A Note On Euler's Theorem

Euler's theorem in the Elementary Number Theory claimed that if a,n be positive integer,n>1 and(a,n)= 1,then a?(n)? 1(mod n),where cp(n)is the Euler function.The main result of this paper is the following:there is a prime number p,such that p|a?(n)-1,but p|n,except for the followings:(?)n = 2,a = 2m + 1,m for any positive integer;(?)n = 3,a = 2;(?)n = 4,a = 3;(?)n = 6,a = 5,7,17;(?)n = 10,a = 3.