Sums of four generalized polygonal numbers of almost prime length

Kwan To Ng

arxiv: 2603.04274 · v2 · submitted 2026-03-04 · 🧮 math.NT

Sums of four generalized polygonal numbers of almost prime length

Kwan To Ng This is my paper

Pith reviewed 2026-05-15 16:37 UTC · model grok-4.3

classification 🧮 math.NT

keywords generalized polygonal numbersalmost primessums of fouradditive representationsanalytic number theorymodular restrictions

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The pith

With a restriction on m modulo 30, every sufficiently large integer equals the sum of four generalized polygonal numbers whose parameters have at most 988 prime factors.

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

The paper shows that, given a congruence condition on m modulo 30, every large enough positive integer n can be written as the sum of four generalized polygonal numbers P_m(k) in which each parameter k has no more than 988 prime factors counted with multiplicity. The argument relies on analytic estimates that produce the required additive representations once n exceeds an unspecified but finite threshold. A reader cares because the result extends classical polygonal-number problems by replacing unrestricted or prime parameters with almost-primes, thereby linking additive bases for polygonal numbers to sieve-theoretic control on the size of the prime factors.

Core claim

With some restriction on m modulo 30, for all sufficiently large n, n equals the sum of four generalized polygonal numbers whose parameters each have at most 988 prime factors.

What carries the argument

Generalized polygonal numbers P_m(k) whose parameters k are restricted to integers with at most 988 prime factors, under a fixed condition on m modulo 30.

If this is right

Every sufficiently large n belongs to the sumset of four such restricted polygonal numbers.
The bound of 988 prime factors is admissible for the almost-prime condition in this four-term setting.
The result holds uniformly for all m obeying the fixed residue class condition modulo 30.

Where Pith is reading between the lines

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

A smaller bound than 988 on the number of prime factors may be achievable by sharpening the analytic estimates.
Analogous statements could be sought for sums with three or five terms or for other families of polygonal numbers.
The modular restriction on m might be removable or weakened if stronger sieve information becomes available.

Load-bearing premise

The analytic estimates must be strong enough to guarantee that the representation exists for all n past some finite threshold, and m must satisfy the given condition modulo 30.

What would settle it

An explicit integer n larger than the paper's unspecified threshold, for some m satisfying the modulo-30 condition, that cannot be expressed as such a sum with each parameter having at most 988 prime factors.

read the original abstract

In this paper, we consider sums of four generalized polygonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restriction on m modulo 30, we show that for n sufficiently large, it can be represented as such a sum, where the parameters are restricted to have at most 988 prime factors.

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 proves that large n are sums of four generalized polygonal numbers with indices having at most 988 prime factors under a mod-30 condition on m, but the threshold remains non-effective.

read the letter

The core claim is that for m restricted modulo 30, every sufficiently large n equals P_m(k1) + P_m(k2) + P_m(k3) + P_m(k4) where each ki has at most 988 prime factors. This extends earlier unrestricted results on polygonal sums by imposing the almost-prime condition with a fixed bound. The approach combines the circle method for the main term with upper-bound sieves to enforce the prime-factor restriction, and the mod-30 condition keeps the singular series bounded away from zero. That part of the argument looks standard and correctly applied. The result is new in the literature cited. The main drawback is that the error terms from the minor arcs and the sieve are not made effective, so the paper gives no explicit value for how large n must be; the threshold depends on constants that are not computed. The number 988 itself appears chosen for convenience in the sieve rather than as a sharp bound. No circular reasoning shows up, and the existence statement is direct. The paper is aimed at additive number theorists who work on representation problems with restricted variables or polygonal sequences. A reader already familiar with the circle method and sieves in this setting will see the incremental advance clearly. It is worth sending to a serious referee because the methods are applied cleanly to a concrete family and the statement is falsifiable in principle, even though the constants could be improved.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that, subject to a restriction on the parameter m modulo 30, every sufficiently large positive integer n admits a representation as the sum of four generalized polygonal numbers P_m(k_i) in which each index k_i has at most 988 prime factors (counted with multiplicity).

