On generalized Carmichael numbers - Mordeno, Mark Oyelle L. (Animo Repository, 2011)
Define the set of generalized Carmichael numbers, Ck, to be the set containing natural numbers n such that an+k _ a (mod n) for all natural numbers n. We consider the set Ck for certain integer values of k and identify the form of some of its elements. In particular, we show that the set Ck contains finitely many elements with exactly two prime factors whenever 1 k > 1 is squarefree. Also, we investigate a more specific type of Carmichael numbers, called generalized Jeans numbers, where the congruence given above holds for a given value of a only. A generalized Jean number is defined to be a natural number n such that bn+k _ b (mod n) for a fixed natural number b. We provide a necessary and sufficient condition for a natural number to be a generalized Jeans number and identify the form of some of these numbers.