The 96th power of 2 minus 1 can be divided by two integers between 60 and 70

The 96th power of 2 minus 1 can be divided by two integers between 60 and 70

The square difference formula is mainly used
2^96-1
=(2^48+1)*(2^48-1)
=(2^24-1)*(2^24+1)*(2^48+1)
=(2^12-1)*(2^12+1)*(2^24+1)*(2^48+1)
=(2^6-1)*(2^6+1)*(2^12+1)*(2^24+1)*(2^48+1)
=63*65*(2^12+1)*(2^24+1)*(2^48+1)
So these two numbers are 63 and 65

Given that XY is a positive number, it is proved that (x + y) (the square of X + the square of Y) (the third power of X + the third power of Y) is greater than or equal to the third power of 8x

∵(x-y)^2≥0
∴x^2+y^2 ≥ 2xy -----1
1 on both sides with 2 XY
x^2+y^2+2xy ≥ 4xy -----2
That is, (x + y) ^ 2 ≥ 4xy
1 both sides equal minus XY
x^2+y^2-xy ≥ xy -----3
1, 2 and 3 are multiplied on the left side to obtain:
(x^2+y^2) *(x+y)^2*(x^2+y^2-xy)≥2xy *4xy*xy
(x^2+y^2) *(x+y)*(x^3+y^3)≥8x^3y^3
The original formula is established

The second coefficient of (1 / 2x-1) ^ 8 expansion is

The second term is C (1,8) (1 / 2x) ^ (8-1) * - 1 ^ 1
=8*(2x)^(-7)*-1
=-1/16 *x^(-7)
The second coefficient is - 1 / 16