On the Distribution of Kurepa’s Function

Round 1

Reviewer 1 Report

The article presented by the authors is quite interesting and provide important hints in favour of a Kurepa's conjecture about his own function, by investigating a random distribution associated to a strictly related conjecture.

Nowadays learning about statistical distribution properties of numbers is becoming of growing importance, both in mathematics and in physics. I think that the present paper is suitable for publication, but the authors should first extend a little bit the introductory section one to explain to the readers the relevance of their contribution and/or to prove/disprove the conjecture (beyond just solving the problem itself). 

Author Response

Dear reviewer,
Thank you for your valuable (mostly encouraging) remarks. Based on your and other comments, we have made a revision to the original text. All the revised parts are highlighted in red. Below we include responses to the specific comments.

Sincerely,
Authors.
--
Comment: The article presented by the authors is quite interesting and provide important hints in favor of a Kurepa's conjecture about his own function, by investigating a random distribution associated to a strictly related conjecture.
Nowadays learning about statistical distribution properties of numbers is becoming of growing importance, both in mathematics and in physics. I think that the present paper is suitable for publication, but the authors should first extend a little bit the introductory section one to explain to the readers the relevance of their contribution and/or to prove/disprove the conjecture (beyond just solving the problem itself).

Response: In the revised version we have made several changes, the ones pertinent to your comments are rewriting of the first introductory section and providing more references valuable for understanding the status of the problem and of the position of current work. We have also revised the second section where the present results are discussed. The pointer to the software used is given, aiding to the reproducibility of the results. In the conclusions and outlook we have added the comment why is the present result is significant.

Reviewer 2 Report

The paper presents a numerical investigation of the asymptotic distribution of (K(n) mod n)/n for numbers n up to 4x10^6, which they conjecture to approach the uniform distribution, where K(n) is Kurepa’s left factorial. Some parts one would expect in such an investigation appear to be missing. They will have to be added, along with clarifications detailed below, for the paper to become publishable.

The literature review features only Kurepa’s own papers, the authors’ own previous numerical studies, and references to a flawed proof of Kurepa’s conjecture that was later withdrawn. What about other work on the conjecture, theoretical and especially numerical? Did the authors build on some other numerical work on this or other number-theoretic conjectures? Did they utilize known properties of the left factorial to optimize their numerical calculations?

There is no description of the methods used to produce the numerical results, or the software used in computations. Without that it is hard to judge how reliable they are, or for others to reproduce them to confirm.
Plots on Figure 1 are hard to interpret. First, the labels are so small one needs a magnifying glass to read them. Second, they are dot plots and it is not entirely clear in what sense they represent the distributions. And, consequently, what one should be looking for to ascertain that they are close to uniform, which is what the conjecture states. There is no explanation provided in the text. I am guessing the point is that the dots appear scattered ‘uniformly’. But all the plots look alike visually, so it is hard to judge if they are getting more ‘uniform’ when n grows. There are no distribution functions or histograms, perhaps the authors could add something like that, along with a verbal explanation, for greater clarity.

Figure 2 is more to the point, but the “fluctuation” used in it is never defined in the text. The caption says “difference from a random uniform distribution, in percentage”, but it is equally unclear what the “difference” is. It is a single number, so in what metric on distributions is it measured?

Conclusions mention in passing that “The present work…, for the first time, confirmed the conjecture, and Kurepa’s hypothesis as well, for the values up to n = 4 × 10^6”. “The conjecture” is the authors’ distribution conjecture, but there is nothing in the main text about testing “Kurepa’s hypothesis”, which is that K(n) ≢ 0 (mod n) for n>2. If it somehow follows from the distribution conjecture it is not obvious, and would be worth describing in detail. I think it is more likely that in the course of testing the distributions the authors also tested the congruence. Considering that this is “for the first time”, and that Kurepa’s conjecture is the better known, omitting the methods and results on it is odd.

Speaking of Kurepa’s conjecture, instead of the standard congruence notation the authors write mod(K(n), n) ≢ 0. I had to look up Kurepa’s conjecture to figure out what that meant. Even when mod is used as an operation (typically, in computer science) it is written between the operands, not in front of them. The authors should either use standard notation or, at least, define their own before using it.

The title is rather ambiguous. There are several things called “Kurepa function”, and what is meant is more specifically called “Kurepa’s left factorial”. But it is not even its distribution that the authors are studying, but rather of a related ratio. I suggest changing the title to something more cogent like “On asymptotic distribution of remainders in Kurepa’s left factorial, a numerical study”.

Author Response

Dear reviewer,
Thank you for your valuable remarks. Based on your and other comments, we have made a revision to the original text. All the revised parts are highlighted in red. Below we include responses to the specific comments.

Sincerely,
Authors.
--
The paper presents a numerical investigation of the asymptotic distribution of (K(n) mod n)/n for numbers n up to 4x10^6, which they conjecture to approach the uniform distribution, where K(n) is Kurepa’s left factorial. Some parts one would expect in such an investigation appear to be missing. They will have to be added, along with clarifications detailed below, for the paper to become publishable.

Comment: The literature review features only Kurepa’s own papers, the authors’ own previous numerical studies, and references to a flawed proof of Kurepa’s conjecture that was later withdrawn. What about other work on the conjecture, theoretical and especially numerical? Did the authors build on some other numerical work on this or other number-theoretic conjectures? 

