On a family of unit equations over simplest cubic fields

Let a∈ℤ and ρ be a root of f (x)=x 3 -ax 2 -(a+3)x-1, then the number field K =ℚ(ρ) is called simplest cubic field. In this paper we consider family unit equations u 1 +u =n where ,u ∈ℤ[ρ] * n∈ℤ. We completely solve under restriction |n|≤max{1,|a| 1/3 }.