Proof of a conjecture of Andrews and El Bachraoui on the parity of two-color partitions

Technical Summary: Proof of a Conjecture of Andrews and El Bachraoui on the Parity of Two-Color Partitions

Problem Statement
This paper addresses a conjecture proposed by Andrews and El Bachraoui regarding the parity of the coefficients $to(n)$ of a specific $q$-series. The series $To(q)$ is defined as:
$$To(q) := \sum_{n \ge 0} to(n)q^n = \sum_{n \ge 0} \frac{(-q)^{2n}}{(q; q^2)_{n+1}} q^{2n}$$
Andrews and El Bachraoui previously established that if the coefficients $c(n)$ of the related series $C(q) = (q)_\infty To(q)$ are non-zero, then $24n + 28$ is represented by the binary quadratic form $x^2 + 3y^2$. Motivated by this, they conjectured a similar representation property for the parity of $to(n)$: specifically, that if $8n + 9$ cannot be represented by the binary quadratic form $x^2 + 2y^2$, then $to(n) \equiv 0 \pmod 2$.

Methodology
The authors employ elementary $q$-series identities and modular arithmetic techniques to analyze the generating function $To(q)$. The proof strategy proceeds as follows:

1. Decomposition of Generating Functions: The authors utilize a known identity relating $C(q)$ to theta functions $\Psi(q)$, $T(q)$, and $\Theta(q)$:
   $$2qC(q) = \Psi(q)T(q) - (q)_\infty \Theta(q)$$
   where $\Psi(q)$, $T(q)$, and $\Theta(q)$ are defined by specific summations involving triangular and pentagonal numbers.

2. Algebraic Manipulation: By expressing $To(q) = C(q)/(q)_\infty$, the authors derive an expression for $2qTo(q)$ in terms of these theta functions. They then decompose $T(q)$ and $\Theta(q)$ into sums over even and odd indices, introducing auxiliary series $S(q)$ and $R(q)$ to isolate specific parity contributions.

3. Modular Reduction: The core of the proof involves reducing the resulting expressions modulo 2. The authors utilize Euler's Pentagonal Number Theorem and properties of eta-quotients to show that:
   $$\frac{\Psi(q)}{(q)_\infty} \equiv \sum_{m \in \mathbb{Z}} q^{m(3m+1)} \pmod 2$$
   This allows the expression for $qTo(q)$ to be rewritten as a sum of products of series involving only odd indices.

4. Coefficient Analysis: The authors analyze the coefficient of $q^{n+1}$ in the resulting series modulo 2. They demonstrate that if $to(n)$ is odd (i.e., $to(n) \not\equiv 0 \pmod 2$), then $n+1$ must arise from one of three specific algebraic forms involving odd integers $r$ and arbitrary integers $m$.

Key Contributions and Results
The primary contribution of the paper is the rigorous proof of Conjecture 1.2 (stated as Theorem 1.3 in the text). The authors establish that:
$$to(n) \not\equiv 0 \pmod 2 \implies 8n + 9 \text{ is represented by } x^2 + 2y^2$$
The proof achieves this by explicitly constructing the representation of $8n+9$ in the form $x^2 + 2y^2$ for each of the three cases where the coefficient $to(n)$ is non-zero modulo 2:

• Case 1: If $n+1 = \frac{m(m+1)}{2}$ for odd $m$, then $8n+9 = (2m+1)^2 + 2(0)^2$.
• Case 2: If $n+1 = \frac{r(3r+1)}{2} + m(3m+1)$ for odd $r$, then $8n+9 = (2r+4m+1)^2 + 2(2r-2m)^2$.
• Case 3: If $n+1 = \frac{r(3r+1)}{2} + (2r+1) + m(3m+1)$ for odd $r$, then $8n+9 = (2r-4m+1)^2 + 2(2r+2m+2)^2$.

Consequently, if $8n+9$ is not representable by $x^2 + 2y^2$, none of these cases can occur, implying $to(n) \equiv 0 \pmod 2$.

Significance and Claims
The paper claims to resolve the conjecture of Andrews and El Bachraoui regarding the parity of these two-color partitions. The authors note that their work was carried out independently and simultaneously with another work (cited as [4]) that also proves the same theorem using substantially different methods. The significance lies in establishing a precise connection between the arithmetic properties of partition coefficients and the representation numbers of binary quadratic forms, extending the known congruence properties of restricted two-color partitions. The paper does not propose new applications or future implications beyond the resolution of this specific conjecture.