指数型丢番图方程x^4-1=2~yn^z

On the Exponential Form Diophantine Equation x^4-1=2~yn^z

研究了指数型丢番图方程 x4- 1=2 ynz(n为正奇数 )的非负整数解 ,证明了 :(1) x为偶数时仅有平凡解 x=2 m,y=0 ,z=1,n=16m4- 1;(2 ) z为偶数时无解 ;(3) x为奇数且 z=1时仅有解为 x=2 y-2 n0 ± 1,y≥ 4 ,z=1,n=n0 (2 y-3n0 ± 1) [2 y-2 n0 (2 y-3n0 ± 1) + 1],其中 n0 为正奇数 ;(4 ) (y- 2 ,z)≥ 3或 (y- 3,z)≥ 3时无解 ;(5 ) n为奇素数时仅有唯一解 x =3,y=4 ,z=1。

The un\|negative integer solutions of the exponential form diophantine equation x 4-1=2 yn z(n be positive odd number) is researched. Following five points is proved:(1) If x is an even, then only there are the common solutions x=2m,y=0,z=1,n=16m 4-1 ; (2) If y is an even, then there isn't solution; (3) If x is an odd and z=1 , then the solations are x=2 y-2 n 0±1,y≥4,z=1, n=n 0(2 y-3 n 0±1)[2 y-2 n 0(2 y-3 n 0±1)+1] ,where n 0 is an positive odd number; (4) If (y-2,z)≥3 or (y-3,z)≥3 , then there isn't solution; (5) If n is an odd prime number, then there is the unique solution x=3, y=4, z=1, n=5 .