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A proof of Andrews-El Bachraoui's conjecture on the parity of coefficients of a $q$ -series

This paper confirms Andrews and El Bachraoui's conjecture regarding the parity of the coefficients of the $q$ -series $T_o(q)$ , while also establishing an infinite family of congruences modulo 8 for the coefficients of $S_1(q)$ and proving that $s_1(n)$ is divisible by 8 with natural density 1.

Original authors: Eric H. Liu, Ernest X. W. Xia

Published 2026-07-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Eric H. Liu, Ernest X. W. Xia

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, infinite box of building blocks. These blocks come in two colors: Red and Blue. You want to build towers using these blocks, but there are some very specific rules you must follow.

This paper is about solving a puzzle regarding how many different ways you can build these towers for a specific number of blocks, and whether that number is "odd" or "even."

Here is the breakdown of the story, the rules, and the solution, explained simply:

1. The Game: Special Towers

The authors are studying a specific type of tower called a "two-color partition into distinct parts."

Distinct Parts: You can't use the same size block twice in a single tower. If you use a size-5 block, you can't use another size-5 block.
Two Colors: Every block size can be Red, Blue, or both (Red and Blue).
The Special Rule: The smallest block in your tower is picky. It must be only one specific color (say, only Red). However, every single block larger than the smallest one can be Red, Blue, or both.

The authors are counting how many different valid towers you can build for a number $n$ . Let's call this count $s_1(n)$ .

2. The Mystery: The "Odd" Companion

The researchers (Andrews and El Bachraoui) had already figured out some patterns about these tower counts. But they noticed something strange happening with a related mathematical object, which we can think of as a "shadow" or "companion" to the main tower count. Let's call this shadow the Odd Companion ( $T_o$ ).

They made a guess (a conjecture) about this shadow:

"If a certain number related to our tower size has a specific 'prime ingredient' that appears an odd number of times, then the count for this shadow will always be an even number."

In math terms, they guessed that under certain conditions, the answer is always divisible by 2.

3. The Solution: Proving the Guess

Eric Liu and Ernest Xia (the authors of this paper) stepped in to prove this guess was correct.

The Detective Work: They used a mathematical toolkit involving "q-series" (which are like infinite recipes for numbers) and "theta functions" (special formulas that count solutions to equations).
The Breakthrough: They translated the problem of counting towers into a problem of fitting numbers into a specific shape (like fitting pieces into a puzzle of the form $x^2 + 2y^2$ ).
The Result: They proved that if the "prime ingredient" condition is met, the shadow count is indeed always even. They confirmed the guess was 100% true.

4. The Deeper Patterns: Dividing by 8

After solving the "even/odd" mystery, they looked deeper. They asked: "What if we look at the main tower count ( $s_1$ ) and see if it's divisible by 8?"

They discovered a whole family of rules (congruences).

The Analogy: Imagine you have a machine that takes a number, does some math, and spits out a tower count. They found that if you feed the machine specific types of numbers (numbers related to prime numbers like 5 or 7), the machine always spits out a number that is perfectly divisible by 8.
Example: If you pick the prime number 5, they proved that for a huge list of numbers (like $25n + 8$ , $25n + 13$ , etc.), the number of towers is always a multiple of 8.

5. The Big Picture: How Rare are the Exceptions?

Finally, they asked a statistical question: "As we look at bigger and bigger numbers, how often do we find a tower count that is not divisible by 8?"

The Finding: They proved that as you go to infinity, the number of "exceptions" (counts not divisible by 8) becomes so rare that they effectively disappear.
The Metaphor: Imagine a beach with infinite grains of sand. If you pick a grain at random, the chance of it being a "special" grain (one that breaks the rule) is zero. Almost every single number you pick will result in a tower count that is perfectly divisible by 8.

Summary

In short, this paper is a mathematical victory lap.

Confirmed a Guess: They proved a specific rule about whether a related number is even or odd.
Found New Rules: They discovered that for many specific numbers, the main count is always divisible by 8.
Showed Rarity: They proved that "exceptions" to this "divisible by 8" rule are so rare they don't matter in the grand scheme of infinite numbers.

The paper doesn't talk about building actual towers or using this for engineering; it is purely about understanding the hidden, rhythmic patterns in how numbers can be broken down and counted.

Technical Summary: A Proof of Andrews-El Bachraoui's Conjecture on the Parity of Coefficients of a q-Series

Problem Statement
This paper addresses a conjecture regarding the parity of the coefficients of a specific $q$ -series, $T_o(q)$ , introduced by Andrews and El Bachraoui. The series $T_o(q)$ is defined as the "odd companion" to the generating function $S_1(q)$ for a partition function $s_1(n)$ . The function $s_1(n)$ counts the number of two-color partitions of an integer $n$ into distinct parts, subject to the constraint that the smallest part occurs in only one prescribed color, while all larger parts may appear in either color or both.

