Problems & Puzzles: Collection 20th

Coll.20th-016. "Good residues" r mod p

On May 17, 2018, Jaroslaw Wroblewski, wrote:

Let p be a prime number. It is a straightforward consequence of Fermat's
little theorem that for any integer r: (r+1)^p=r^p+1 (mod p).

We will call r a "good residue" mod p if 0<=r<p and the following
stronger condition holds:(r+1)^p=r^p+1 (mod p^2).

Q. Prove that for any prime p, the number of good residues mod p is NOT
divisible by 3.


 


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