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A Theorem on Congruence
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The object of this note is to prove the following
THEOREM 1. Let p be an odd prime. Then
1p-1+2p-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
(1p-1-1)+(2p-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.
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