Comments on Questions regarding the Kurepa Conjecture
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Questions regarding the Kurepa Conjecture
Kurepa's conjecture states that for any 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 $p$. We do not know any infinite set of prime numbers for which the conjecture holds. Moreover, Barsky and Benzaghou failed to prove it.
Kurepa’s conjecture/hypothesis for the left factorial has been an unsolved problem for more than 50 years now. Kurepa’s hypothesis, was formulated in 1971 by Duro Kurepa (1907–1993) and is a long-standing difficult conjecture.
Kurepa 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
$!0 = 0, \quad !n = \sum_{k=0}^{n-1} k!, \quad n \in \mathbb{N}$
In the same paper, Kurepa gave an equivalent reformulation of the hypothesis that:
$!p \not\equiv 0 \pmod{p}$ for any odd prime $p$
Over the past fifty years, there have been many attempts to find a solution to Kurepa’s conjecture, and the problem still remains open. This problem is listed in Guy’s (Prob lem B44), Koninck–Mercier’s (Problem 37), and in Sandor–Cristici’s books and has been studied by numerous researchers. Most recently, Vladica Andrejić, A. Bostan, and M. Tatarevic, in their paper, showed that Kurepa’s conjecture is valid for $p < 2^{40}$. There were several announcements about the final solution of Kurepa’s conjecture, even papers with incorrect proof were proposed.
My questions are:
- Why is Kurepa's conjecture, also known as the Left Factorial Hypothesis, so less commonly known and relatively less studied in the field of Number Theory and mathematics as a whole?
- What makes it so hard to prove that it's a long-standing difficult problem even after half a century?
- Does it have any significant 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 (it might be even related to continued fractions, I guess).
There is only one question about this conjecture on MSE: https://math.stackexchange.com/a/1808977/1379223. Besides that, there is no proper Wiki page on it and hardly few papers (in my opinion less than 30) are there which discuss about this conjecture. So, there is negligible literature available on internet which talks about Kurepa’s conjecture, and I think this question would largely help people who want to learn more or understand about it.
I understand this is not a rigorous mathematical question, however if such requests for conjectures and theorems are permitted, then I humbly ask this question to be considered and approved here. I will make sure to include appropriate tags. Since this is my first question here, any suggestions are welcome. Thanks in advance.
Note - I also asked a similar question on Math Stack Exchange: https://math.stackexchange.com/q/4961268/1379223, however, I felt that the community here might be more interested in answering my question, so I posted it here as well for better insights.
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