关于Diophantine方程x^n+2~ky^n=pz^2

On the Diophantine equation x^n+2~ky^n=pz^2

设p为奇素数且对任意的整数m,d,p≠(2m±1)/d2,则对任意的素数n>p8p2,方程xn+2kyn=pz2,k≥2没有整数解(x,y,z)使得x,y,z两两互素且均不为0.

In this note,we show that if p is an odd prime and p= （2m ± 1）/d2 for any integers m and d,then the equation xn ＋ 2kyn=pz2,k≥2 has no solutions in nonzero pairwise coprime integers x,y,z and prime n with np8p2.