On Thue’s theorem. (English) Zbl 0544.10014

Mahler, Kurt

Math. Scand. 55, 188-200 (1984).

Let F(x,y) be an irreducible binary form with integral coefficients, of degree \(n\geq 3\) and height a; let further w be an integer satisfying \(| w| \geq(450 a^ 4n^ 4)^{n/(n-2)}.\) Let U be an integer satisfying the condition F(U,1)\(\equiv 0 (mod w)\), \(0\leq U\leq | w| -1,\quad(U,w)=1,\) and let \(\upsilon\) (w) be the number of all such U. Then the conditions \(F(u,v)=w,\quad(u,w)=(v,w)=1\) have fewer than 32 n \(\upsilon\) (w) integral solutions u,v. The proof is based on A. Thue’s original paper [”Über Annäherungwerte algebraischer Zahlen”, J. Reine Angew. Math. 135, 284-305 (1909)], uses only real numbers, and is elementary.

MSC:

11D61	Exponential Diophantine equations
11J68	Approximation to algebraic numbers

Keywords:

irreducible binary form; integral solutions