{"title":"组合素性检验","authors":"M. R. Valluri","doi":"10.1145/3465002.3465004","DOIUrl":null,"url":null,"abstract":"In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then [EQUATION] This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If an integer of the form n = 4k + 1, k > 0 is prime, then ([EQUATION]) and (2) If an integer of the form n = 4k + 3, k ≥ 0 is prime, then [EQUATION]. In addition, the author proposes important conjectures based on the converse of the above theorems which aim to establish primality of n. These conjectures are scrutinized by the given combinatorial primality test algorithm which can also distinguish patterns of prime n whether it is of the form 4k + 1 or 4k + 3.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"129 - 133"},"PeriodicalIF":0.4000,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3465002.3465004","citationCount":"0","resultStr":"{\"title\":\"Combinatorial primality test\",\"authors\":\"M. R. Valluri\",\"doi\":\"10.1145/3465002.3465004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then [EQUATION] This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If an integer of the form n = 4k + 1, k > 0 is prime, then ([EQUATION]) and (2) If an integer of the form n = 4k + 3, k ≥ 0 is prime, then [EQUATION]. In addition, the author proposes important conjectures based on the converse of the above theorems which aim to establish primality of n. These conjectures are scrutinized by the given combinatorial primality test algorithm which can also distinguish patterns of prime n whether it is of the form 4k + 1 or 4k + 3.\",\"PeriodicalId\":41965,\"journal\":{\"name\":\"ACM Communications in Computer Algebra\",\"volume\":\"54 1\",\"pages\":\"129 - 133\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2020-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1145/3465002.3465004\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACM Communications in Computer Algebra\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3465002.3465004\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Communications in Computer Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3465002.3465004","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then [EQUATION] This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If an integer of the form n = 4k + 1, k > 0 is prime, then ([EQUATION]) and (2) If an integer of the form n = 4k + 3, k ≥ 0 is prime, then [EQUATION]. In addition, the author proposes important conjectures based on the converse of the above theorems which aim to establish primality of n. These conjectures are scrutinized by the given combinatorial primality test algorithm which can also distinguish patterns of prime n whether it is of the form 4k + 1 or 4k + 3.
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