##
**A parametric family of quintic Thue equations. II.**
*(English)*
Zbl 0995.11024

From the authors’ introduction: In a recent paper [Part I, Math. Comput. 69, 851-859 (2000; Zbl 0983.11014)] the authors showed that for \(|t|\geq 3.28\cdot 10^{15}\) the family of quintic equations
\[
\begin{split} F(x,y)= x^5+ (t-1)^2 x^4y- (2t^3+4t+4) x^3y^2\\ +(t^4+t^3+2t^2+4t-3) x^2y^3+ (t^3+t^2+5t+3) xy^4+y^5= \pm 1 \end{split} \tag{1}
\]
has only the trivial solutions with \(x=0\) or \(y=0\).

To prove this result, some standard techniques for Thue equations were used to obtain rational linear forms in 5 logarithms of real algebraic numbers. Using the asymptotic expansions (with respect to the parameter \(t\)) of the involved expressions and constructing suitable “small” linear combinations of the exponents of the units, a “large” lower bound for any further solution \((x,y)\) of (1) of the shape \(\log|y|> c_1t^2\log|t|\) was obtained … .

In the present paper, we introduce a new idea to refine the above method to end up with linear forms in two logarithms, as was similarly done in [M. Mignotte, A. Pethő and R. Roth, Math. Comput. 65, 341-354 (1996; Zbl 0853.11022) and M. Mignotte, in: Number theory: Diophantine, computational and algebraic aspects, de Gruyter, Berlin, 383-399 (1998; Zbl 0916.11016)]. In this situation far better lower bounds are at hand, which enable us to prove that there exist no further solutions of (1) for \(|t|\) exceeding some bound, \(t_0\approx 10^6\). This bound is small enough to handle the remaining Thue equation with the help of computers. So we arrived at the following main result:

Theorem. The integral solutions of (1) are \(\pm(x,y)= (1,0),(0,1)\) for integer \(t\neq -1,0\) and \(\pm(x,y)= (1,0), (0,1), (\pm 1,1), (-2,1)\) for \(t=-1,0\).

To prove this result, some standard techniques for Thue equations were used to obtain rational linear forms in 5 logarithms of real algebraic numbers. Using the asymptotic expansions (with respect to the parameter \(t\)) of the involved expressions and constructing suitable “small” linear combinations of the exponents of the units, a “large” lower bound for any further solution \((x,y)\) of (1) of the shape \(\log|y|> c_1t^2\log|t|\) was obtained … .

In the present paper, we introduce a new idea to refine the above method to end up with linear forms in two logarithms, as was similarly done in [M. Mignotte, A. Pethő and R. Roth, Math. Comput. 65, 341-354 (1996; Zbl 0853.11022) and M. Mignotte, in: Number theory: Diophantine, computational and algebraic aspects, de Gruyter, Berlin, 383-399 (1998; Zbl 0916.11016)]. In this situation far better lower bounds are at hand, which enable us to prove that there exist no further solutions of (1) for \(|t|\) exceeding some bound, \(t_0\approx 10^6\). This bound is small enough to handle the remaining Thue equation with the help of computers. So we arrived at the following main result:

Theorem. The integral solutions of (1) are \(\pm(x,y)= (1,0),(0,1)\) for integer \(t\neq -1,0\) and \(\pm(x,y)= (1,0), (0,1), (\pm 1,1), (-2,1)\) for \(t=-1,0\).

Reviewer: Nikos Tzanakis (Iraklion)