Ever since I was introduced to modular arithmetic, I've had some trouble with it. I think it uses a part of my brain that I haven't used often. Anyways, I've been thinking about this specific equivalence: $$a^3 \equiv 5 \, (\text{mod } 7)$$ And I have a hunch that no $a$ exists s.t. this equivalence is true. Simulating it, it's clear that there is a pattern: 6, 1, 6, 6, 0, 1, 1, 6, 1, 6, 6, 0, 1, 1, 6, 1, 6, 6, 0...
But I can't figure out how to formally prove 1. that this pattern is the actual pattern and as an extension, 2. that the equivalence above doesn't hold (it should be trivial if I can prove 1).
Can anyone help? Thanks so much.