1
2^48 - 1=(2^24+1)(2^24-1)=(2^24+1)(2^12+1)(2^12-1)=(2^24+1)(2^12+1)(2^6+1)(2^6-1)=(2^24+1)(2^12+1)(2^6+1)(2^6-1)=)=(2^24+1)(2^12+1)×65×63
这两个数是63和65.
2.
当a=1或2时,上式分别为质数7和13;a>2时上式是合数,可通过因式分解证明：
原式=(a^4+6a^2+9)-9a^2
=(a^2+3)^2-(3a)^2
=(a^2+3+3a)*(a^2+3-3a)
使a^2+3-3a>1
推出a>2
所以当a=1或2时,上式为质数,a>2时上式是合数
3.
3^24 - 1
=(3^12 - 1)(3^12 + 1)
=(3^6 - 1)(3^12 + 1)
=(3^3 - 1)(3^3 + 1)(3^12 + 1)
=26×28×(3^12 + 1)
=2×(13×7)×(3^12 + 1)
=2×91×(3^12 + 1)
∴3*24 - 1能被91整除
4.
x^4 + 6x^3y + 8x^2 y*2- 6xy^3 - 9y^4
=(x^4 - y^4)+(6x^3y - 6xy^3)+(8x^2 y^2 - 8y^4)
=(x^2 + y^2)(x^2 - y^2)+6xy(x^2 - y^2)+8y^2(x^2 - y^2)
=(x^2 + y^2+6xy+8y^2)(x^2 - y^2)
=(x^2 +6xy+9y^2)(x - y)(x + y)
=(x+3y)^2 (x - y)(x + y)