When is x^n congruent to x^m (mod 3) for all x in Z^+?

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The discussion centers on determining the conditions under which x^n is congruent to x^m modulo 3 for all positive integers x. The key finding is that n - m must be an even integer. Participants suggest analyzing three specific cases for x to derive the solution. A simplification is noted, indicating that x^m modulo 3 can be expressed as (x modulo 3)^m modulo 3. The conversation emphasizes the importance of understanding modular arithmetic to solve the problem effectively.
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Hello, I need some help. For which ##m,n \in \mathbb{Z^+}## is ##x^n \equiv x^m \space \text{(mod 3)}## for all ##x \in \mathbb{Z^+}##? I have no clue how to solve this. According to the answer, ##n - m ## is an even integer. Anyone who can point me in the right direction? Thanks.
 
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There are just three relevant cases for x, check them individually and you should get the right answer.
 
A quick simplification is to note:
##x^m \mod 3 \equiv (x \mod 3)^m \mod 3. ##
 

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