[(1025^4+1025^2+1)/(1024^4+1024^2+1)]x[(1027^4+1027^2+1)/(1026^4+1026^2+1)]x.......x[(2047^4+2047^2+1)/(2046^4+2046^2+1)]之值最接近哪一個整數?

Sol
x^4+x^2+1
=(x^4+2x^2+1)－x^2
=(x^2+1)^2－x^2
=(x^2+x+1)(x^2－x+1)
(x+1)^4+(x+1)^2+1
=[(x+1)^2+(x+1)+1]*[(x+1)^2－(x+1)+1]
=[(x+1)^2+(x+1)+1]*(x^2+x+1)
So
[(x+1)^4+(x+1)^2+1]/(x^4+x^2+1)=[(x+1)^2+(x+1)+1]/(x^2+x+1)
So
[(1025^4+1025^2+1)/(1024^4+1024^2+1)]*[(1027^4+1027^2+1)/(1026^4+1026^2+1)]*…*[(2047^4+2047^2+1)/(2046^4+2046^2+1)]
=[(2047^4+2047^2+1)/(2046^4+2046^2+1)]*[(2046^4+2046^2+1)/(2045^4+2045^2+1)]*…*[(1025^4+1025^2+1)/(1024^4+1024^2+1)]
=(2047^2+2047+1)/(1024^2+1024+1)
=4192257/1049601
=3.994
最接近整數4