{"docId":138,"paperId":138,"url":"https:\/\/hrj.episciences.org\/138","doi":"10.46298\/hrj.1999.138","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":33,"name":"Volume 22 - 1999"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01109575","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01109575v1","dateSubmitted":"2015-03-03 16:13:52","dateAccepted":"2015-06-12 16:05:41","datePublished":"1999-01-01 08:00:00","titles":{"en":"On generalised Carmichael numbers."},"authors":["Halbeisen, L","Hungerb\u00fchler, N"],"abstracts":{"en":"For arbitrary integers $k\\in\\mathbb Z$, we investigate the set $C_k$ of the generalised Carmichael number, i.e. the natural numbers $n< \\max\\{1, 1-k\\}$ such that the equation $a^{n+k}\\equiv a \\mod n$ holds for all $a\\in\\mathbb N$. We give a characterization of these generalised Carmichael numbers and discuss several special cases. In particular, we prove that $C_1$ is infinite and that $C_k$ is infinite, whenever $1-k>1$ is square-free. We also discuss generalised Carmichael numbers which have one or two prime factors. Finally, we consider the Jeans numbers, i.e. the set of odd numbers $n$ which satisfy the equation $a^n\\equiv a \\mod n$ only for $a=2$, and the corresponding generalizations. We give a stochastic argument which supports the conjecture that infinitely many Jeans numbers exist which are squares."},"keywords":[{"en":" Korselt's criterion"},{"en":" Fermat congruence"},{"en":" square-free numbers"},{"en":"generalised Carmichael numbers"},"[MATH] Mathematics [math]"]}