Another reason that 2 is a strange prime is that for a prime $p$, the divided powers $\frac{p^n}{n!}$ tend to zero $p$-adically unless $p=2$. This makes many things in the theory of crystalline cohomology, $p$-divisible groups and integral $p$-adic Hodge theory more subtle (and in some cases just false) when $p=2$.
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