Congruences via Partitions with Exactly Two Part Sizes

Kirati Sriamorn; Kraiwich Kongsiri; Nitipon Moonwichit; Sittinon Jirattikansakul; Teeradej Kittipassorn

arxiv: 2604.25394 · v1 · submitted 2026-04-28 · 🧮 math.NT · math.CO

Congruences via Partitions with Exactly Two Part Sizes

Sittinon Jirattikansakul , Teeradej Kittipassorn , Kraiwich Kongsiri , Nitipon Moonwichit , Kirati Sriamorn This is my paper

Pith reviewed 2026-05-07 15:08 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords partitions with exactly two part sizesdivisor functioncongruences modulo 4arithmetic progressionscombinatorial identitiesmodular arithmeticnumber theory

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The pith

The sum of the number of divisors of N minus each k squared is congruent to 0 modulo 4 for N in any of five arithmetic progressions.

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

The paper proves that ∑_{1 ≤ k < √N} σ_0(N - k²) ≡ 0 mod 4 when N takes any of the forms 16n + 14, 36n + 30, 72n + 42, 196n + 70 or 252n + 114. It does so by establishing a combinatorial link between this particular divisor sum and the number of partitions of N that use exactly two distinct part sizes, then applying Keith's prior result that the latter count is always divisible by 4. Readers may care because the argument turns an apparently analytic sum into a purely combinatorial object whose modular behavior is already settled, giving a new family of explicit congruences that mix divisor functions with partition counts.

Core claim

We prove the congruence ∑_{1 ≤ k < √N} σ_0 (N - k²) ≡ 0 (mod 4) for N = An + B with (A,B) in {(16,14), (36,30), (72,42), (196,70), (252,114)}, by showing via combinatorial arguments that the left-hand side equals ν₂(N) modulo 4, where ν₂(N) is the number of partitions of N into exactly two part sizes, and invoking the known fact that ν₂(N) is divisible by 4.

What carries the argument

The combinatorial identification, for the listed arithmetic progressions, between the divisor sum ∑ σ_0(N - k²) and the count ν₂(N) of partitions with exactly two part sizes.

If this is right

The stated congruence holds for every positive integer n in each of the five infinite arithmetic progressions.
The same modular relation can be checked or extended by direct computation on small members of each progression.
The result supplies explicit infinite families in which a weighted divisor sum is controlled by the known 4-divisibility of a partition function.
The proof technique combines direct bijective counting with modular arithmetic and therefore applies to other small moduli once analogous identifications are found.

Where Pith is reading between the lines

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

If the identification between the divisor sum and ν₂(N) turns out to be an equality rather than merely a congruence modulo 4, then the sum itself would be divisible by 4 without any restriction on the modulus.
The same linking method could be tested on other residue classes or on sums involving higher powers instead of k².
The five chosen progressions may be the complete list for which the identification holds, or they may be the first members of a larger parametric family.

Load-bearing premise

The divisor sum must match the two-part-size partition count exactly modulo 4 for every N in the five listed arithmetic progressions.

What would settle it

Any single N belonging to one of the five forms for which the computed value of ∑_{1 ≤ k < √N} σ_0(N - k²) is not divisible by 4.

Figures

Figures reproduced from arXiv: 2604.25394 by Kirati Sriamorn, Kraiwich Kongsiri, Nitipon Moonwichit, Sittinon Jirattikansakul, Teeradej Kittipassorn.

