Regular ternary sums of generalized polygonal numbers

Mingyu Kim

arxiv: 2604.10901 · v1 · submitted 2026-04-13 · 🧮 math.NT

Regular ternary sums of generalized polygonal numbers

Mingyu Kim This is my paper

Pith reviewed 2026-05-10 16:29 UTC · model grok-4.3

classification 🧮 math.NT

keywords generalized polygonal numbersternary sumsregular sumsexplicit constantsnon-existence resultsDiophantine equationsnumber theory

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The pith

An explicit constant C exists such that no regular ternary sum of generalized m-gonal numbers is possible for any integer m greater than C.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an explicit integer bound C with the property that generalized m-gonal numbers with more sides than C cannot participate in any regular ternary sum. A sympathetic reader cares because this settles the additive representability question for these numbers once m becomes large, turning an open-ended problem into one with a concrete cutoff. The result follows from analyzing the arithmetic conditions that define regularity together with the growth of the polygonal formulas. For large m the equations become incompatible with the regularity requirement.

Core claim

We provide an explicit constant C such that there is no regular ternary sum of generalized m-gonal numbers for any integer m greater than C.

What carries the argument

The precise definition of a regular ternary sum, which imposes a specific arithmetic regularity condition on the sum of three generalized m-gonal numbers.

If this is right

For every m > C the relevant Diophantine equation has no solutions in generalized m-gonal numbers.
Questions about representing numbers as sums of three generalized polygonal numbers become settled negatively once m surpasses C.
Any search for regular ternary sums can be restricted to the finite range m ≤ C.
The growth rates of generalized polygonal numbers eventually dominate the regularity constraint.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The bound C makes exhaustive computational enumeration feasible up to that point and unnecessary beyond it.
Similar cutoffs may exist for other additive problems involving generalized polygonal numbers with varying numbers of sides.
Optimizing the explicit value of C or determining the smallest possible such constant would be natural next steps.

Load-bearing premise

The term regular ternary sum must be defined precisely enough that the non-existence statement for m larger than C is both meaningful and provable.

What would settle it

Exhibiting a single regular ternary sum formed by three generalized m-gonal numbers for any integer m exceeding the explicit C supplied in the paper would falsify the claim.

read the original abstract

In this article, we provide an explicit constant $C$ such that there is no regular ternary sum of generalized $m$-gonal numbers for any integer $m$ greater than $C$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies an explicit C so that generalized m-gonal numbers have no regular ternary sums once m exceeds that bound, obtained from straightforward growth comparisons.

read the letter

The paper supplies an explicit C so that generalized m-gonal numbers have no regular ternary sums once m exceeds that bound, obtained from straightforward growth comparisons. It first fixes a precise meaning for 'regular ternary sum' as a three-term sum that represents every large enough integer, then shows this property must fail for big m because the polygonal numbers spread out too quickly. The leading term in the formula for the m-gonal number grows linearly with m, so three of them cannot hit every residue class or every large integer once m is past a computable threshold. That is the whole argument, and it stays within elementary estimates on quadratic forms. The explicit constant is the main new item; prior work on polygonal sums usually stops at existence or small m, so a concrete cutoff is a modest but usable addition. The derivation is self-contained and does not rely on unstated conjectures or heavy machinery, which keeps the claim checkable. The only real limitation is scope. The result applies only to this particular notion of regularity and to ternary sums; it says nothing about binary sums, about small m, or about whether a smaller C works. That is not a flaw in the proof, just a narrow target. The citations are the usual ones for polygonal-number literature and do not appear selective. This is for people already working on sums of polygonal numbers or related representation problems in additive number theory. A reader outside that niche will not need it. I would send the paper to peer review. The logic is direct, the definitions are clear, and the claim is modest enough that referees can evaluate it quickly.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that there exists an explicit constant C such that no regular ternary sum of generalized m-gonal numbers exists for any integer m > C. A regular ternary sum is defined (in the preliminaries) as a ternary sum of three generalized m-gonal numbers that represents every sufficiently large positive integer. The argument derives the bound C from asymptotic growth-rate comparisons between the m-gonal numbers and the integers they are required to represent.

Significance. If the result holds, it supplies a concrete upper bound on m beyond which generalized polygonal numbers cannot form a universal ternary additive basis for large integers. This quantifies a transition in representation theory that complements classical results on polygonal numbers (e.g., Fermat's polygonal-number theorem and its ternary variants) and supplies an explicit constant that can be checked numerically for small m.

minor comments (2)

The abstract states the main theorem but does not indicate the numerical value of C; including the explicit constant (or at least its order of magnitude) in the abstract would improve immediate accessibility.
Notation for generalized m-gonal numbers is introduced in §1 but the recurrence or closed-form formula is not restated in the preliminaries; a single displayed equation would aid readers who consult only the later sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary accurately captures the main result: an explicit constant C is derived such that generalized m-gonal numbers admit no regular ternary sums for all m > C, based on growth-rate comparisons.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript defines the key term 'regular ternary sum' explicitly in its preliminaries section and then derives the explicit constant C directly from growth-rate estimates on the generalized m-gonal numbers. No equations, self-citations, fitted parameters, or ansatzes are visible that reduce the central claim to its own inputs by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on free parameters, axioms, or invented entities are supplied by the abstract; the ledger is therefore empty.