Response: Dear reviewer, we have made several changes to the introduction, providing more references valuable for understanding the status of the problem and of the position of current work. In this revision we have included the reference to the software PEGI, conventionally used for number theoretical computations, which implies that this work can be regarded as one building on past experience in computational approaches to number theory, starting from the need for multiple precision arithmetic, etc.

Comment: Did they utilize known properties of the left factorial to optimize their numerical calculations?

Response: In this particular paper we were not concerned with optimizing the calculations, although as the part of the work we noticed some deficiencies of the currently implemented algorithms, such as not optimal parallelization. Improving the algorithm by using the said properties and the implementation of the algorithm may be the subject of the future work.

Comment: There is no description of the methods used to produce the numerical results, or the software used in computations. Without that it is hard to judge how reliable they are, or for others to reproduce them to confirm.

Response: We have corrected the manuscript in the revised version to accommodate these observations. We have provided the pointer to the software used and gave more details aiming at better reproducibility of the present work.

Comment: Plots on Figure 1 are hard to interpret. First, the labels are so small one needs a magnifying glass to read them. 

Response: We have now changed the labels to improve the visibility of the labels based in this comment. The labels are roughly of the same size as the main text.

Comment: Second, they are dot plots and it is not entirely clear in what sense they represent the distributions. And, consequently, what one should be looking for to ascertain that they are close to uniform, which is what the conjecture states. There is no explanation provided in the text. I am guessing the point is that the dots appear scattered ‘uniformly’. But all the plots look alike visually, so it is hard to judge if they are getting more ‘uniform’ when n grows. There are no distribution functions or histograms, perhaps the authors could add something like that, along with a verbal explanation, for greater clarity.

Response: As will be pointed out in the next response, the revision of the manuscript now contains the discussion on the fluctuation and its significance to the present problem (highlighted text). Hopefully it will aid in clarifying raised questions.

Comment: Figure 2 is more to the point, but the “fluctuation” used in it is never defined in the text. The caption says “difference from a random uniform distribution, in percentage”, but it is equally unclear what the “difference” is. It is a single number, so in what metric on distributions is it measured?

Response: The manuscript is corrected accordingly. See highlighted part related to fluctuation in Section 2.

Comment: Conclusions mention in passing that “The present work…, for the first time, confirmed the conjecture, and Kurepa’s hypothesis as well, for the values up to n = 4 × 10^6”. “The conjecture” is the authors’ distribution conjecture, but there is nothing in the main text about testing “Kurepa’s hypothesis”, which is that K(n) ≢ 0 (mod n) for n>2. If it somehow follows from the distribution conjecture it is not obvious, and would be worth describing in detail. I think it is more likely that in the course of testing the distributions the authors also tested the congruence. Considering that this is “for the first time”, and that Kurepa’s conjecture is the better known, omitting the methods and results on it is odd.

Response: We have made some clarifications regarding this in the main text, please see the highlighted text in the revised version.

Comment: Speaking of Kurepa’s conjecture, instead of the standard congruence notation the authors write mod(K(n), n) ≢ 0. I had to look up Kurepa’s conjecture to figure out what that meant. Even when mod is used as an operation (typically, in computer science) it is written between the operands, not in front of them. The authors should either use standard notation or, at least, define their own before using it.

Response: For clarification we have added a comment what this notation represents, please see line 16.

Comment: The title is rather ambiguous. There are several things called “Kurepa function”, and what is meant is more specifically called “Kurepa’s left factorial”. But it is not even its distribution that the authors are studying, but rather of a related ratio. I suggest changing the title to something more cogent like “On asymptotic distribution of remainders in Kurepa’s left factorial, a numerical study”.

Response: We have made a minor modification of the title to accommodate this remark, still aiming at a concise title. We hope the referee will find it satisfying.

Reviewer 3 Report

The paper, which presents a numerical study of Kurepa's conjecture , is intended to verify it up to very large integer values n. Moreover, it is shown that the differences between  Kurepa's distribution and a random uniform distribution are exponentially suppressed for large n. 

 

The paper is well written and the results are well presented. As minor issues, which could improve the quality of the manuscript, I suggest to the authors to say a bit more about the methods they used and to add more details in introducing such very specialized subject. After such revision, I will recommend the paper for the publication.

Author Response

Dear reviewer,
Thank you for your valuable (mostly encouraging) remarks. Based on your and other comments, we have made a revision to the original text. All the revised parts are highlighted in red. Below we include responses to the specific comments.

Sincerely,
Authors.

---
The paper, which presents a numerical study of Kurepa's conjecture , is intended to verify it up to very large integer values n. Moreover, it is shown that the differences between  Kurepa's distribution and a random uniform distribution are exponentially suppressed for large n.

Comment: The paper is well written and the results are well presented. As minor issues, which could improve the quality of the manuscript, I suggest to the authors to say a bit more about the methods they used and to add more details in introducing such very specialized subject. After such revision, I will recommend the paper for the publication.

Response: In the revised version we have made several changes, the ones pertinent to your comments are rewriting of the first introductory section and providing more references valuable for understanding the status of the problem and of the position of current work. We have also revised the second section where the present results are discussed. The pointer to the software used is given, aiding to the reproducibility of the results. In the conclusions and outlook we have added the comment why is the present result is significant.

Round 2

Reviewer 2 Report

The authors mostly addressed the flaws in the exposition, and the paper can be published in this form.