{"title":"关于形式为X^2+c的数的因数分解","authors":"M. Wolf, Franccois Wolf","doi":"10.14738/tmlai.104.12959","DOIUrl":null,"url":null,"abstract":"We study the factorization of the numbers N=X^2+c, where c is a fixed constant, and this independently of the value of gcd(X,c). We prove the existence of a family of sequences with arithmetic difference (Un,Zn) generating factorizations, i.e. such that: (Un)^2+c= ZnZn+1. The different properties demonstrated allow us to establish new factorization methods by a subset of prime numbers and to define a prime sieve. An algorithm is presented on this basis and leads to empirical results which suggest a positive answer to Landau's 4th problem.","PeriodicalId":119801,"journal":{"name":"Transactions on Machine Learning and Artificial Intelligence","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Factorization of Numbers of the Form X^2+c\",\"authors\":\"M. Wolf, Franccois Wolf\",\"doi\":\"10.14738/tmlai.104.12959\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the factorization of the numbers N=X^2+c, where c is a fixed constant, and this independently of the value of gcd(X,c). We prove the existence of a family of sequences with arithmetic difference (Un,Zn) generating factorizations, i.e. such that: (Un)^2+c= ZnZn+1. The different properties demonstrated allow us to establish new factorization methods by a subset of prime numbers and to define a prime sieve. An algorithm is presented on this basis and leads to empirical results which suggest a positive answer to Landau's 4th problem.\",\"PeriodicalId\":119801,\"journal\":{\"name\":\"Transactions on Machine Learning and Artificial Intelligence\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-07-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions on Machine Learning and Artificial Intelligence\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.14738/tmlai.104.12959\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions on Machine Learning and Artificial Intelligence","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.14738/tmlai.104.12959","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We study the factorization of the numbers N=X^2+c, where c is a fixed constant, and this independently of the value of gcd(X,c). We prove the existence of a family of sequences with arithmetic difference (Un,Zn) generating factorizations, i.e. such that: (Un)^2+c= ZnZn+1. The different properties demonstrated allow us to establish new factorization methods by a subset of prime numbers and to define a prime sieve. An algorithm is presented on this basis and leads to empirical results which suggest a positive answer to Landau's 4th problem.
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