Abstract
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 < 10 12 , 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 which are not Sylow p-pseudoprimes to two bases only, where p = 2 or 3.
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