Hardy-Ramanujan Journal |
For arbitrary integers k∈Z, we investigate the set Ck of the generalised Carmichael number, i.e. the natural numbers n<max{1,1−k} such that the equation an+k≡amodn holds for all a∈N. We give a characterization of these generalised Carmichael numbers and discuss several special cases. In particular, we prove that C1 is infinite and that Ck 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 an≡amodn 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.