. For arbitrary integers k 2 Z we investigate the set C k of the generalized Carmichael numbers, i.e. the natural numbers n ? maxf1; 1 \Gamma kg such that the equation a n+k j a mod n holds for all a 2 N. We give a characterization of these generalized Carmichael numbers and discuss several special cases. In particular, we prove that C 1 is finite and that C k is infinite, whenever 1 \Gamma k ? 1 is square-free. We also discuss generalized 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 j 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. 1 Introduction: Historical Background On October 18th, 1640, Pierre de Fermat wrote in a letter to Bernard Frenicle de Bessy that if p is a prime number, then p divides a p\Gamma1 \Gamma 1 for all in...