Suppose we have a vector bundle $E$ over a smooth manifold $M$, and let $F$ be a vector bundle endomorphism $E$. Is there anyway to now generalize the theory of eigenspaces in linear algebra to vector bundles? In particular, I am wondering when it is possible to construct sub bundles which corres...

While researching the topic of odd perfect numbers, I came across the following related subproblem: PROBLEM: Determine congruence conditions (on $n > 1$) such that $$\frac{5^n - 1}{4}$$ is squarefree. MY ATTEMPT Set $$m_n := \frac{5^n - 1}{4}.$$ I noticed that $m = m_1 = 1$ is squarefree for $n...