New results similar to Lagrange's four-square theorem

Zhi-Wei Sun

REVIEW 1 major objections 5 minor 1 cited by

New results similar to Lagrange's four-square theorem

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that every sufficiently large integer is a sum of four values of a single quadratic polynomial from a broad family, generalizing the four-square theorem.

desk verdict Broadens the Lagrange-style four-square family to a two-parameter quadratic sequence, and the central theorems hold up; Theorem 1.5 has a localized but fixable congruence slip. read the letter →

arxiv 2411.14308 v3 pith:NJP6KT7D submitted 2024-11-21 math.NT

Zhi-Wei Sun This is my paper · ORCID

classification math.NT MSC 11B1311E2511D8511E2011P05

keywords additivebasisasymptoticpolygonalnumberssecondpentagonalquadraticpolynomialsfour-squaretheoremsumsoffourvaluesnumbertheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes new four-term additive results for families of quadratic numbers, in the spirit of the classical theorem that every nonnegative integer is a sum of four squares. The main theorem says that for coprime parameters $a,b$ with $ab$ odd, the set of numbers $x(ax+b)/2$ with $x$ nonnegative is an asymptotic basis of order four, with an explicit threshold; the analogue for $x(ax+b)$ holds when $a$ is even and $b$ is odd. The headline instance is the second pentagonal numbers $w(5w+1)/2$: every integer $n>1$ is a sum of four such generalized numbers with integer coefficients. The proof selects an auxiliary integer $d$ and, by a congruence condition, finds $c$ so that $n=ac+bd$, then invokes a four-squares-with-fixed-sum lemma to complete the representation.

What carries the argument

The machinery is a reduction to a four-square-with-prescribed-sum lemma. If $c$ and $d$ have the same parity, $4\nmid c$, and $d^2/4<c<(d^2+2d+4)/3$, then $c$ is a sum of four squares $w^2+x^2+y^2+z^2$ and $d$ is the sum $w+x+y+z$. The paper writes the target $n$ as $ac+bd$, chooses $d$ in a carefully positioned interval whose length exceeds the modulus step, and solves a congruence modulo $2a$ (or $4a$ or $6a$) so the resulting $c,d$ satisfy the lemma. The identity $n=a(w^2+x^2+y^2+z^2)+b(w+x+y+z)$ then becomes the desired sum $w(aw+b)/2+x(ax+b)/2+y(ay+b)/2+z(az+b)/2$. The weighted variants use analogous interval and congruence arguments with $5$ and $9$ divisors.

What would settle it

For $a=3$, $b=2$, and $n=187353$, the construction in the proof yields $d=585$ and $c\equiv6\pmod9$, so the required $c\equiv3\pmod9$ fails; checking whether this $n$ is actually representable as $3w(3w+2)+x(3x+2)+y(3y+2)+z(3z+2)$ would show whether Theorem 1.5 is false or merely unproved for $b\equiv2\pmod3$.

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Extended reading notes

Core claim

The central claim is that a single quadratic family $\{x(ax+b)/2 : x\in\mathbb{N}\}$, for coprime $a,b$ with the right parity, represents all sufficiently large integers as a sum of four elements. The flagship case is the second pentagonal numbers: with $(a,b)=(5,1)$, every integer $n>1$ can be written as $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb{Z}$. The same method yields weighted variants, including sums where one summand carries a factor $2$ or where the first coefficient is $3$, each with explicit thresholds and short exception lists. The theorems are effective: they give bounds rather than only a statement of eventual representability.

Load-bearing premise

The proof of the coefficient-three theorem assumes a divisibility identity that only holds for $b=1$; for many allowed values of $b$ the constructed numbers violate the condition the final lemma needs, so the claim is not established for those parameters.

