高中數學質數問題？

p = n^4 - 2n^3 + n^2 - 25
= n^2(n^2 - 2n + 1) - 5^2
= n^2(n - 1)^2 - 5^2
= (n^2 - n)^2 - 5^2
= (n^2 - n + 5)(n^2 - n - 5)

因為質數 = 1*本身
所以可令 n^2 - n + 5 = 1 or n^2 - n - 5 = 1

( i )
n^2 - n + 5 = 1
n^2 - n + 4 = 0
n的解非整數

( ii )
n^2 - n - 5 = 1
n^2 - n - 6 = 0
n = - 2 or 3

兩解代入 n^2 - n + 5 皆等於 11



所以 n = - 2 or 3 ; p = 11

p=n^4-2n^3+n^2-25=(n^2-n-5)(n^2-n+5)
其一必等於1 , 而 (n^2-n-5)較小 , 故(n^2-n-5)=1
(n^2-n-5)可分解成(n+2)(n-3)=0 , 故n=-2 (捨去) 或3
由此推得p=11