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 can be proved combinatorially by counting necklaces (15[1])whose beads must not all be the same color, or algebraically by applying (-5[1])Lagrange's theorem to (*) Z-sub-p. This theorem and its generalizations are used to prove the correctness (-5[1])of RSA. Since the Carmichael numbers are never detected as composite by a test based on this theorem, they are called pseudoprimes. (10[1])This theorem published in 1640 states that, 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])