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Congruences via Partitions with Exactly Two Part Sizes

This paper establishes a congruence modulo 4 for a sum involving the divisor function $\sigma_0$ by leveraging Keith's result on the parity of partitions with exactly two part sizes, thereby connecting combinatorial partition theory with modular arithmetic.

Original authors: Sittinon Jirattikansakul, Teeradej Kittipassorn, Kraiwich Kongsiri, Nitipon Moonwichit, Kirati Sriamorn

Published 2026-04-29

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CC BY 4.0

Original authors: Sittinon Jirattikansakul, Teeradej Kittipassorn, Kraiwich Kongsiri, Nitipon Moonwichit, Kirati Sriamorn

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant pile of Lego bricks. Your goal is to build towers using these bricks, but with a very specific rule: you can only use two different sizes of Lego bricks in any single tower. Maybe you use some big 4-block bricks and some small 1-block bricks, but you can't mix in a medium 2-block brick.

Mathematicians call these towers "partitions." The paper you're looking at is about counting how many different ways you can build these "two-size" towers for a specific number of bricks, let's call that number N.

Here is the story of what the authors discovered, broken down into simple steps:

1. The Mystery of the "Two-Size" Towers

The authors are interested in a special number, let's call it $\nu_2(N)$ . This number counts exactly how many ways you can build a tower with $N$ bricks using only two distinct sizes.

For a long time, mathematicians knew a complicated formula to calculate this. But recently, a mathematician named Keith found something strange: for certain special numbers $N$ (specifically numbers that fit patterns like $16n+14$ or $36n+30$ ), the count of these towers is always a multiple of 4. It's like saying, "No matter how you build these towers, you can always group them into perfect sets of four."

2. The "Gluing" Trick

The authors of this paper wanted to understand why this happens. They came up with a clever visual trick using Young diagrams.

Think of a Young diagram as a shape made of squares (like a Tetris piece).

If you have a tower with only one size of brick, the shape is a perfect rectangle.
If you have a tower with two sizes of bricks, the shape looks like an L.

The authors realized that every "L-shaped" tower (two sizes) can be thought of as two rectangles glued together vertically. One rectangle sits on top of the other.

They created a giant "multiset" (a bag) of all possible ways to glue two rectangles together to make the number $N$ . They then sorted these glued shapes into four different buckets:

Bucket B: Shapes that are definitely "L" shapes (the two-size towers we care about).
Bucket C: Shapes where one of the rectangles is a perfect square.
Bucket D: Shapes that are actually just one big rectangle (one-size towers).
Bucket E: Shapes where the two rectangles are mirror images of each other.

3. The Great Balancing Act

The magic of the paper is this: When you look at the total number of items in this giant bag, the math works out so that the total is always divisible by 4.

Because the total is divisible by 4, and the authors could prove that the items in Buckets C, D, and E also follow specific rules (often being divisible by 4 or having a known relationship to the number of divisors), they could deduce something about Bucket B.

They found that for their special numbers $N$ , the number of "L-shapes" (Bucket B) plus a specific sum involving the divisors of $N$ must equal a multiple of 4.

4. The Main Discovery

The paper proves a new, simpler rule. They showed that for those special numbers $N$ (like $16n+14$ ), if you take a specific sum:

Add up the number of divisors for every number you get by subtracting a square number from N.

...the result is always divisible by 4.

In plain English:
If you pick a number $N$ from their special list, and you subtract $1^2$ , $2^2$ , $3^2$ , etc., from it, and count how many factors (divisors) the remaining numbers have, the total count of all those factors will always be a multiple of 4.

5. Why Does This Matter? (According to the Paper)

The paper doesn't claim this will cure diseases or build better bridges. Instead, it's a piece of a larger puzzle in number theory.

It connects the world of partitions (building towers) with divisors (counting factors).
It confirms a pattern Keith found earlier but explains it using a new "gluing" method.
It leads to two smaller "Corollaries" (side conclusions) that tell us exactly how many odd numbers fit into these patterns.

6. The "What If?" (Conjectures)

At the end, the authors say, "We think this might work for even more numbers than we proved."
They ran computer tests and noticed a pattern: if you pick numbers like $8n + 6$ , the rule seems to hold true, but they couldn't fully prove it yet. They also guess that for this rule to work, the "step size" of the number pattern (the $A$ in $An+B$) must be divisible by 4, and the starting number ( $B$ ) must be even but not divisible by 4.

Summary Analogy:
Imagine you have a machine that sorts Lego towers. The authors proved that for a specific set of inputs, the machine always outputs a number of "L-shaped" towers that is a multiple of 4. They did this by showing that the "L-shapes" are just part of a larger, balanced system of shapes that naturally cancels itself out in groups of four. This gives us a new, simpler way to calculate a complex sum involving divisors.

1. Problem Statement

The paper investigates a specific congruence relation involving the divisor function $\sigma_0(m)$ (the number of positive divisors of $m$ ) and partitions of integers. Specifically, the authors aim to prove that for integers $N$ in certain arithmetic progressions, the following sum is congruent to 0 modulo 4:
$\sum_{1 \le k < \sqrt{N}} \sigma_0(N - k^2) \equiv 0 \pmod 4$
This sum counts the number of representations of $N$ as the sum of a square and a product (related to Hooley's work), but the authors approach it through the lens of integer partitions with exactly two part sizes.

