On the simplest sextic fields and related Thue equations

Abstract

We consider the parametric family of sextic Thue equationswhere m ∈ Z is an integer and λ is a divisor of 27(m 2 + 3m + 9). We show that the only solutions to the equations are the trivial ones with xy(x + y)(x − y)(x + 2y)(2x + y) = 0.

Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields

“…In our proof we shall use Lemma 3 and the corresponding results in the absolute case. G.Lettl, A.Pethő, and P.Voutier [17] and A.Hoshi [12] gave all solutions in rational integers of the equation F For larger right hand sides we shall use the statement of G.Lettl, A.Pethő and P.Voutier [18]. Lemma 11 Let t ∈ Z, t ≥ 89 and consider the primitive solutions of |F (6) t (x, y)| ≤ 120t + 323 in x, y ∈ Z.…”

“…A couple of other infinite parametric families of Thue equations were completely solved, see [9], [5], among others the parametric family of simplest quartic Thue equations [16], [4] and the parametric family of simplest sextic Thue equations [17], [12].…”

A generalization of simplest number fields and their integral basis

“…According to D.Shanks [27], A.J.Lazarus [18], G.Lettl, A.Pethő and P.Voutier [21] and A.Hoshi [14] the polynomials of degrees 3,4 and 6 with these properties are called simplest polynomials, and the corresponding number fields are called simplest number fields. These fields have an extensive literature.…”

“…E.Thomas [28], M.Mignotte [22], G.Lettl, A.Pethő and P.Voutier [21], [20] and G.Lettl and A.Pethő [19], I.Gaál [5] solved Thue equations corresponding to the simplest polynomials in absolute case, and C.Heuberger [11], I.Gaál, B.Jadrijević and L.Remete [6] in certain relative cases. A.Hoshi [12], [13], [14] gave a correspondence between solutions of a family of Thue equations and the isomorphism classes of the simplest number fields. He extended his results to a family of polynomials of degree 12, which has similar properties as the simplest polynomials, but over Q( √ −3).…”