广义商高数的纯指数Diophantine方程ax+by=cz的解

Solution of Pure Exponential Diophantine Equations ax+by=cz for Generalized Pythagorean Triplets

关键词:  丢番图方程  Terai猜想  正整数解  Gel'fond-Baker方法

Key words:  Diophantine equation  Terai conjecture  positive interger solution  Gel'fond-Baker method

摘要:

运用Gel'fond-Baker方法证明,在m ≥ 10⁵r³时,丢番图方程a^x+b^y=c^z仅有正整数解(x,y,z)=(2,2,r).其中r和m为正偶数,(a,b,c)=(|V(m,r)|,|U(m,r)|,m²+1),V(m,r)+U(m,r)√-1=(m+√-1)^r.

Abstract:

Using the Gel'fond-Baker method, we prove that if m ≥ 10⁵r³, then the Diophantine equation a^x+b^y=c^z has only one positive integer solution (x, y, z)=(2, 2, r).Let r and m be a positive even integer, (a, b, c)=(|V(m, r)|,|U(m, r)|, m²+1), V(m, r)+U(m, r)√-1=(m+√-1)^r.