关于丢番图方程X^(2)-(a^(2)+1)Y^(4)=k^(2)-1-2ka

On the Diophantine Equation X^(2)-(a^(2)+1)Y^(4)=k^(2)-1-2ka

令a、k为正整数.运用代数有效逼近的某些结果,证明不定方程X^(2)-(a^(2)+1)Y^(4)=k^(2)-1-2ka在一定条件下最多只有2组互素的正整数解(X,Y).

Let a,k be positive integers.Using some results on the effective algebraic approximations of algebraic numbers,we prove that,under some conditions the Diophantine equation X^(2)-(a^(2)+1)Y^(4)=k^(2)-1-2ka has at most two coprime positive integer solutions(X,Y).