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” (15[1])to extend it to arbitrary integers (15[2])in (15[1])Euler's (15[1])theorem. (15[6])This theorem (15[2])can be proved combinatorially (15[2])by (15[1])counting necklaces (15[1])whose beads must not (15[1])all be the same (15[1])color, (15[1])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 (10[1])numbers (10[1])are never detected as composite (-5[1])by a test based on this theorem, (10[2]-5[1])they are called (10[1]-5[1])pseudoprimes. (10[1])This theorem published in 1640 (10[1])states that, (-5[1])for all a and prime (-5[1])p, (-5[1])“a to the p (-5[2])is equivalent to a mod p.” For 10 points, name this theorem in number theory by a French mathematician (10[1])who also names (10[1])a “last” theorem. (10[1])■END■ (10[11]0[3])