  The proliferation of probable prime tests in recent years has produced a
plethora of definitions with the word ``pseudoprime'' in them. Examples include
pseudoprimes, Euler pseudoprimes, strong pseudoprimes, Lucas pseudoprimes,
strong Lucas pseudoprimes, extra strong Lucas pseudoprimes and Perrin
pseudoprimes. Though these tests represent a wealth of ideas, they exist as a
hodge-podge of definitions rather than as examples of a more general theory. It
is the goal of this paper to present a way of viewing many of these tests as
special cases of a general principle, as well as to re-formulate them in the
context of finite fields. One aim of the reformulation is to enable the
creations of stronger tests; another is to aid in proving results about large
classes of pseudoprimes.

Number Theory