Abstract
1. Basic theorems. In this paper, we discuss some of the known results concerning the use of the converse of Fermat's theorem as a test for primeness, and prove several results, believed to be new, which clarify certain points and also delimit to some extent the nature of possible improvements. The proofs of the known results are included, so that only the standard material of elementary number theory is presupposed. Fermat's theorem states that if N is prime and (a, N) = 1, then
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