Objects that violate the converse of this theorem can be detected via Korselt's criterion. The exponent in this theorem is replaced with “phi-of-n” to extend it to arbitrary integers in Euler's theorem. (15[1])This theorem (15[1])can be proved combinatorially (15[1])by counting necklaces whose beads must not all be the same color, or algebraically by applying Lagrange's theorem to (*) Z-sub-p. This theorem and its generalizations are used to prove the correctness of RSA. Since the Carmichael numbers are never detected as composite (-5[1])by a test based on this theorem, (10[1])they are called pseudoprimes. This theorem published in 1640 states that, (-5[1])for all a and prime p, “a to the p is equivalent to a mod p.” For 10 points, name this theorem in number theory by a French mathematician who also names a “last” theorem. ■END■ (10[2])