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On the matrix negative Pell equation

Aleksander Grytczuk; Izabela Kurzydło — 2009

Discussiones Mathematicae - General Algebra and Applications

Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) $\sum_{i = 1}^{n} X ₂_{i} - d \sum_{i = 1}^{n} Y ²_{i} = - I$ with d ∈ N for nonsingular $X_{i}, Y_{i} \in M ₂ (Z)$ , i=1,...,n.

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