On the practical solution of the Thue equation

Abstract

This paper gives in detail a practical general method for the explicit determination of all solutions of any Thue equation. It uses a combination of Baker's theory of linear forms in logarithms and recent computational diophantine approximation techniques. An elaborated example is presented.

On a family of Thue equations of degree 16

“…Hence log b = max{log b + 0.14, 21 16 , 1 2 } ≤ log 2.070725565 · 10 13 (log log 10 7 ) 1/3 + 0.14 < 30.46069518, and therefore we have log r 1 t 2 δ Λ > −5.064759008 · 10 15 log t(log log t) 2/3 .…”

“…This method is based on Baker's theorems on linear forms in logarithms [1,3]. Baker's method was further developed by Tzanakis and de Weger [21] and by Bilu and Hanrot [6,7]. So we have efficient algorithms to solve single Thue equations.…”

On the diophantine equations x2 + 74 = y5 and x2 + 86 = y5

“…We write the left hand side of (14) as e A -1, and thus we find that |A| is small. In fact, following [3], we obtain, assuming m > y ; , that…”

“…First we reduce equation (2) to a set of quintic Thue equations. Then, following classical arguments as outlined in [3], and using the theory of linear forms in logarithms of algebraic numbers as in [1], we derive large upper bounds for the unknowns in these Thue equations. Finally, by computational diophantine approximation techniques, following [3], and using a new idea of Yuri Bilu to improve efficiency, we reduce these large upper bounds to small upper bounds, and thus are able to find all the solutions.…”

Solving Thue equations without the full unit group

“…Despite numerous recent improvements, the algorithmic solution of this equation still relies on the algorithm described in [21].…”