{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T09:28:30Z","timestamp":1770456510223,"version":"3.49.0"},"reference-count":17,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,5]],"date-time":"2020-03-05T00:00:00Z","timestamp":1583366400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Y. S. T. N. S. P","award":["2019KJXX-076"],"award-info":[{"award-number":["2019KJXX-076"]}]},{"name":"the N. S. B. R. P.","award":["2019JM-207"],"award-info":[{"award-number":["2019JM-207"]}]},{"name":"the N. S. F.","award":["11771351"],"award-info":[{"award-number":["11771351"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we give some interesting identities and asymptotic formulas for one kind of counting function, by studying the computational problems involving the symmetry sums of one kind quadratic residues and quadratic non-residues     mod \u00a0 p    . The main methods we used are the properties of the Legendre\u2019s symbol      mod \u00a0 p     , and the estimate for character sums. As application, we solve two open problems proposed by Zhiwei Sun.<\/jats:p>","DOI":"10.3390\/sym12030421","type":"journal-article","created":{"date-parts":[[2020,3,6]],"date-time":"2020-03-06T09:26:41Z","timestamp":1583486801000},"page":"421","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["The Quadratic Residues and Some of Their New Distribution Properties"],"prefix":"10.3390","volume":"12","author":[{"given":"Tingting","family":"Wang","sequence":"first","affiliation":[{"name":"College of Science, Northwest A&amp;F University, No.22 Xinong Road, Yangling 712100, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7785-5638","authenticated-orcid":false,"given":"Xingxing","family":"Lv","sequence":"additional","affiliation":[{"name":"School of Mathematics, Northwest University, Xi\u2019an 710127, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Apostol, T.M. 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