The Dushnik–Miller Theorem states that every infinite countable linear ordering has a nontrivial self-embedding. We examine computability-theoretical aspects of this classical theorem.

Article history: Received 28 August 2015 Accepted 6 February 2017 Available online 16 February 2017 MSC: 03E15 03E17 03E35 03E75

We define some new kinds of pseudoprimes to several bases, which generalize strong pseudoprimes. We call them Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. It turns out that every n < 1012, which is a strong pseudoprime to bases 2, 3 and 5, is not a Sylow p-pseudoprime to two of these bases for an appropriate prime p|n− 1. We also give examples of strong pseudoprimes to many bases...