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In this article, we provide criteria for the Cohen-Macaulayness of theassociated graded ring of a Gorenstein noncomplete intersection monomial curve$C$ in the 4-dimensional affine space. We also give an explicit formula for theminimal number of generators of the defining ideal of the tangent cone of $C$at $0$.

Cohen-Macaulayness of associated graded rings of Gorenstein monomial curves