1. f(x) is monic
2. f(x) has degree 1000
3. f(x) has real coefficients
4. f(x) divides (x-1)f(2x)
5. f(2) is an integer
Thanks!! Please see http://ca.answers.yahoo.com/question/index;_ylt=ArYAwHh46uF5FrFEwH8_X_TBFQx.;_ylv=3?qid=20130412223057AAKzQPS I have no idea what is going on.
"Suppose z is a point such that f(z)=0, and z is not 1. Then f(2z)(z-1) = 0, so f(2z) = 0. Thus, if z is a zero of f(x) and z is not 1, then 2z is also a zero." What does it mean?
"all roots are either 0, 1, or of the form 1/2^k (else, f(x) would have an infinite number of roots since we can always double and get new roots, until such doubling gives us 1)." I dont understand. Why 0, 1 or 1/2^k?
f(2) = 2^(1000-r)(2-1)(2-1/2)…(2-1/2^(r-1)) = 2^(1000-r) 2^r (1-1/2)(1-1/4)(…)(1-1/2^r) = 2^1000 (1-1/2)(1-1/4)(1-1/8)…(1-1/2^r). = 2^(1000) (1/2)(3/4)(7/8)…((2^r-1)/2^r) = 2^(1000) Prod_{k=1}^r (2^k-1)/2^k
What is going on here? What is Prod? How can he conclude that "This is integer if and only if r <= 44. So there are 45 solutions."?
sorry, could you answer the questions above as well? THANKS!!!!
