Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions
Round 1
Reviewer 1 Report
The author deals with the special case of the famous Tary-Escott problem. I think the results are interesting and their proofs are correct. I like the computer-aided presentation which is important for theoretical mathematics including number theory.
Two remarks: Please use alphabetical order for reference. The equation in row 58 is a very special and trivial case of equation (1). For F=1, equation (1) leads to xyz=x+y+z, which is more general than xyz=x+y+z=1.
Author Response
Point 1: The author deals with the special case of the famous Tary-Escott problem. I think the results are interesting and their proofs are correct. I like the computer-aided presentation which is important for theoretical mathematics including number theory.
Response 1: Thank you for the positive view of the work and for checking the proofs. I am glad that the efforts devoted to the presentation of computation results paid off.
Point 2: Two remarks: Please use alphabetical order for reference.
Response 2: I would be glad to comply but the instructions of MDPI forces me to list and number the references in the order in which they are cited.
Point 3: The equation in row 58 is a very special and trivial case of equation (1). For F=1, equation (1) leads to xyz=x+y+z, which is more general than xyz=x+y+z=1.
Response 3: Thank you for pointing it out. In the revised version, the text is improved as follows:
"For F=1 and n=3, (1) becomes xyz=x+y+z. A special case of this equation is
...,
which does not have any solution in Z...
Reviewer 2 Report
In this article, the author faces the problem of studying the set of solutions of certain Diophantine equation, F(∑xi)=∏xi, F being a fixed natural number. In particular, the author studies the solvability of the equation, as well as the upper and lower bounds of the sums of the terms of such solutions. When F equals 1, this problem has been largely studied. Thus, the problem proposed by the author to be studied is a natural and very interesting generalization of the classical one.
Despite of this, I think the results presented in the article are not novel enough considering the standards of the journal, and I would encourage the author to get deeper inside the problem. For instance, the problem of solvability of this equation can be presented as a particular case of the problem stated in Zhang et al 2013 (and related works). Also, most of the explicit solutions obtained concern the particular cases n=2,3 which can be solved by inspection, as the author does. The most interesting part is the one regarding the explicit upper and lower bounds of the sums of the terms of the solutions. It would be certainly valuable if the author gave the proof of any of them, apart from supporting the conjectures by computational methods.
Author Response
Point 1: In this article, the author faces the problem of studying the set of solutions of certain Diophantine equation, F(∑xi)=∏xi, F being a fixed natural number. In particular, the author studies the solvability of the equation, as well as the upper and lower bounds of the sums of the terms of such solutions. When F equals 1, this problem has been largely studied. Thus, the problem proposed by the author to be studied is a natural and very interesting generalization of the classical one.
Response 1: Thank you!
Point 2: Despite of this, I think the results presented in the article are not novel enough considering the standards of the journal, and I would encourage the author to get deeper inside the problem. For instance, the problem of solvability of this equation can be presented as a particular case of the problem stated in Zhang et al 2013 (and related works).
Response 2: I cited Zhang et al. (2013) because the method they used to prove Lemma 2.1 on n-tuples in Q inspired the construction of a universal parametric solution in Section 5. Zhang et al. (2013) results on sets of n-tuples in Z are however not applicable because they counted solutions of (1.2) (No. in their paper) for suitable A,B > 0 while we counted solutions of (1) (No. in our manuscript) for all F, corresponding to integral values of B/A of Zhang's et al. paper. Furthermore, the current manuscript also shows results on distinct-term solutions while most of the solutions provided by Zhang et al. (2013) for n = 4 and none of their solutions for n = 5 and n= 6 consist of distinct terms.
Point 3: Also, most of the explicit solutions obtained concern the particular cases n=2,3 which can be solved by inspection, as the author does.
Response 3: Yes, most but not all of them. These solutions are provided for the sake of completeness and also because they are used in the proof of Theorem 1, which is used in the proofs of Theorems 3 and 4 on the relation between the primality of F and solvability of (1) with n=2, and also in the proof of Theorem 2, which established an upper bound for n=2 independently of Theorem 6.
Point 4: The most interesting part is the one regarding the explicit upper and lower bounds of the sums of the terms of the solutions. It would be certainly valuable if the author gave the proof of any of them, apart from supporting the conjectures by computational methods.