Editorial extensions

If this is right

For $(a,b)=(5,1)$, every integer $n>1$ is a sum of four generalized second pentagonal numbers, settling the second-pentagonal analogue of the four-square theorem.
For each admissible $(a,b)$, the explicit threshold in Theorem 1.1 turns the asymptotic statement into a finite verification problem; the corollaries show such verifications produce clean exception lists such as $\{1,3,5,10,12,20,25,27,38,53,65,153,165\}$.
The weighted theorems show that replacing one coefficient by $2$ or $3$ preserves four-term universality over the integers for all large targets, matching the earlier coefficient-$1$ results.
In the even-$b$ case, Theorem 1.3 implies the same family is an asymptotic base of order five, because one of $n$ or $n-(a+b)$ is never divisible by $4$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The explicit bounds are likely far from sharp: the interval-length method typically pays large constants, and the paper's own numerical corollaries suggest the true thresholds are much smaller for small $a,b$.
The congruence difficulty in the coefficient-$3$ theorem appears repairable by a different residue choice; if repaired, the coefficient-$3$ statement would likely hold for all odd coprime pairs, matching the exception lists the paper derives.
The same reduction template could apply to any quadratic whose values pair with four-square-sum data through $n=ac+bd$, suggesting that the class of four-term asymptotic bases of order four may be much broader than the families listed in the theorems.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: Theorems 1.1–1.4 look right to me, and the uniform treatment of the two-parameter family x(ax+b)/2 and x(ax+b) is genuinely new. Theorem 1.5 has a concrete algebra slip in Case 2, but it is a fixable proof gap, not a counterexample.

What the paper actually does: for coprime a,b with parity conditions, it shows the fourfold sums of the quadratic sequence form an asymptotic basis, with explicit thresholds. The interval arguments in Section 2 are detailed and check out; the extension of Krachun–Sun to general a,b in Theorem 1.4 is also plausible. The corollaries with explicit exceptional sets are nice, though the finite verifications are asserted without code or data.

The soft spot is real. In Section 4, Case 2 (3|a, 3|n), the paper writes c=(n−bd)/a = −d0 + 6q − 6r, but the last term should be −6br. With b≡2 (mod 3), the chosen r does not force c≡3 (mod 9), so Lemma 4.1's condition can fail. The stress-test instance a=3, b=2, n=187353 gives d=585 and c≡6 (mod 9), which confirms the failure. This affects Theorem 1.5 for all parameters in that residue class and hence the general Corollary 1.9. It is easily repaired: choose r ≡ b^{-1}(q+1+d0/3) (mod 3) instead, and the same interval-length argument keeps d in K. I'd ask the author to fix that and to double-check the congruence display.

The self-citation load is heavier than I'd like—three lemmas come from the author's earlier papers—but those are published and the lemmas are used as black boxes, which is normal for this area. The main theorems do not assume their own conclusion.

Recommendation: send it to review. The gap is localized and fixable, the central results are worth having, and the explicit bounds are a useful addition. For the finite corpus checks, ask for machine-readable verification or at least a reproducible script.

Referee Report

1 major / 5 minor

Summary. The paper proves several effective asymptotic-basis statements for families of quadratic polynomials. Theorem 1.1 shows that for a>0, b>-a, gcd(a,b)=1, and 2∤ab, the set {x(ax+b)/2 : x∈N} is an asymptotic base of order four, with an explicit bound in (1.1); Theorems 1.2 and 1.3 give analogous statements in the even-a and even-b cases for the unhalved polynomials; Theorem 1.4 handles a coefficient 2 on one variable with gcd(a,5b)=1; and Theorem 1.5 treats a coefficient 3 under a parity hypothesis. The paper derives concrete corollaries, including the headline statement that every integer n>1 is a sum of four generalized second pentagonal numbers, and several exact descriptions of representable sets after finite computer checks. The proofs are elementary, based on variants of Cauchy's four-square lemma and interval arguments.

Significance. If the proofs are completed, the paper makes a meaningful contribution to additive number theory: it provides uniform effective four-square-type theorems for a two-parameter family of quadratic polynomials, extending classical results of Cauchy and Legendre as well as earlier work of Meng-Sun and Krachun-Sun. The bounds are explicit, the finite verifications in the corollaries are checkable, and Theorems 1.1-1.4 appear internally consistent. The principal weakness is the gap in Theorem 1.5, Case 2, which is load-bearing for that theorem and for Corollary 1.9. The gap is localized and appears repairable, so the central thesis of the paper remains plausible, but the manuscript as written is not fully correct.

major comments (1)

[Section 4, Case 2 (proof of Theorem 1.5, p. 16)] The displayed computation of c is algebraically wrong for general b. From n=(b-a)d0+6aq and d=d0+6ar, the correct value is c=(n-bd)/a = 6q - d0 - 6br, not 6q - d0 - 6r. Therefore Lemma 4.1's requirement c≡3 (mod 9) is equivalent to br ≡ q + 1 + d0/3 (mod 3). The paper's choice r ≡ q + 1 + d0/3 (mod 3) works only when b≡1 (mod 3). For b≡2 (mod 3) the construction can produce a pair (c,d) with d≡0 (mod 3) but c not ≡ 3 (mod 9); a concrete instance is a=3, b=2, n=187353, where the construction gives d0=567, q=10440, r=1, d=585, and c=(187353-2·585)/3=62061≡6 (mod 9). Since this step is used for all parameters with 3|a, 3|n, and b≡2 (mod 3), Theorem 1.5 and Corollary 1.9 are not proved as written. The gap is local: choosing r with br≡q+1+d0/3 (mod 3) would restore the congruence, and d0+6ar remains in K because the interval length is >18a.

