关于丢番图方程x^4+mx^2y^2+ny^4=z^2

On the Diophantine Equation x^4+mx^2y^2+ny^4=z^2

利用初等数论和 Fermat无穷递降法证明了方程 x4+ mx2 y2 + ny4=z2在 ( m,n) =( 1 8,2 7) ,( -9,-2 7) ,(± 9,2 7) ,(± 1 8,-2 7) ,( 1 8,1 89) ,( -36 ,2 1 6 )时均无正整数解 ,并且获得了方程在 ( m,n) =(± 6 ,2 4 ) ,(± 1 2 ,-6 0 ) ,( 9,-2 7) ,( -1 8,1 89) ,( 36 ,2 1 6 ) ,( -1 8,2 7)时的无穷多组正整数解的通解公式 ,从而完善了Aubry等人的结果 .

We use the elementary theory of number and Fermat method of infinite descents to show that the Diophantine equation x 4+mx 2y 2+ny 4=z 2 has no positive Interger solution when (m,n)=(18,27) etc;to get infinite class of general solutions of the Diophtine equation when (m,n)=(±6,24),(±12,-60),(9,-27),(-18,189),(36,216),(-18,27);to make perfect the theory of Aubry et al.