## On Thue’s theorem.(English)Zbl 0544.10014

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
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