episciences.org_138_1653158322
1653158322
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
01
01
1999
Volume 22  1999
On generalised Carmichael numbers.
L
Halbeisen
N
Hungerbühler
For arbitrary integers $k\in\mathbb Z$, we investigate the set $C_k$ of the generalised Carmichael number, i.e. the natural numbers $n< \max\{1, 1k\}$ 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 $1k>1$ is squarefree. 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.
01
01
1999
138
https://hal.archivesouvertes.fr/hal01109575v1
10.46298/hrj.1999.138
https://hrj.episciences.org/138

https://hrj.episciences.org/138/pdf