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Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...

For the last 10+ years, as a math amateur, I worked nightly on understanding the distribution of primes and the classic results in the history of Fermat's Last Theorem. I have made numerous mistakes ...

I am reading the paper MR0029924: Neville, E. H. The structure of Farey series. Proc. London Math. Soc. (2) 51, (1949). 132–144. (Reviewer: W. H. Simons) and by now two questions raised for me; ...