This is a continuation of my previous question: Questions regarding the Kurepa Conjecture, which you can refer to for more detailed information on what is Kurepa's Conjecture and its brief history. I am making this a separate question as I was informed that MSE format wants questions to be focused.
Kurepa's conjecture states that for any (odd) prime number $p >2$, we have
$0! + 1! + \ldots + (p - 1)! \not\equiv 0 \pmod{p}$
We let $!p$ denote the expression on the left-hand side. We call it the left factorial of $p$.
Kurepa originally proposed that: For every natural number $n > 1$, it holds
$gcd(!n, n!) = 2$
where $gcd(a, b)$ is the greatest common divisor of integers $a$ and $b$ and the left factorial $!n$ is defined by
$!0 = 0, \quad !n = \sum_{k=0}^{n-1} k!, \quad n \in \mathbb{N}$
My question is - Does Kurepa's conjecture, also known as the Left Factorial Hypothesis, have any (significant or not) implications?
Since I am not much aware about the uses of Left Factorial function, so please share (if you can) atleast some general consequences that will follow if the conjecture in question is proved or disproved. I guess it might even be related to continued fractions.
In my opinion, this question may help those people who are struggling to find any good research paper on this conjecture (and considering that it doesn't even have a proper Wiki page).
Thanks in advance and edit suggestions are always welcome.
