The Casas Alvero conjecture: Let $f \in \mathbb{C}[x]$ be a monic polynomial of degree $n$. Suppose that for each $k = 1, \ldots, n-1$, there is a common root of $f$ and $f^{(k)}$. Then $f = (x-a)^n$ for some $a \in \mathbb{C}$. It is known only for the case that $n$ is a prime power or two times a prime power (see for example, this). At some point I thought I proved it :-)
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