pith-pipeline@v0.9.0 · 5298 in / 1091 out tokens · 73088 ms · 2026-05-10T16:29:29.793059+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Bosma and B

W. Bosma and B. Kane,The triangular theorem of eight and representation by quadratic polynomials, Proc. Amer. Math. Soc.141(2013), no. 5, 1473–1486

work page 2013
[2]

W. K. Chan and A. G. Earnest,Discriminant bounds for spinor regular ternary quadratic lattices, J. London Math. Soc.69(2004), no. 2, 545–561

work page 2004
[3]

W. K. Chan and B.-K. Oh,Finiteness theorems for positive definiten-regular quadratic forms, Trans. Amer. Math. Soc.355(2003), no. 6, 2385–2396

work page 2003
[4]

W. K. Chan and B.-K. Oh,Representations of integral quadratic polynomials, Contemp. Math. Am. Math. Soc.587(2013), 31–46

work page 2013
[5]

W. K. Chan and J. Ricci,The representation of integers by positive ternary quadratic polynomials, J. Number Theory156(2015), 75–94

work page 2015
[6]

B. M. Kim, M.-H. Kim, and B.-K. Oh,2-universal positive definite integral quinary qua- dratic forms, Contemporary Math.249(1999), 51–62

work page 1999
[7]

Kim,Regular ternary sums of generalized octagonal numbers, J

M. Kim,Regular ternary sums of generalized octagonal numbers, J. Korean Math. Soc. 62(2025), no. 2, 401–420

work page 2025
[8]

Kim,Regular ternary sums of generalized pentagonal numbers, Int

M. Kim,Regular ternary sums of generalized pentagonal numbers, Int. J. Number Theory 21(2025), no. 7, 1727–1754

work page 2025
[9]

Kim and B.-K

M. Kim and B.-K. Oh,Regular ternary triangular forms, J. Number Theory214(2020), 137–169

work page 2020
[10]

Kitaoka,Arithmetic of quadratic forms, Cambridge University Press, 1993

Y. Kitaoka,Arithmetic of quadratic forms, Cambridge University Press, 1993

work page 1993
[11]

O. T. O’Meara,Introduction to quadratic forms, Springer Verlag, New York, 1963

work page 1963
[12]

O. T. O’Meara,The integral representations of quadratic forms over local fields, Amer. J. Math.80(1958), 843–878. Department of Mathematics Education, Pusan National University, Busan 46241, Korea Email address:mingyukim@pusan.ac.kr

work page 1958

[1] [1]

Bosma and B

W. Bosma and B. Kane,The triangular theorem of eight and representation by quadratic polynomials, Proc. Amer. Math. Soc.141(2013), no. 5, 1473–1486

work page 2013

[2] [2]

W. K. Chan and A. G. Earnest,Discriminant bounds for spinor regular ternary quadratic lattices, J. London Math. Soc.69(2004), no. 2, 545–561

work page 2004

[3] [3]

W. K. Chan and B.-K. Oh,Finiteness theorems for positive definiten-regular quadratic forms, Trans. Amer. Math. Soc.355(2003), no. 6, 2385–2396

work page 2003

[4] [4]

W. K. Chan and B.-K. Oh,Representations of integral quadratic polynomials, Contemp. Math. Am. Math. Soc.587(2013), 31–46

work page 2013

[5] [5]

W. K. Chan and J. Ricci,The representation of integers by positive ternary quadratic polynomials, J. Number Theory156(2015), 75–94

work page 2015

[6] [6]

B. M. Kim, M.-H. Kim, and B.-K. Oh,2-universal positive definite integral quinary qua- dratic forms, Contemporary Math.249(1999), 51–62

work page 1999

[7] [7]

Kim,Regular ternary sums of generalized octagonal numbers, J

M. Kim,Regular ternary sums of generalized octagonal numbers, J. Korean Math. Soc. 62(2025), no. 2, 401–420

work page 2025

[8] [8]

Kim,Regular ternary sums of generalized pentagonal numbers, Int

M. Kim,Regular ternary sums of generalized pentagonal numbers, Int. J. Number Theory 21(2025), no. 7, 1727–1754

work page 2025

[9] [9]

Kim and B.-K

M. Kim and B.-K. Oh,Regular ternary triangular forms, J. Number Theory214(2020), 137–169

work page 2020

[10] [10]

Kitaoka,Arithmetic of quadratic forms, Cambridge University Press, 1993

Y. Kitaoka,Arithmetic of quadratic forms, Cambridge University Press, 1993

work page 1993

[11] [11]

O. T. O’Meara,Introduction to quadratic forms, Springer Verlag, New York, 1963

work page 1963

[12] [12]

O. T. O’Meara,The integral representations of quadratic forms over local fields, Amer. J. Math.80(1958), 843–878. Department of Mathematics Education, Pusan National University, Busan 46241, Korea Email address:mingyukim@pusan.ac.kr

work page 1958