关于丢番图方程ax^4+bx^2y^2+cy^4=dz^2

ON THE DIOPHANTINE EQUATIONS ax^4+bx^2y^2+cy^4=dz^2

该文利用初等数论和Fermat无穷递降法 ,证明了丢番图方程ax4+bx2 y2 +cy4=dz2 在 (a ,b ,c,d) =(1,± 5 0 ,12 5 ,1) ,(1,± 2 5 ,12 5 ,1) ,(1,- 10 ,5 ,1) ,(1,5 ,5 ,1) (1,± 10 ,5 ,5 ) ,(1,± 5 ,5 ,5 ) ,(1,- 5 0 ,12 5 ,5 )和 (1,2 5 ,12 5 ,5 )时均无满足 (x,y) =(x,z) =(y ,z) =1的正整数解。

The use of elementary theory of number and Fermat method of infinite descent shows that the Diophantine equations ax 4+bx 2y 2+cy 4=dz 2 has no positive integer solutions when (x,y)=(x,z)=(y,z)=1 and (a,b,c,d)=(1,±50,125,1),(1,±25,125,1),(1,-10,5,1),(1,5,5,1),(1,±10,5,5),(1,±5,5,5),(1,-50,125,5),(1,25,125,5).