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Q&A

Comments on Questions regarding the Kurepa Conjecture

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Questions regarding the Kurepa Conjecture

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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:

  1. 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?
  2. What makes it so hard to prove that it's a long-standing difficult problem even after half a century?
  3. 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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Areas for improvement (2 comments)
Areas for improvement
Peter Taylor‭ wrote 3 months ago

I think there may be an interesting question in here somewhere, but it needs some work to bring it out. My first impression when reading it was that it is rather confusing. Who are Barsky and Benzaghou, and why is it relevant that they in particular haven't proved something that no-one has proved? Why is there so much repetition? That makes the first half of the question look like ChatGPT asked it rather than a person.

To make three concrete suggestions for improvement: firstly, pick one question and ask it. (FWIW my guesses are 1: It's not actually understudied, but see 3. 2: the interplay between the additive and multiplicative structure of finite fields is non-trivial. 3: not really). Secondly, rewrite to reduce repetition. Thirdly, add more details to the references: at the very least, titles and dates of papers; ideally also links (Google Scholar may help here).

Math_Maven‭ wrote 3 months ago

Thank you very much Peter Taylor‭ for your constructive feedback and detailed suggestions. I will surely work on them and edit the question as soon as possible to make it better.

Also, can we consider the non-trivial interplay between the additive and multiplicative structures of finite fields (as mentioned) as a reason to work on Kurepa’s conjecture? Just because it bridges that gap to some extent?