Uniquely clean 2 × 2 invertible integral matrices
DOI:
https://doi.org/10.24193/subbmath.2017.3.02Keywords:
Clean, uniquely clean, class number, Diophantine equation, reduced matrix.Abstract
While units in any unital ring are strongly clean by definition, which units are uniquely clean, is a far from being simple question, even in particular rings. In this paper, the question is solved for 2 × 2 integral matrices. It turns out that uniquely clean invertible matrices are scarce: only the matrices similar I 1 0 to 0 −1l. The study is splitted into three cases: the elliptic, the parabolic and the hyperbolic cases, according to the discriminant of their characteristic polynomial. In the first two cases, units are not uniquely clean.
Mathematics Subject Classification (2010): 15B36, 16U99, 11D09, 11D45.
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