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We show that there is no classical regular Sturm-Liouville problem on afinite interval whose spectrum consists of infinitely many distinct primesnumbers. In particular, this answers in the negative a question raised by Zettlin his book on Sturm-Liouville theory. We also show that there {\it may} existsuch a problem if the parameter dependence is nonlinear.

A note on Sturm-Liouville problems whose spectrum is the set of prime numbers