abstract
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On generalized Carmichael numbers

Lorenz Halbeisen and Norbert Hungerbühler

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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
C_{1} 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 *a*^{n} = *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.