The study focuses on $N = An + B$ where $(A, B)$ belongs to the set:
$\{(16, 14), (36, 30), (72, 42), (196, 70), (252, 114)\}$

2. Methodology

The authors employ a hybrid methodology combining combinatorial partition theory and modular arithmetic.

A. Combinatorial Framework (Young Diagrams and Rectangles)

Partition Definition: Let $\nu_2(N)$ be the number of partitions of $N$ with exactly two distinct part sizes.
Geometric Interpretation: The authors interpret partitions with exactly two part sizes as L-shaped Young diagrams. These diagrams can be uniquely decomposed into a pair of rectangles glued vertically (the wider base on top).
Canonical Pairs: They define a "canonical pair" of rectangles $\{X, Y\}$ for a given $N$ such that the total number of cells is $N$ .
Multiset Construction: They construct a multiset $A$ $A$ containing all possible gluings of rectangles and their transposes ( $X \star Y$ $X ⋆ Y$ ). This multiset is partitioned into four subsets ( $B, C, D, E$ $B, C, D, E$ ) based on geometric properties:
- $B$ : Diagrams corresponding to partitions with exactly two part sizes ( $\nu_2(N)$ ).
- $C$ : Diagrams where at least one rectangle is a square (related to $\sum \sigma_0(N-k^2)$ ).
- $D$ : Diagrams corresponding to partitions with exactly one part size (rectangular shapes).
- $E$ : Diagrams where the two rectangles are transposes of each other ( $X = Y^T$ ).

B. Algebraic and Number-Theoretic Analysis

Auxiliary Theorem (Theorem 4): The authors first prove a general congruence for $N$ that cannot be written as a sum of two squares and is of the form $N=2m$ (where $m$ is odd):
$\nu_2(N) + \sum_{1 \le k < \sqrt{N}} \sigma_0(N - k^2) + \frac{1}{2}\sigma_1(N) \equiv 0 \pmod 4$
Here, $\sigma_1(N)$ is the sum of divisors of $N$ .
Modular Arithmetic: To prove the main theorem, they rely on a result by Keith stating that for the specific arithmetic progressions listed, $\nu_2(N) \equiv 0 \pmod 4$ .
Divisor Sum Analysis: The proof reduces to showing that $\frac{1}{2}\sigma_1(N) \equiv 0 \pmod 4$ (i.e., $\sigma_1(N)$ is divisible by 8) for the specific cases of $N$ . This is achieved by analyzing the prime factorization of $N$ modulo 8 and 7, demonstrating that the sum of divisors contains factors that force divisibility by 8.

3. Key Contributions

New Congruence Proof: The paper provides a rigorous proof for the congruence $\sum \sigma_0(N-k^2) \equiv 0 \pmod 4$ for five specific arithmetic progressions, a result previously unproven in this specific form.
Combinatorial-Number Theoretic Bridge: The authors establish a novel link between the count of partitions with two part sizes ( $\nu_2(N)$ ) and the sum of divisor functions over quadratic differences. This is achieved through the geometric decomposition of Young diagrams into rectangle pairs.
Refinement of Keith's Work: While Keith previously proved $\nu_2(N) \equiv 0 \pmod 4$ for these sequences, this paper extends the analysis to derive the specific congruence for the sum involving $\sigma_0(N-k^2)$ .
Derivation of Corollaries:
- Corollary 2: Proves that for $N \equiv 14 \pmod{16}$ or $N \equiv 70 \pmod{196}$ , the number of odd integers $k$ such that $k^2 < N$ and $\sigma_0(N-k^2) \equiv 2 \pmod 4$ is always even.
- Corollary 3: Proves that for $N \equiv B \pmod A$ (for the remaining pairs), the sum excluding multiples of 3 is also $0 \pmod 4$ .

4. Main Results

Theorem 1: For $N = An + B$ with $(A, B) \in \{(16, 14), (36, 30), (72, 42), (196, 70), (252, 114)\}$ , the congruence $\sum_{1 \le k < \sqrt{N}} \sigma_0(N - k^2) \equiv 0 \pmod 4$ holds.
Theorem 4: Establishes the fundamental identity relating $\nu_2(N)$ , the divisor sum, and $\sigma_1(N)$ modulo 4 for non-sum-of-two-squares integers of the form $2m$ .
Conjectures:
- Conjecture 7: The congruence likely holds for the broader class $N = 16n + 6$ .
- Conjecture 8: If the congruence holds for all $n$ in $N=An+B$, then $A \equiv 0 \pmod 4$ and $B \equiv 2 \pmod 4$ .

5. Significance

Theoretical Depth: The paper advances the understanding of partition functions with restricted part sizes, a topic rooted in the work of Euler, MacMahon, and Andrews.
Methodological Innovation: By using Young diagrams and rectangle gluing operations to count partitions, the authors provide a fresh combinatorial perspective on classical number-theoretic sums.
Computational Insight: The authors performed computational experiments up to $N < 100,000$ , suggesting that the structural properties of the congruence (specifically regarding $N \equiv 6 \pmod 8$ ) are more general than the proven cases, motivating future research into the necessary and sufficient conditions for the congruence to hold.
Interdisciplinary Connection: The work connects the geometry of partitions (Ferrers/Young diagrams) with analytic number theory (divisor sums and quadratic forms), offering a template for similar investigations.