minor comments (5)

[Theorem 1.5 statement] The hypothesis 'with ab or n even' should read 'with ab even or n even'; as printed the parity condition is ungrammatical and could be misread.
[Corollary 1.6] The statement 'any integer /greaterorequalslant...' appears to have lost its variable; the bound should be explicitly attached to n.
[Corollaries 1.1, 1.3, 1.5, 1.10, 1.11] Several corollaries rely on finite computer verifications described only as 'easy computations via a computer'; for reproducibility, the author should specify the verification range and, ideally, provide the checking code or the full list of small exceptional values.
[Global presentation] The arXiv text contains numerous typographical artifacts, including 'RESUL TS' in the header, 'an y' in the abstract, and converted LaTeX symbols such as '/greaterorequalslant' and 'suﬃciently'; these should be cleaned in the final version.
[Section 5, Theorems 5.1 and 5.2] These theorems are asserted without proof, with the note that they follow from Lemma 5.1 and arguments similar to Section 2; since they are not used elsewhere, this is not blocking, but the author should either provide the proofs or clearly label the results as sketched.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: target theorems are proved from independent Cauchy-type lemmas; the algebraic gap in Theorem 1.5 is a correctness issue, not a circular step.

full rationale

Theorems 1.1–1.5 are derived by reducing n = ac + bd to a four-square (or weighted four-square) representation of c with a prescribed digit sum d. The reduction is carried out in the paper: Theorem 2.1 chooses d in an interval I and then invokes Lemma 2.1; Theorem 1.4 reduces to Lemma 3.1; Theorem 1.5 reduces to Lemma 4.1. None of these lemmas states or assumes the target theorem. Lemma 2.1 is essentially Cauchy's classical lemma (Remark 2.1) with a parity variant from [4]; Lemma 4.1 is the analogous 3w² + x² + y² + z² lemma from [4]; Lemma 3.1 is a standalone characterization from [2]. These are parameter-free auxiliary results whose assumptions do not include the conclusion being proved, so citing them is independent evidence rather than circularity. The explicit bounds (1.1), (1.2), and (1.5) are derived from interval-length estimates, not fitted to the conclusions. No equation in the paper is equal to its target by construction, and no fitted value is renamed as a prediction. The reader's algebraic concern about Section 4, Case 2 (the congruence c ≡ 3 mod 9) is a proof-correctness gap for parameters with b ≡ 2 mod 3; it does not make the theorem an input to itself and therefore does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; all bounds are explicit constants. The paper's results rest on three imported lemmas from prior work (two from papers coauthored by the author) and on unshown finite computations.

assumptions (4)

standard math Lemma 2.1 (Cauchy's lemma variant): if c≡d (mod 2), 4∤c, 4c>d² and 3c<d²+2d+4, then c is a sum of four squares with sum d.
Stated without proof, attributed to [4, Lemma 2.4]; it underlies the core interval argument in Theorem 2.1 and hence Theorems 1.1-1.3.
domain assumption Lemma 3.1: every even n outside E={5^{2k+1}m: m≡±2 (mod 5)} is representable as x²+y²+z²+(x+y+z)²/2.
Imported from [2, Lemma 2.1], a paper coauthored by the present author; Theorem 1.4 and its corollaries depend entirely on this lemma.
domain assumption Lemma 4.1: under the stated congruence and inequalities, the system 3w²+x²+y²+z²=c and 3w+x+y+z=d is solvable.
Imported from [4, Lemma 3.1]; it is the engine of Theorem 1.5.
ad hoc to paper Finite computer verifications listed in corollaries are correct.
The manuscript reports exact exception sets and small-case checks without supplying code or certificates; the central 'n>1' claim in Corollary 1.2 depends on the check that 2..775 are representable.

