n=1時a^2+(a+1) 滿足
n=k時滿足a^(k+1) *a+(a+1)^(2k-1) * a 能被a^2+a+1整除
n=k+1時
a^(k+1+1) +(a+1)^(2k+2-1)
= a^(k+1) *a + (a+1)^(2k-1)(a+1)^2
= a^(k+1) *a +(a+1)^(2k-1) (a^2+2a+1)
=a^(k+1) *a+(a+1)^(2k-1) * a + (a+1)^(2k-1) (a^2+a+1)
顯然,上式左邊部分和右邊部分都能被a^2+a+1整除,所以整個式子能被整除
因此得證