{"title":"Backward Touchard congruence","authors":"G. Serafin","doi":"10.36045/j.bbms.210412a","DOIUrl":null,"url":null,"abstract":"The celebrated Touchard congruence states that Bn+p ≡ Bn + Bn+1 modulo p, where p is a prime number and Bn denotes the Bell number. In this paper we study divisibility properties of Bn−p and their generalizations involving higher powers of p as well as the r-Bell numbers. In particular, we show a closely relation of the considered problem to the Sun-Zagier congruence, which is additionally improved by deriving a new relation between r-Bell and derangement numbers. Finally, we conclude some results on the period of the Bell numbers modulo p.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"1 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Belgian Mathematical Society-Simon Stevin","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.36045/j.bbms.210412a","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The celebrated Touchard congruence states that Bn+p ≡ Bn + Bn+1 modulo p, where p is a prime number and Bn denotes the Bell number. In this paper we study divisibility properties of Bn−p and their generalizations involving higher powers of p as well as the r-Bell numbers. In particular, we show a closely relation of the considered problem to the Sun-Zagier congruence, which is additionally improved by deriving a new relation between r-Bell and derangement numbers. Finally, we conclude some results on the period of the Bell numbers modulo p.
期刊介绍：
The Bulletin of the Belgian Mathematical Society - Simon Stevin (BBMS) is a peer-reviewed journal devoted to recent developments in all areas in pure and applied mathematics. It is published as one yearly volume, containing five issues.
The main focus lies on high level original research papers. They should aim to a broader mathematical audience in the sense that a well-written introduction is attractive to mathematicians outside the circle of experts in the subject, bringing motivation, background information, history and philosophy. The content has to be substantial enough: short one-small-result papers will not be taken into account in general, unless there are some particular arguments motivating publication, like an original point of view, a new short proof of a famous result etc.
The BBMS also publishes expository papers that bring the state of the art of a current mainstream topic in mathematics. Here it is even more important that at leat a substantial part of the paper is accessible to a broader audience of mathematicians.
The BBMS publishes papers in English, Dutch, French and German. All papers should have an abstract in English.
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