heath-Brown, D. R.: The Pjateckii-Šapiro prime number theorem.J. Number Theory 16 (1983), 242-266. Zbl 0513.10042, MR 0698168, 10.1016/0022-314X(83)90044-6 |
| Reference: | [9] Hua, L.-K.: Some results in the additive prime-number theory.Q. J. Math., Oxf. Ser. 9 (1938), 68-80. Zbl 0018.29404, MR 3363459, 10.1093/qmath/os-9.1.68 |
| Reference: | [10] Huxley, M. N.: Exponential sums and the Riemann zeta function. V.Proc. Lond. Math. Soc., III. Ser. 90 (2005), 1-41. Zbl 1083.11052, MR 2107036, 10.1112/S0024611504014959 |
| Reference: | [11] Li, S.: On a Diophantine equation with prime numbers.Int. J. Number Theory 15 (2019), 1601-1616. Zbl 1462.11084, MR 3994149, 10.1142/S1793042119300011 |
| Reference: | [12] Linnik, Y. V.: An asymptotic formula in an additive problem of Hardy-Littlewood.Izv. Akad. Nauk SSSR, Ser. Mat. 24 (1960), 629-706 Russian. Zbl 0099.03501, MR 0122796 |
| Reference: | [13] Robert, O., Sargos, P.: Three-dimensional exponential sums with monomials.J. Reine Angew. Math. 591 (2006), 1-20. Zbl 1165.11067, MR 2212877, 10.1515/CRELLE.2006.012 |
| Reference: | [14] Sargos, P., Wu, J.: Multiple exponential sums with monomials and their applications in number theory.Acta Math. Hung. 87 (2000), 333-354. Zbl 0963.11045, MR 1771211, 10.1023/A:1006777803163 |
| Reference: | [15] Zhang, M., Li, J.: On a Diophantine equation with five prime variables.Available at https://arxiv.org/abs/1809.04591v2 (2019), 17 pages. 10.48550/arXiv.1809.04591 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
