如何用數學歸納法證明m^n+2+(m+1)^2n+1能被m^2+m+1整除。?

如何用數學歸納法證明m^(n+2)+(m+1)^(2n+1)能被m^2+m+1整除?
Sol
當n=1時
m^(n+2)+(m+1)^(2n+1)
= m^3+(m+1)^3
=m^3+m^3+3m^2+3m+1
=2m^3+3m^2+3m+1
=(2m^3+2m^2+2m)+(m^2+m+1)
=2m(m^2+m+1)+(m^2+m+1)
=(2m+1)(m^2+m+1)
So n=1時為真
設n=k時為真
存在g(m)使得
m^(k+2)+(m+1)^(2k+1)=g(m)(m^2+m+1)
(m+1)^(2k+1)=g(m)(m^2+m+1)－m^(k+2)
m^(k+3)+(m+1)^(2k+3)
=m^(k+3)+(m+1)^2*(m+1)^(2k+1)
=m^(k+3)+(m+1)^2*[g(m)(m^2+m+1)－-m^(k+2)]
=m^(k+3)+g(m)(m+1)^2*(m^2+m+1)-(m^2+2m+1)m^(k+2)
=m^(k+2)*(m─m^2－2m－1)+g(m)(m+1)^2*(m^2+m+1)
=－m^(k+2)*(m^2+m+1)+g(m)(m+1)^2*(m^2+m+1)
=[－m^]k+2)+g(m)(m+1)^2[(m^2+m+1)
So
N=k+1時為真