Andrews and El Bachraoui previously established a complete description of $s_1(n)$ modulo 4. However, they left open the question of the parity (modulo 2) of the coefficients $t_o(n)$ of the series $T_o(q)$ . Specifically, they conjectured that $t_o(n) \equiv 0 \pmod 2$ if the integer $8n + 9$ possesses a prime divisor $p \equiv 5, 7 \pmod 8$ raised to an odd exponent (equivalently, if $8n + 9$ cannot be represented by the quadratic form $x^2 + 2y^2$ ).

Methodology
The authors employ a combination of classical $q$ -series identities, modular arithmetic, and the theory of quadratic forms to resolve the conjecture and derive further congruences.

Series Manipulation and Modular Reduction:
The proof begins by utilizing Euler's pentagonal number theorem and identities established by Andrews and El Bachraoui to express $2qT_o(q)$ in terms of infinite products and sums. By working modulo 2, the authors simplify the expression for $T_o(q)$ , utilizing the identity $(q; q)_\infty^2 \equiv (q^2; q^2)_\infty \pmod 2$ .
Connection to Quadratic Forms:
Through algebraic manipulation, the coefficient $t_o(n)$ is related to the number of integer solutions to specific Diophantine equations involving quadratic forms. Specifically, the parity of $t_o(n)$ is shown to depend on the solvability of equations of the form $2(6k+1)^2 + (12m+1)^2 = 3(8n+9)$ and related variants.
Application of Representation Theory:
The authors invoke a classical result concerning the number of representations of an integer by the quadratic form $x^2 + 2y^2$ . This result states that an integer $N$ has zero representations by $x^2 + 2y^2$ if and only if its prime factorization contains a prime $q \equiv 5, 7 \pmod 8$ raised to an odd power. By analyzing the prime factorization of $3(8n+9)$ , the authors demonstrate that the condition on $8n+9$ implies the non-existence of solutions to the derived Diophantine equations, thereby forcing $t_o(n)$ to be even.
Extension to Modulo 8 and Density:
To prove the infinite family of congruences modulo 8 for $s_1(n)$ , the authors combine the expression for $S_1(q)$ with the derived expansion of $T_o(q)$ . They analyze the term $u(n)$ , which counts representations of $24n+3$ by specific quadratic forms. Using properties of the Legendre symbol and the valuation of primes $p \equiv 5, 7 \pmod 8$ , they show that for specific arithmetic progressions, the relevant quadratic forms have no solutions. Finally, to establish the density result, the authors apply a theorem by Landau regarding "lacunary" power series (series with zero density of non-zero coefficients), arguing that the components of $s_1(n)$ are lacunary.

Key Contributions and Results

Confirmation of Conjecture 1.2: The primary result is the proof of Andrews and El Bachraoui's conjecture. The authors establish that if $8n + 9$ is not representable by $x^2 + 2y^2$ (i.e., it has a prime divisor $p \equiv 5, 7 \pmod 8$ with an odd exponent), then $t_o(n) \equiv 0 \pmod 2$ .
Infinite Family of Congruences Modulo 8: The paper proves Theorem 1.4, which provides an infinite family of congruences for $s_1(n)$ . For any prime $p \equiv 5, 7 \pmod 8$ and integers $n, k \ge 0$ with $p \nmid n$ , the following holds:
$s_1\left(p^{2k+1}n + \frac{p^{2k+2}-1}{8}\right) \equiv 0 \pmod 8.$
The authors provide a concrete example with $p=5$ and $k=0$ , showing $s_1(25n + 8 + 5j) \equiv 0 \pmod 8$ for $j \in \{0, 5, 10, 15\}$ .
Natural Density Result: The paper proves Theorem 1.5, demonstrating that the set of integers $n$ for which $s_1(n)$ is not divisible by 8 has a natural density of zero. In other words, $s_1(n)/8$ takes integer values with natural density 1.

Significance
The paper confirms a specific conjecture regarding the arithmetic properties of a partition function related to two-color distinct partitions and concave compositions. By resolving the parity of the coefficients of the odd companion series $T_o(q)$ , the authors provide a deeper understanding of the structure of $S_1(q)$ . Furthermore, the establishment of congruences modulo 8 and the density result extends the known arithmetic behavior of $s_1(n)$ beyond the previously known modulo 4 results.

The authors conclude by noting that a natural direction for future research is to extend these congruences to higher powers of 2 (moduli 16, 32, etc.) and to determine the arithmetic density of the set of integers where $s_1(n) \equiv 0 \pmod{2^k}$ for fixed $k \ge 4$ .