Abstract
Let $d(n)$ denote the corank of $I + A$ over the field with two elements, where $A$ is the adjacency matrix of the discrete torus $C_n × C_n$, and $I$ is the identity matrix. We shall prove that $d(2n) = 2d(n)$ and $d(2^r + 1) = d(2^r − 1) + 4$. For the proof of the latter result, we use an elliptic curve. Our motivation for this study is the “lights out” puzzle.
Citation
Masato Goshima. Masakazu Yamagishi. "On the Dimension of the Space of Harmonic Functions on a Discrete Torus." Experiment. Math. 19 (4) 421 - 429, 2010.
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