Article
| Title: | On an additive problem of unlike powers in short intervals (English) |
| Author: | Zhang, Qingqing |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 1167-1174 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We prove that almost all positive even integers $n$ can be represented as $p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}$ with $|p_{k}^{k}-\tfrac 14 N|\leq N^{1-1/54+\varepsilon }$ for $2\leq k\leq 5$. As a consequence, we show that each sufficiently large odd integer $N$ can be written as $p_{1}+p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}$ with $|p_{k}^{k}- \tfrac 15 N|\leq N^{1-1/54+\varepsilon }$ for $1\leq k\leq 5$. (English) |
| Keyword: | Waring-Goldbach problem |
| Keyword: | exponential sum over prime in short interval |
| Keyword: | circle method |
| MSC: | 11P05 |
| MSC: | 11P32 |
| MSC: | 11P55 |
| idZBL: | Zbl 07655791 |
| idMR: | MR4517604 |
| DOI: | 10.21136/CMJ.2022.0417-21 |
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| Date available: | 2022-11-28T11:42:09Z |
| Last updated: | 2023-04-11 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151138 |
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| Reference: | [9] Roth, K. F.: A problem in additive number theory.Proc. Lond. Math. Soc., II. Ser. 53 (1951), 381-395. Zbl 0044.03601, MR 0041874, 10.1112/plms/s2-53.5.381 |
| Reference: | [10] Zhang, M.: Waring-Goldbach problems for unlike powers with almost equal variables.Front. Math. China 11 (2016), 449-460. Zbl 1335.11083, MR 3473745, 10.1007/s11464-016-0512-4 |
| Reference: | [11] Zhao, L. L.: The exceptional set for sums of unlike powers of primes.Acta Math. Sin., Engl. Ser. 30 (2014), 1897-1904. Zbl 1302.11080, MR 3266786, 10.1007/s10114-014-3661-y |
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