**Figure 1.** Figure 1: The Young diagrams for the partitions: (a) (3, 3, 3, 3) of 12, and (b) (5, 5, 5, 2, 2) of 19 3. Proof of Theorem 1 Let N be a natural number that cannot be written as a sum of two squares. A canonical pair for N is an unordered pair of rectangles {X, Y } such that the numbers of rows of X and Y are fewer than or equal to the numbers of columns of X and Y , respectively, and the sum of numbers of cells of X… view at source ↗

**Figure 2.** Figure 2: Young diagrams and their corresponding canonical pairs for N = 6 B. A similar argument applies to C, D, and E. Therefore, at N = 6, |A| = 16 = 6 + 4 + 5 + 1 = |B| + |C| + |D| + |E|. Note that any two multisets X ⋆ Y and Z ⋆ W are either identical or disjoint. Let mS (A) be the multiplicity of A in a multiset S. Let {X, Y } ∈ A. Since N is not a sum of two squares, we know that if X = Y T , then X and Y are… view at source ↗

read the original abstract

We prove the congruence $\sum_{1 \leq k < \sqrt{N}} \sigma_0 (N - k^2) \equiv 0 \pmod 4$, where $\sigma_0(m)$ denotes the number of positive divisors of $m$, for $N = An + B$ with $(A,B) \in \{ (16,14),$ $(36,30),$ $(72,42),$ $(196,70),$ $(252,114) \}$. Our proof relies on a result of Keith which states that $\nu_2 (N) \equiv 0 \pmod 4$, where $\nu_2(N)$ is the number of partitions of $N$ with exactly two part sizes. Inspired by Dewitt and Keith, our approach combines combinatorial arguments with modular arithmetic techniques.

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 gives five concrete new congruences for that divisor sum by tying it to Keith's two-part-size partition count, but the advance stays small and incremental.

read the letter

The core result is five explicit arithmetic progressions where the sum over k of the number of divisors of N minus k squared is always divisible by 4. The authors reach this by identifying the sum with the number of partitions of N into exactly two part sizes, then invoking Keith's theorem that this count is 0 mod 4 for all N. They add some combinatorial and modular steps to make the identification hold inside the listed residue classes. That is the actual new content: the specific pairs (16,14), (36,30), (72,42), (196,70), and (252,114) had not appeared before with this congruence. The approach is straightforward and re-uses an existing external result rather than inventing new machinery, which keeps the paper short and focused. The combinatorial link between the divisor sum and the partition function is the part that does the real work, and the abstract suggests they handle the necessary counting arguments carefully for those five cases. The main soft spot is that the whole claim rests on that identification being exact on the chosen progressions. If the counting step introduces any hidden restriction or off-by-one that fails outside the verified examples, the congruences would not follow even though Keith's theorem holds. Without the full derivation in front of me I cannot spot an actual gap, but the dependence on the external result and the precise matching step is the place where a referee would want to check the details line by line. This is a paper for specialists in combinatorial number theory who already know Keith's work and want to see it applied to divisor sums. A reader looking for small, checkable extensions will find it useful; someone hunting for new frameworks or resolutions of open problems will not. It is solid enough on its own terms to deserve a serious referee rather than a desk reject, even if the overall contribution remains modest.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that ∑_{1 ≤ k < √N} σ₀(N - k²) ≡ 0 (mod 4) for N ≡ B (mod A) where (A, B) belongs to the five listed pairs. The argument identifies the left-hand side combinatorially with ν₂(N), the number of partitions of N into exactly two part sizes, and invokes Keith's theorem that ν₂(N) ≡ 0 (mod 4) for all positive integers N, supplemented by modular-arithmetic reductions that hold inside the specified arithmetic progressions.

Significance. The result supplies explicit, verifiable congruences that link a classical divisor sum to a partition function whose divisibility properties are already known. By making the combinatorial identification explicit and restricting attention to five arithmetic progressions where the modular reductions close, the paper demonstrates a concrete application of Keith's theorem. The approach is falsifiable by direct computation for small N and credits the external result appropriately.

minor comments (3)

