When is the Conway–Maxwell–Poisson distribution infinitely divisible?

An essential characteristic for a distribution to play a central role in limit theory is infinite divisibility. In this note, we prove that the Conway–Maxwell–Poisson (CMP) distribution is infinitely divisible iff it is the Poisson or geometric distribution. This explains that, despite its applications in a wide range of fields, there is no theoretical foundation for the CMP distribution to be a natural candidate for the law of small numbers.