Response 4: The proofs for both bounds are provided. For the lower bound, the proof was obtained by Lagrange multiplier in Section 6. For the upper bound, the proof was obtained by variable mixing (Theorem 6). Merely the upper bounds for the sum of distinct-term solutions has not been proved. Because of interesting patterns generated by these sums (Figure 6), I hope that the problem will attract others to prove the conjecture. To address the importance of bounds for future work, I added the following sentence to the Discussion:
"The fact that the upper bound for distinct-term solutions have not been proved and a lower bound is unknown will not hinder future computational studies because weak bounds for the sum of non-distinct solution terms can be used instead."
Reviewer 3 Report
This paper is interesting for at least two reasons: for its computation-aided approach to discovering mathematics and for opening some new questions/conjectures in Diophantine equations relating sums and products of integers. Computation-aided approach is probably more common than it is reported, while the author of this paper explicitly and honestly represents his methods, results and approach. In general, author first shows an upper bound for the sum of solution terms, then uses this bound to enumerate and graphically display the solutions. Such enumeration and figure(s) then suggest some results which he successfully proves. The whole process also motivates three conjectures which are supported by the data and will certainly prompt further investigation. Aside from some comments that are stated in the next paragraphs, I can suggest the acceptance of this paper after minor revision.
There is one short part that seems completely unnecessary -- using Dolan's method to obtain an upper bound for sum of solutions that turns out to be weaker than the trivial one. I would suggest replacing this (lines 119-130) with a short comment that Dolan's method results with an even weaker bound.
On the other hand, computation-aided approach of this paper suggests that author should try and place this work within the field of computation-aided discovery and automated conjecturing. Fajtlowicz's Graffiti is perhaps the most significant example, but there have been few papers using this approach in number theory/combinatorics, even some in the past five years. That is why I would suggest adding a paragraph at the end of the introduction with a few references on such papers.
Minor typos:
lines 165, 374, 409 : shell -> shall
line 237: v \leq sqrt(F) makes no sense, shouldn't there be one factor larger that sqrt(F)?
line 409: "of consisting of" -> "consisting of"
line 458: I think (n+k+2)(n-2)! should be (n+k-2)(n-2)!
(If not, then Figure 6 has a sign error)
line 491: "[e.g. 27]" -> e.g. [27]
Author Response
Point 1: This paper is interesting for at least two reasons: for its computation-aided approach to discovering mathematics and for opening some new questions/conjectures in Diophantine equations relating sums and products of integers. Computation-aided approach is probably more common than it is reported, while the author of this paper explicitly and honestly represents his methods, results and approach. In general, author first shows an upper bound for the sum of solution terms, then uses this bound to enumerate and graphically display the solutions. Such enumeration and figure(s) then suggest some results which he successfully proves. The whole process also motivates three conjectures which are supported by the data and will certainly prompt further investigation. Aside from some comments that are stated in the next paragraphs, I can suggest the acceptance of this paper after minor revision.
Response 1: Thank you.
Point 2: There is one short part that seems completely unnecessary -- using Dolan's method to obtain an upper bound for sum of solutions that turns out to be weaker than the trivial one. I would suggest replacing this (lines 119-130) with a short comment that Dolan's method results with an even weaker bound.
Response 2: The derivation by Dolan's method was removed entirely and the text was replaced with the following comment:
"Application of the method of Stan Dolan [19] led to an upper (2F2+2), which is even weaker than the upper bound derived from the bounds for both terms according to [18]..."
Point 3: On the other hand, computation-aided approach of this paper suggests that author should try and place this work within the field of computation-aided discovery and automated conjecturing. Fajtlowicz's Graffiti is perhaps the most significant example, but there have been few papers using this approach in number theory/combinatorics, even some in the past five years. That is why I would suggest adding a paragraph at the end of the introduction with a few references on such papers.
Response 3: Thank you for this advice. The following paragraph was added to the Introduction, with new references 18-20:
"Computing has been used in mathematics mainly for four purposes, in an increasing level of sophistication: (i) to disprove or support conjectures by brute-force computation [18]; (ii) provide insight and inspiration by revealing patterns and relationships; (iii) generate new conjectures [19]; and (iv) provide formal proofs [20]. While testing conjectures dominated computing in number theory after the advent of machine computing, automatic conjecturing and theorem proving are subjects of vigorous current research. The second application mentioned, in which computing inspires rather than generates conjectures, is elusive, hard to capture, and rarely discussed. It may vaguely be described as a pattern-supported trial-and-error approach. In this work, apart from exhaustive enumeration of solutions, we make a heavy use of computing as a discovery tool, especially in sections 8 and 9."