[Abstract] The abstract states the range as 1 ≤ k < √N; the manuscript should clarify whether this is intended as k = 1 to ⌊√N⌋-1 and whether the upper limit is strict or inclusive when N is a perfect square (even though the listed APs avoid squares for small n).
[Introduction or §3] A short computational table verifying the claimed congruence for the first few terms of each arithmetic progression would strengthen the presentation and allow readers to check the combinatorial identification directly.
[§2] The notation ν₂(N) is introduced without an explicit definition in the abstract; the manuscript should restate the definition (number of partitions into exactly two distinct part sizes, or with multiplicity) at the first use in the body.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and accurate summary of our manuscript, as well as for the positive assessment of its significance. The description correctly captures both the combinatorial identification of the sum with ν₂(N) and the role of Keith's theorem together with the arithmetic-progression reductions. No major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity; central proof uses external Keith theorem plus independent combinatorial link

full rationale

The paper states that it proves the target sum ≡ 0 mod 4 by combining a combinatorial identification of the divisor sum with ν₂(N) and Keith's external theorem that ν₂(N) ≡ 0 mod 4 on the listed arithmetic progressions. No equation inside the manuscript defines the target sum in terms of a fitted parameter, renames a known result, or reduces the congruence to a self-citation chain whose load-bearing premise is unverified. The combinatorial step is presented as an independent argument (inspired by but not copied from Dewitt-Keith), and Keith's result is cited as an external benchmark rather than derived internally. This satisfies the default expectation of a non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on one external theorem and standard background facts; it introduces no new free parameters or postulated entities.

axioms (2)

domain assumption Keith's theorem: the number of partitions of N with exactly two part sizes is divisible by 4
Explicitly invoked in the abstract as the foundation for the proof.
standard math Basic properties of the divisor function and modular arithmetic hold
Used throughout the combinatorial and modular arguments.

pith-pipeline@v0.9.0 · 5465 in / 1322 out tokens · 82841 ms · 2026-05-07T15:08:20.757729+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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[10]

N. B. Tani and S. Bouroubi , Enumeration of the partitions of an integer into parts of a specified number of different sizes and especially two sizes , Jour. Integer Seq., 14 (2011). Department of Mathematics and Computer Science, F aculty of Science, Chulalongkorn University, Bangkok 10330, Thailand, and Centre of Excellence in Mathematics, Ministry of H...

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[1] [1]

Andrew , Stacked lattice boxes , Annals of Combinatorics, 3 (1999), pp

G. Andrew , Stacked lattice boxes , Annals of Combinatorics, 3 (1999), pp. 115–130

work page 1999

[2] [2]

G. E. Andrews , The theory of partitions , no. 2, Cambridge university press, 1998

work page 1998

[3] [3]

E. R. DeWitt and W. J. Keith , Combinatorial proof of a congruence for partitions into two sizes of part , arXiv preprint arXiv:2507.13566, (2025)

work page arXiv 2025

[4] [4]

Euler , Introductio in analysin infinitorum , vol

L. Euler , Introductio in analysin infinitorum , vol. 2, Apud Bernuset, Delamolliere, Falque & soc., 1797

work page

[5] [5]

G. H. Hardy and S. Ramanujan , Asymptotic formulaæ in combinatory analysis , Pro- ceedings of the London Mathematical Society, 2 (1918), pp. 75–115

work page 1918

[6] [6]

Hooley , On the representation of a number as the sum of a square and a product , Mathematische Zeitschrift, 69 (1958), pp

C. Hooley , On the representation of a number as the sum of a square and a product , Mathematische Zeitschrift, 69 (1958), pp. 211–227

work page 1958

[7] [7]

W. J. Keith , Partitions into a small number of part sizes , International Journal of Number Theory, 13 (2017), pp. 229–241

work page 2017

[8] [8]

P. A. MacMahon , Combinatory analysis: By Percy A. Macmahon , vol. 1, CUP Archive, 1915

work page 1915

[9] [9]

P. A. MacMahon , Divisors of numbers and their continuations in the theory of partitions , Proceedings of the London Mathematical Society, 2 (1921), pp. 75–113

work page 1921

[10] [10]

N. B. Tani and S. Bouroubi , Enumeration of the partitions of an integer into parts of a specified number of different sizes and especially two sizes , Jour. Integer Seq., 14 (2011). Department of Mathematics and Computer Science, F aculty of Science, Chulalongkorn University, Bangkok 10330, Thailand, and Centre of Excellence in Mathematics, Ministry of H...

work page 2011