Prove or disprove: for every positive integer n, the congruence equation must hold.
Prove or disprove: if is an integer greater than 2, is composite.
Here are my thoughts:
A1: Let n be any positive integer. Suppose does not hold. Thus, it is not the case that . Now, notice that . We can see that is even whenever is odd and vice-versa. Hence, either expression will at least be divisible by two, which is a factor of 6. Thus, which is the same as . This contradicts our original claim.
I don't know much about modular arithmetic, so I am not sure if I am doing this right.
A2: Again, I begin a proof by contradiction. Although, I am not sure if it is true or not. I have tested some numbers and it seems to hold. I am having trouble finding a good starting point. There is not much infromation.