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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Why are the only numbers $m$ for which $n^{m+1}\equiv n \bmod m$ also the only numbers such ...

It can be seen here that the only numbers for which $n^{m+1}\equiv n \bmod m$ is true are 1, 2, 6, 42, and 1806. Through experimentation, it has been found that $\displaystyle\sum_{n=1}^{m}{n^m}\equiv …
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