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This is so close to the Kurepa conjecture which asserts that $\gcd\left(\sum_{k=0}^{n-1}k!,n!\right)=2$ for all $n\geq 2$, which was settled in 2004 by D. Barsky and B. Benzaghou "Nombres de Bell et somme de factorielles". So what they proved is that $K(p)=1!+\cdots+(p-1)!\neq -1\pmod{p}$ for any odd prime $p$. This goes against Kevin Buzzard's heuristic that $K(p)$ is random mod $p$. Let me mention two ways you can restate the fact $p|K(p)$:

a) It is equivalent to $K(\infty)=\sum_{k=1}^{\infty}k!$ not being a unit in $\mathbb Z_p$.

b) It is equivalent to $\mathcal B_{p-1}=2\pmod{p}$ where $\mathcal{B} _n$ is the $n$th Bell number. (It is easy to show that $\mathcal B _{p}=2\pmod{p}$)

I forgot to mention that the conjecture that $p>11$ doesn't divide $K(p)$ is in question B44 of R. Guy's "Unsolved Problems in Number theory".

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This is so close to the Kurepa conjecture which asserts that $\gcd\left(\sum_{k=0}^{n-1}k!,n!\right)=2$ for all $n\geq 2$, which was settled in 2004 by D. Barsky and B. Benzaghou "Nombres de Bell et somme de factorielles". So what they proved is that $K(p)=1!+\cdots+(p-1)!\neq -1\pmod{p}$ for any odd prime $p$. This goes against Kevin Buzzard's heuristic that $K(p)$ is random mod $p$. Let me mention two ways you can restate the fact $p|K(p)$:

a) It is equivalent to $K(\infty)=\sum_{k=1}^{\infty}k!$ being a unit in $\mathbb Z_p$.

b) It is equivalent to $\mathcal B_{p-1}=2\pmod{p}$ where $\mathcal{B} _n$ is the $n$th Bell number. (It is easy to show that $\mathcal B _{p}=2\pmod{p}$)

I forgot to mention that the conjecture that $p>11$ doesn't divide $K(p)$ is in question B44 of R. Guy's "Unsolved Problems in Number theory".

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This is so close to the Kurepa conjecture which asserts that $\gcd\left(\sum_{k=0}^{n-1}k!,n!\right)=2$ for all $n\geq 2$, which was settled in 2004 by D. Barsky and B. Benzaghou "Nombres de Bell et somme de factorielles". So what they proved is that $K(p)=1!+\cdots+(p-1)!\neq -1\pmod{p}$ for any odd prime $p$. This goes against Kevin Buzzard's heuristic that $K(p)$ is random mod $p$. Let me mention two ways you can restate the fact $p|K(p)$:

a) It is equivalent to $K(\infty)=\sum_{k=1}^{\infty}k!$ being a unit in $\mathbb Z_p$.

b) It is equivalent to $\mathcal B_{p-1}=2\pmod{p}$ where $\mathcal{B} _n$ is the $n$th Bell number. (It is easy to show that $\mathcal B _{p}=2\pmod{p}$)