18. Borwein, J.; Bailey, D. Mathematics by Experiment, 2nd ed.; CRC Press: Boca Raton, USA, 2008; pp. 1-365.
19. Fajtlowicz, S. On Conjectures of Graffiti. Discrete Mathematics 1988, 72, 113-118.
20. Kaliszyk, J.; Urban, J. Learning-Assisted Automated Reasoning with Flyspeck. J. Autom. Reasoning 2014, 53, 173-213.
Point 4: Minor typos:
lines 165, 374, 409 : shell -> shall
line 237: v \leq sqrt(F) makes no sense, shouldn't there be one factor larger that sqrt(F)?
line 409: "of consisting of" -> "consisting of"
line 458: I think (n+k+2)(n-2)! should be (n+k-2)(n-2)!
(If not, then Figure 6 has a sign error)
line 491: "[e.g. 27]" -> e.g. [27]
Response 4: Thank you very much for catching these errors! In the revised version they were all corrected.
Round 2
Reviewer 2 Report
Major concerns:
I have some doubts about the relevance of the obtained results. Sections 2, 3 and 4 deal only with the particular cases of degree 2 and 3. The method for proving the results belonging to that sections is by inspection, so no novel methods are introduced. Section 5 deals with the general equation (degree larger that 3). However, the equation is proved to be solvable again by inspection, once all indeterminates except two are made to be equal 1. This is how the classical equation, where F=1, is proved to be solvable (obtaining the solution (1,...,1,2,n)), so no novel method is introduced.
Sections 6 and 7 are certainly interesting since something about the "space of solutions" is said and, apprently, they should be the core of the article.
Minor concerns:
Please, enounce the main theorem(s) of the article in the introduction. For instance, state the two results concerning the lower and upper bounds for the sum of the terms of the solutions.
Author Response
Thank you for your time and the comments.
Point 1: I have some doubts about the relevance of the obtained results. Sections 2, 3 and 4 deal only with the particular cases of degree 2 and 3. The method for proving the results belonging to that sections is by inspection, so no novel methods are introduced.
Response 1: These solutions are easy to prove but not easy to discover. Besides, introducing novel methods was not the purpose here. These solutions are provided for their own sake, to shed light on the relationship among solutions, and to provide access to upper bounds and solvability. The key result is a proof that (5) generates all solutions of (1) of 2nd degree, which enabled a proof of the upper bound that is independent of Theorem 6, and also a proof of the relation between the primality of F and solvability. These results are novel and important, regardless of how novel were the methods used for their proofs.
Point 2: Section 5 deals with the general equation (degree larger that 3). However, the equation is proved to be solvable again by inspection, once all indeterminates except two are made to be equal 1. This is how the classical equation, where F=1, is proved to be solvable (obtaining the solution (1,...,1,2,n)), so no novel method is introduced.
Response 2: Our result for any F and the classical result for F=1 are related but the relationship is the other way around. Our solution (1, 1, ..., F+1, F*(F+n-1)) for any F implies the classical solution (1, 1, ..., 2, n) for F=1. The classical solution for F=1 does not imply our general solution for any F. Thus, our solution is novel, as is the proof that equation (1) of any degree with any F is solvable, which is based on the solution.
Point 3 Sections 6 and 7 are certainly interesting since something about the "space of solutions" is said and, apparently, they should be the core of the article.
Response 3: Thank you for this assessment. I also see these results as the core of the work but other people might find other results more interesting: for instance the surprising distinct-term solution for 5th degree when (F+9) is a multiple of 23; or the fact equations of 4th degree with any F are solvable in distinct terms but equations of 5th to 9th degree not; or the intriguing upper bound (18) for the sum of terms of distinct-term solutions. Many results generated by this work are interesting, and I suppose that people would differ in their judgement which results are the most interesting.
Point 4: Please, enounce the main theorem(s) of the article in the introduction. For instance, state the two results concerning the lower and upper bounds for the sum of the terms of the solutions.
Response 4: Thank you for this advice. Major theorems are declared at the end of Introduction in the revised version. The first part of Introduction now received a subheading "1.1. Background", and the second part a subheading 1.2:
"1.2. Scope of the work and main theorems"
...
"Main theorems provide the following results. Equation (1) of any degree with any F has a solution (1, 1, ..., F+1, F(F+n-1)). Equation of the 2nd degree has exactly two solutions if and only F is a prime, and an upper bound for the sum of solution terms is (F2+2F+1). Lower and upper bounds for the sum of solution terms of (1) of any degree with any F are \sqrt[n-1]{Fn^n} and (F+1)(F+n-1), respectively. Equations of the 4th degree with any F are solvable in distinct terms but equations of the 5th to 9th degree are not."
