A Theorem on Congruence

The object of this note is to prove the following

THEOREM 1. Let p be an odd prime. Then

1^p-1+2^p-1+........+(p-1)^p-1-p-(p-1)=0 (mod p²).                          (1)

1.1. The point in the above theorem is seen if we write the left side of (1) in the form

(1^p-1-1)+(2^p-1-1)+......+[(p-1)^p-1-1]-[(p-1)+1]

which is=0 (mod p) by Fermat's and Wilson's theorems. The above theorem states that the left side of (1) is divisible by p².