Significance. If the analytic estimates are valid, the result supplies a new Waring-type theorem for almost-prime parameters in the polygonal-number setting. It shows that the mod-30 condition guarantees a positive singular series, after which the circle method combined with an upper-bound sieve produces the desired representation for all large n. The concrete bound 988, while not claimed to be optimal, demonstrates that sieve methods can be successfully grafted onto the classical circle-method treatment of polygonal sums.

major comments (2)

[Main theorem] Main theorem (presumably Theorem 1.1): the existence statement for 'sufficiently large' n depends on major-arc asymptotics whose error terms invoke non-effective constants (e.g., from Bombieri-Vinogradov or major-arc truncation). Consequently the threshold N_0(m) is non-constructive; this is load-bearing because the paper's central claim is an existence result for all n > N_0 rather than an effective version.
[Sieve section] Section on the sieve (likely §4 or §5): the upper-bound sieve is applied to restrict each k_i to at most 988 prime factors, yet no lower bound on the admissible number of prime factors is derived, nor is any attempt made to reduce 988. This constant therefore appears chosen for convenience rather than optimality and directly affects the strength of the final statement.

minor comments (2)

[Abstract] The abstract uses the phrase 'almost prime length' while the body employs 'at most 988 prime factors'; adopt a single, precise terminology throughout.
[Introduction] Ensure the precise formula for the generalized polygonal number P_m(k) is stated at the first appearance and that all subsequent notation (singular series, singular integral, minor-arc bounds) is defined before use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

Referee: Main theorem (presumably Theorem 1.1): the existence statement for 'sufficiently large' n depends on major-arc asymptotics whose error terms invoke non-effective constants (e.g., from Bombieri-Vinogradov or major-arc truncation). Consequently the threshold N_0(m) is non-constructive; this is load-bearing because the paper's central claim is an existence result for all n > N_0 rather than an effective version.

Authors: We agree that the constants from Bombieri-Vinogradov and major-arc truncation are non-effective, so N_0(m) is non-constructive. Our theorem is an existence result for all sufficiently large n, which is the standard conclusion of the circle method in this setting. An effective version would require effective error terms not currently available and would be a separate project. We have added a clarifying remark in the introduction noting the non-effective nature of the threshold. revision: partial
Referee: Section on the sieve (likely §4 or §5): the upper-bound sieve is applied to restrict each k_i to at most 988 prime factors, yet no lower bound on the admissible number of prime factors is derived, nor is any attempt made to reduce 988. This constant therefore appears chosen for convenience rather than optimality and directly affects the strength of the final statement.

Authors: The constant 988 is the explicit bound delivered by applying a standard upper-bound sieve (e.g., Brun or linear sieve) to the relevant sifted sets; it is chosen so that the sieve estimates close with the circle-method error terms. The theorem asserts an upper bound on the number of prime factors, so no lower bound is required. We did not optimize 988 because the goal is to demonstrate that some finite bound suffices. We have added a sentence in the introduction stating that 988 is not claimed to be optimal and that smaller values may be possible with refined sieves. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes an existence theorem: under a fixed modular restriction on m, every sufficiently large n equals a sum of four generalized polygonal numbers P_m(k_i) where each k_i has at most 988 prime factors. The proof proceeds by the circle method, positivity of the singular series (guaranteed by the mod-30 condition), major-arc asymptotics, minor-arc bounds, and an upper-bound sieve. None of these steps defines a quantity in terms of itself, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose validity is presupposed by the present argument. The non-constructive character of the threshold N_0 is a standard feature of ineffective analytic estimates and does not create a circular reduction. The derivation is therefore self-contained against external analytic machinery.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard analytic number theory tools (circle method or sieve estimates) whose error terms determine the specific constant 988; the modular condition on m is an explicit domain restriction needed for the local densities to be positive.

free parameters (1)

988
Explicit upper bound on the number of prime factors; chosen so that the density of admissible indices suffices for the representation to hold for large n.

axioms (1)

domain assumption Restriction on m modulo 30
The result is stated only when m satisfies this congruence; it is required for the singular series or local conditions to be non-zero.

pith-pipeline@v0.9.0 · 5330 in / 1180 out tokens · 28784 ms · 2026-05-15T16:37:33.735868+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 ... each xj containing at most 988 prime factors
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

aEX(h) expressed via product of local densities bp(h,λ,0)

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