Predicting Maximal Gaps in Sets of Primes
Round 1
Reviewer 1 Report
The article is an exercise in "experimental mathematics" --- extensive numerical computations have been carried out and are used to test various conjectures concerning variants on the notion of prime gaps over certain numerical ranges.
The topic of the article, prime gaps, is certainly interesting --- so readers will definitely be interested in this article.
The numerical computations are sufficiently well described that the reader could see how to in principle check them.
Regarding presentation I have two points:
1) The computations mostly seem limited to integers less than 10^{12}; though figure 3 suggests some computations ranged as far as 10^{14}. Maybe a sentence somewhat along these lines could be added to both the abstract and the introduction.
2) I find figures 6 to 9 somewhat confusing: For small integers the plots are zero, then there is some messy transition region, and for large integers the plots asymptote to parabolas. I could not seem to find any real discussion of why the transition regions switched on where they did, nor any discussion of why these transition regions seem to be so relatively messy? A little more discussion on these points would be useful.
Overall, I do like the article, but I feel the authors should carefully consider the two points of presentation I raise above.
Author Response
> The article is an exercise in "experimental mathematics" --- extensive numerical computations have been carried out and are used to test various conjectures concerning variants on the notion of prime gaps over certain numerical ranges. The topic of the article, prime gaps, is certainly interesting --- so readers will definitely be interested in this article. The numerical computations are sufficiently well described that the reader could see how to in principle check them.
Authors' reply:
Thank you very much for reading the paper and making valuable suggestions. We greatly appreciate your insights helping make the paper better. In Latex source, each major change is marked with a comment line beginning with %%% edit April 22 --
> Regarding presentation I have two points:
1) The computations mostly seem limited to integers less than 10^{12}; though figure 3 suggests some computations ranged as far as 10^{14}. Maybe a sentence somewhat along these lines could be added to both the abstract and the introduction.
Authors' reply:
This is now addressed in both the abstract and the introduction. In the abstract, the word "Computations" is now replaced with "Extensive computations for primes up to 10^{14}". In the introduction (which has been extended), subsection 1.2 now ends with remarks; remark (ii) explains the range of computational experiments and the overall strategy.
> 2) I find figures 6 to 9 somewhat confusing: For small integers the plots are zero, then there is some messy transition region, and for large integers the plots asymptote to hyperbolas. I could not seem to find any real discussion of why the transition regions switched on where they did, nor any discussion of why these transition regions seem to be so relatively messy? A little more discussion on these points would be useful.
Authors' reply:
This is now addressed in sect.3.3. The paragraph explaining the transition region is at the end of sect.3.3 on page 18, starting with line 351.
> Overall, I do like the article, but I feel the authors should carefully consider the two points of presentation I raise above.
Authors' reply:
We are glad that overall the paper made a positive impression. Your suggestions helped a lot!
Reviewer 2 Report
The paper investigates the gaps among prime numbers. The paper is well-written and the conclusions make sense. Some improvements should be made. In particular, the bibliography presents some weak points. In recent years, the link between fractal geometry has been applied in prime number theory. Therefore, I suggest adding the references reported below.
1) Batchko, R.G. A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes. arXiv 2014, arXiv:1405.2900v2.
2) E. Guariglia. Primality, Fractality and Image Analysis, Entropy 2019, 21(3), 304.
3) Salas, C. Base-3 repunit primes and the Cantor set. Gen. Math. 2011, 19(2), 103–107.
4) Ares, S.; Castro, M. Hidden structure in the randomness of the prime number sequence? Physica A 2006, 360(2), 285–296.
Moreover, I strongly recommend an additional English review.
p { margin-bottom: 0.1in; line-height: 115%; }Author Response
> The paper investigates the gaps among prime numbers. The paper is well-written and the conclusions make sense.
Authors' reply:
Thanks a lot for refereeing the paper. We are glad that overall the paper made a positive impression.
> Some improvements should be made. In particular, the bibliography presents some weak points. In recent years, the link between fractal geometry has been applied in prime number theory. Therefore, I suggest adding the references reported below.
1) Batchko, R.G. A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes. arXiv 2014, arXiv:1405.2900v2.
2) E. Guariglia. Primality, Fractality and Image Analysis, Entropy 2019, 21(3), 304.
3) Salas, C. Base-3 repunit primes and the Cantor set. Gen. Math. 2011, 19(2), 103–107.
4) Ares, S.; Castro, M. Hidden structure in the randomness of the prime number sequence? Physica A 2006, 360(2), 285–296.
Authors' reply:
All of these references are now added. The introduction (sect.1) has been extended to indicate links to other subsets of primes; namely, the new subsection 1.3 lists gives more areas of applicability of our formulas (1) and (2); the new subsection 1.4 gives examples where formulas (1),(2) are not applicable. In Latex, each major change is marked with a line beginning with "%%% edit April 22 -- "
We greatly appreciate your insights helping make the paper better. A native English speaker has edited the paper, so language should not be a concern in this version. Many thanks for your input!
Round 2
Reviewer 2 Report
The authors investigated the maximal gaps in set of prime numbers. The suggested revision has been applied. As a result, both the bibliography and the English have been improved.
