{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:44:38Z","timestamp":1760237078768,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T00:00:00Z","timestamp":1583107200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The main purpose of this paper is to use the Hardy\u2013Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of prime, and one biquadrate of prime. However, this representation can not be realized for all even integers. In this paper, we establish the exceptional set of this kind of representation and give an upper bound estimate.<\/jats:p>","DOI":"10.3390\/sym12030367","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T07:50:53Z","timestamp":1583135453000},"page":"367","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Exceptional Set for Sums of Symmetric Mixed Powers of Primes"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3204-4046","authenticated-orcid":false,"given":"Jinjiang","family":"Li","sequence":"first","affiliation":[{"name":"Department of Mathematics, China University of Mining and Technology, Beijing 100083, China"}]},{"given":"Chao","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, China University of Mining and Technology, Beijing 100083, China"}]},{"given":"Zhuo","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, China University of Mining and Technology, Beijing 100083, China"}]},{"given":"Min","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Vaughan, R.C. 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