On generalized Carmichael numbers

For arbitrary integers k in Z we investigate the set C_k of the generalized Carmichael numbers, i.e. the natural numbers n > max(1,1-k) such that the equation a^n+k = a (mod n) holds for all a in N. We give a characterization of these generalized Carmichael numbers and discuss several special cases. In particular, we prove that C₁ is finite and that C_k is infinite, whenever 1-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 aⁿ = 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.

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