In this paper, we prove a conjecture of Andrews and El Bachraoui concerning the parity of certain two-color partitions. Precisely, we show that if the Fourier coefficient $t_o(n)$ of the corresponding $q$-series is odd, then $8n+9$ is represented by the binary quadratic form $x^2+2y^2$.
In this short note, we show that the loss of order $N^{c_p/\log\log N}$ in the version of the Ionescu--Wainger multiplier theorem for canonical fractions, recently established by Kosz, Mirek, S{\l}omian, and Zhang in arXiv:2606.12663, is optimal.
We prove that, for every positive integer $n$, the polynomial $C_0+C_1x+\cdots+C_nx^n$ is irreducible over $\mathbb{Q}$, where $C_k$ denotes the $k$th Catalan number.
We show that there exists $C\in\mathbb{N}$ such that $K_t$-minor free graphs are $Ct$-colorable. The proof was found by GPT-6 Astra, following the directions by the authors.
We provide an explicit bound for the image of the Zilber--Pink locus of a curve in an abelian variety under reduction modulo a suitably large prime $p$. We also provide a new proof of finiteness of the Zilber--Pink locus. Under further assumptions on the abelian variety, we prove a bound on the size of the locus.
For every integer $k\ge2$, Chen and Raspaud conjectured that each graph $G$ with odd girth $\og(G)\ge2k+1$ and maximum average degree $\mad(G)<2+1/k$ has a $(2k+1:k)$-coloring. In this paper, we prove the conjecture.
Let $b(n)$ denote the minimum number of perfect matchings in a simple brace of order $n$. De~Carvalho, Lucchesi and Murty (SIAM Journal on Discrete Mathematics, 27(2): (2013) 940-958) proved that $b(n)\ge(n-2)^2/8$ and conjectured that $b(n)=(n-2)^2/4$ for all sufficiently large even $n$. We prove this conjecture.
Xing Feng, Jin-Qiu Zhou· 0 citations
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