On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Let m⩾−1m⩾−1 be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equationsX3−mX2Y−(m+3)XY2−Y3=λX3−mX2Y−(m+3)XY2−Y3=λ where λ>0λ>0 is a divisor of m2+3m+9m2+3m+9 and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazakiʼs result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors λ   of m2+3m+9m2+3m+9. As a consequence, we obtain another proof of Okazakiʼs theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for m=−1,0,1,2,3,5,12,54,66,1259,2389m=−1,0,1,2,3,5,12,54,66,1259,2389.