Fundamental units for a family of totally real cubic orders and the diophantine equation u(u + a)(u + 2a) = v(v + 1)

Stéphane Louboutin (1) , Jun Ho Lee (2)

1 I2M - Institut de Mathématiques de Marseille
2 Mokpo University

We determine a system of fundamental units of the totally real cubic orders Z[alpha], where alpha is a root of X-3 - 4a(2)X + 2, 1 <= a epsilon Z, which arise in our study here of the equation E-a : v(v + 1) = u(u + a)(u + 2a). For potentially infinitely many values of a, we reduce the resolution of Ea to the resolution of the Thue equations of the form a(2)m(4) - 4m(3)n - 6am(2)n(2) + n(4) = d. In particular, we solve E-alpha for a epsilon {1, 2, 3, 5, 15, 34, 35, 50, 62, 63, 71, 79, 87} among 1 <= a <= 100. This extends and corrects the work of Godinho, Porto and Togbe for a = 2, 5, as well as extending earlier work of Mordell for a = 1.

Mots clés: Cubic number fields; cubic units; cubic orders; fundamental units; quartic Thue equations

Domaines: Théorie des nombres [math.NT]

Stéphane Louboutin, Jun Ho Lee. Fundamental units for a family of totally real cubic orders and the diophantine equation u(u + a)(u + 2a) = v(v + 1). International Journal of Number Theory, 2017, 13 (07), pp.1729 - 1746. ⟨10.1142/S1793042117500993⟩. ⟨hal-01620410⟩