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Pith. "Pith review of New results similar to Lagrange's four-square theorem." pith.science (2026). https://pith.science/paper/NJP6KT7D

@misc{pith2026241114308,
  author       = {Pith},
  title        = {Pith review of: New results similar to Lagrange's four-square theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NJP6KT7D}},
  note         = {Machine review of arXiv:2411.14308}
}

abstract

In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer $n>1$ can be written as $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb Z$. Let $a$ and $b$ be integers with $a>0$, $b>-a$ and $\gcd(a,b)=1$. When $2\nmid ab$, we show that any sufficiently large integer can be written as $$\frac{w(aw+b)}2+\frac{x(ax+b)}2+\frac{y(ay+b)}2+\frac{z(az+b)}2$$ with $w,x,y,z$ nonnegative integers. When $2\mid a$ and $2\nmid b$, we prove that any sufficiently large integer can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)$$ with $w,x,y,z$ nonnegative integers.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Universal quaternary mixed sums involving generalized 3-, 4-, 5- and 8-gonal numbers via products of Ramanujan's theta functions
math.NT 2025-07 reject novelty 4.0 of 10

The paper lists quaternary universal sums of generalized polygonal numbers, but its main proof lemma, Corollary 1.3, is false as stated.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

[1]

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work page 1994
[2]

Krachun and Z.-W

D. Krachun and Z.-W. Sun, On sums of four pentagonal numbers with coeﬃcients , Electron. Res. Arch. 28 (2020), 559–566

work page 2020
[3]

Sums of four polygonal numbers: precise formulas

J. Li and H. Wang, Sums of four polygonal numbers: precise formulas , arXiv:2405.14710, 2024

work page Pith review arXiv 2024
[4]

Meng and Z.-W

X.-Z. Meng and Z.-W. Sun, Sums of four polygonal numbers with coeﬃcients , Acta Arith. 180 (2017), 229–249

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[5]

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work page 1996
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Z.-W. Sun, On universal sums of polygonal numbers , Sci. China Math. 58 (2015), 1367–1396

work page 2015
[8]

Sun, A result similar to Lagrange’s theorem , J

Z.-W. Sun, A result similar to Lagrange’s theorem , J. Number Theory 162 (2016), 190–211. NEW RESULTS SIMILAR TO LAGRANGE’S FOUR-SQUARE THEOREM 19

work page 2016

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[9]

Sun, On universal sums x(ax + b)/2 +y(cy + d)/2 +z(ez + f )/2, Nanjing Univ

Z.-W. Sun, On universal sums x(ax + b)/2 +y(cy + d)/2 +z(ez + f )/2, Nanjing Univ. J. Math. Biquarterly , 35(2) (2018), 85–199

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[12]

Wu and Z.-W

H.-L. Wu and Z.-W. Sun, Some universal quadratic sums ov er the integers, Electronic Research Archive, 27 (2019), 69–87. School of Mathematics, Nanjing University, Nanjing 210093 , People’s Re- public of China Email address : zwsun@nju.edu.cn

2019

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

R. K. Guy, Every number is expressible as the sum of how many polygonal n umbers? Amer. Math. Monthly 101 (1994), 169–172

work page 1994

[2] [2]

Krachun and Z.-W

D. Krachun and Z.-W. Sun, On sums of four pentagonal numbers with coeﬃcients , Electron. Res. Arch. 28 (2020), 559–566

work page 2020

[3] [3]

Sums of four polygonal numbers: precise formulas

J. Li and H. Wang, Sums of four polygonal numbers: precise formulas , arXiv:2405.14710, 2024

work page Pith review arXiv 2024

[4] [4]

Meng and Z.-W

X.-Z. Meng and Z.-W. Sun, Sums of four polygonal numbers with coeﬃcients , Acta Arith. 180 (2017), 229–249

work page 2017

[5] [5]

M. B. Nathanson, A short proof of Cauchy’s polygonal number theorem , Proc. Amer. Math. Soc. 99 (1987), 22–24

work page 1987

[6] [6]

M. B. Nathanson, Additive Number Theory: The Classical Bases , Grad. Texts in Math., Vol. 164, Springer, New York, 1996

work page 1996

[7] [7]

Sun, On universal sums of polygonal numbers , Sci

Z.-W. Sun, On universal sums of polygonal numbers , Sci. China Math. 58 (2015), 1367–1396

work page 2015

[8] [8]

Sun, A result similar to Lagrange’s theorem , J

Z.-W. Sun, A result similar to Lagrange’s theorem , J. Number Theory 162 (2016), 190–211. NEW RESULTS SIMILAR TO LAGRANGE’S FOUR-SQUARE THEOREM 19

work page 2016