On a family of cubics

Thomas’ conjecture over function fields

“…These nontrivial solutions have been found by Lee [10], Mignotte and Tzanakis [15] in the rational case and by Ziegler [22] in the imaginary quadratic case (κ = ω 3 , ω 2 3 ). One might conjecture, as Thomas [19] did, that there are only trivial solutions, if deg κ > 0 but this is not true.…”

“…if d = 3 and p 1 = ±1. In this case there exist the non-trivial solutions (1, −(1 + p 2 (a))) respectively (3 + p 2 (a), −2 − p 2 (a)) found by Lee [10] respectively Mignotte and Tzanakis [15]. To the authors knowledge these are the only exceptions known yet in the case of rational integers and d = 3.…”

Complete solutions of a family of quartic Thue and index form equations

“…All the examples treated in [4,6,8] fulfill the conditions above. Obviously, Lemma 1 does not depend on the special choice of δ 1 and δ 2 .…”

“…Apart from the result of Pethö [8] and the references therein, we mention the papers of Mignotte and Tzanakis [6], Lee [4] and Thomas [9]. We also refer to the paper of Mignotte [5], where he proved that for n ≥ 4, n ∈ Z, the diophantine equation…”

On a conjecture of E. Thomas concerning parametrized Thue equations

“…p 1 = 0, p 2 = 1 and deg p 3 ≥ 1, the assertion of the theorem is false since F a (p 3 − 3, p 3 − 2) = −1. See however Lee [11], Mignotte and Tzanakis [13], and Mignotte [12], where all solutions of this family are determined.…”