Article
| Title: | A diophantine equation involving special prime numbers (English) |
| Author: | Dimitrov, Stoyan |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 1 |
| Year: | 2023 |
| Pages: | 151-176 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $[{\cdot }]$ be the floor function. In this paper, we prove by asymptotic formula that when $1<c<\frac {3441}{2539}$, then every sufficiently large positive integer $N$ can be represented in the form $$ N=[p^c_1]+[p^c_2]+[p^c_3]+[p^c_4]+[p^c_5], $$ where $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ are primes such that $p_1=x^2 + y^2 +1$. (English) |
| Keyword: | Diophantine equation |
| Keyword: | prime |
| Keyword: | exponential sum |
| Keyword: | asymptotic formula |
| MSC: | 11L07 |
| MSC: | 11L20 |
| MSC: | 11P32 |
| idZBL: | Zbl 07655760 |
| idMR: | MR4541094 |
| DOI: | 10.21136/CMJ.2022.0469-21 |
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| Date available: | 2023-02-03T11:10:50Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151509 |
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| Reference: | [1] Arkhipov, G. I., Zhitkov, A. N.: On Waring's problem with non-integral exponent.Izv. Akad. Nauk SSSR, Ser. Mat. 48 (1984), 1138-1150 Russian. Zbl 0568.10027, MR 772109 |
| Reference: | [2] 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: | [3] Buriev, K.: Additive Problems with Prime Numbers: Thesis.Moscow State University, Moscow (1989), Russian. MR 0946769 |
| Reference: | [4] Deshouillers, J.-M.: Problème de Waring avec exposants non entiers.Bull. Soc. Math. Fr. 101 (1973), 285-295 French. Zbl 0292.10038, MR 342477, 10.24033/bsmf.1762 |
| Reference: | [5] Deshouillers, J.-M.: Un problème binaire en théorie additive.Acta Arith. 25 (1974), 393-403 French. Zbl 0278.10020, MR 340204, 10.4064/aa-25-4-393-403 |
| Reference: | [6] Dimitrov, S. I.: A ternary Diophantine inequality over special primes.JP J. Algebra Number Theory Appl. 39 (2017), 335-368. Zbl 1373.11032, MR 3846403, 10.17654/NT039030335 |
| Reference: | [7] Dimitrov, S. I.: Diophantine approximation with one prime of the form $p=x^2+y^2+1$.Lith. Math. J. 61 (2021), 445-459. Zbl 07441577, MR 4344100, 10.1007/s10986-021-09538-5 |
| Reference: | [8] Dimitrov, S. I.: A ternary Diophantine inequality by primes with one of the form $p=x^2+y^2+1$.Ramanujan J. 59 (2022), 571-607. Zbl 7589232, MR 4480301, 10.1007/s11139-021-00545-1 |
| Reference: | [9] Dimitrov, S. I.: A quinary Diophantine inequality by primes with one of the form $p=x^2+y^2+1$.Available at https://arxiv.org/abs/2107.04028v2 (2021), 27 pages. MR 4480301 |
| Reference: | [10] Graham, S. W., Kolesnik, G.: Van der Corput's Method for Exponential Sums.London Mathematical Society Lecture Note Series 126. Cambridge University Press, New York (1991). Zbl 0713.11001, MR 1145488, 10.1017/CBO9780511661976 |
| Reference: | [11] Gritsenko, S. A.: Three additive problems.Russ. Acad. Sci., Izv., Math. 41 (1993), 447-464. Zbl 0810.11057, MR 1208161, 10.1070/IM1993v041n03ABEH002271 |
| Reference: | [12] Heath-Brown, D. R.: The Piateckii-Sapiro prime number theorem.J. Number Theory 16 (1983), 242-266. Zbl 0513.10042, MR 698168, 10.1016/0022-314X(83)90044-6 |
| Reference: | [13] Hilbert, D.: Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl $n^{ter}$ Potenzen (Waringsches Problem).Math. Ann. 67 (1909), 281-300 German \99999JFM99999 40.0236.03. MR 1511530, 10.1007/BF01450405 |
| Reference: | [14] Hooley, C.: Applications of Sieve Methods to the Theory of Numbers.Cambridge Tracts in Mathematics 70. Cambridge University Press, Cambridge (1976). Zbl 0327.10044, MR 0404173 |
| Reference: | [15] 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: | [16] 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 1189090, 10.1112/plms/s3-66.1.1 |
| Reference: | [17] Iwaniec, H., Kowalski, E.: Analytic Number Theory.Colloquium Publications. American Mathematical Society 53. AMS, Providence (2004). Zbl 1059.11001, MR 2061214, 10.1090/coll/053 |
| Reference: | [18] Konyagin, S. V.: An additive problem with fractional powers.Math. Notes 73 (2003), 594-597. Zbl 1093.11006, MR 1991912, 10.1023/A:1023279809491 |
| Reference: | [19] 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: | [20] Linnik, Y. V.: An asymptotic formula in an additive problem of Hardy and Littlewood.Izv. Akad. Nauk SSSR, Ser. Mat. 24 (1960), 629-706 Russian. Zbl 0099.03501, MR 122796 |
| Reference: | [21] 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: | [22] Segal, B. I.: On a theorem analogous to Waring's theorem.Dokl. Akad. Nauk. SSSR 1933 (1933), 47-49 Russian. Zbl 0008.24303 |
| Reference: | [23] Segal, B. I.: Waring's theorem for powers with fractional and irrational exponents.Trudy Mat. Inst. Steklov. 5 (1934), 73-86 Russian. Zbl 0009.29905 |
| Reference: | [24] Tolev, D. I.: On a Diophantine inequality involving prime numbers.Acta Arith. 61 (1992), 289-306. Zbl 0762.11033, MR 1161480, 10.4064/aa-61-3-289-306 |
| Reference: | [25] Zhang, M., Li, J.: On a Diophantine equation with five prime variables.Available at https://arxiv.org/abs/1809.04591v2 (2019), 17 pages. |
| Reference: | [26] Zhang, M., Li, J.: On a Diophantine equation with three prime variables.Integers 19 (2019), Article ID A39, 13 pages. Zbl 1461.11133, MR 3997444 |
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