n=1,x-y能被x-y整除成立
假設(shè)n=k時(shí),k>=1
x^k-y^k能被x-y整除成立
則n=k+1
x^(k+1)-y^(k+1)
=x^(k+1)-y^(k+1)-x*y^k+x^k*y+x*y^k-x^k*y
=[x^(k+1)-x^k*y]-[y^(k+1)-x*y^n]+x*y^k-x^k*y
=[x^(k+1)-x^k*y]-[y^(k+1)-x*y^n]+x*y^k-x^k*y
=x^k(x-y)-y^k(x-y)+x*y^k-x^k*y
=(x-y)(x^k-y^k)-x^(k-1)*y^(k-1)*(x-y)
=(x-y)[(x^k-y^k)-x^(k-1)*y^(k-1)]
能被xy整除
綜上
x^n-y^n 可以被 x-y 整除.