{"title":"Advances in additive number theory","authors":"José Alfonso López Nicolás","doi":"10.5269/bspm.51233","DOIUrl":null,"url":null,"abstract":"We obtain sufficient conditions to know if given a positive even integer number and a set of positive integer numbers being all even or all odd, such a number can be expressed as sum of two elements of this set. As consequence we obtain a result which, when applied to the prime numbers set, would prove Goldbach's Conjecture provided that certain conditions are satisfied. These hypothesis include Prime Consecutive Conjecture, which is a generalized form of Twin Prime Conjecture. In addition, we extend these results to sets of positive real numbers, even for two different sets. We also obtain a recurrent approximation of \\pi(x) for enough large real x, being \\pi the distribution function of the prime number set, which uses whichever expression of x as product of enough large factors. We also state this approximation in a more general context, give upper and lower bounds for the error, and show that this approximation is asymptotically equivalent to \\pi(x).","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Boletim Sociedade Paranaense de Matematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5269/bspm.51233","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We obtain sufficient conditions to know if given a positive even integer number and a set of positive integer numbers being all even or all odd, such a number can be expressed as sum of two elements of this set. As consequence we obtain a result which, when applied to the prime numbers set, would prove Goldbach's Conjecture provided that certain conditions are satisfied. These hypothesis include Prime Consecutive Conjecture, which is a generalized form of Twin Prime Conjecture. In addition, we extend these results to sets of positive real numbers, even for two different sets. We also obtain a recurrent approximation of \pi(x) for enough large real x, being \pi the distribution function of the prime number set, which uses whichever expression of x as product of enough large factors. We also state this approximation in a more general context, give upper and lower bounds for the error, and show that this approximation is asymptotically equivalent to \pi(x).