Hello, need your help thanks
Prove that if then
, a polynomial in x of degree 2n with integer coefficients.
Let , a complex cube root of unity. Note that . Let . Then the complex cube roots of 28 are and . Therefore
. . . . .
This is a (real) integer of the form for some integer k, and is therefore an odd multiple of 3.
Next, notice that , and . Hence and so , and similarly . So both these quantities are very small (certainly less than 1/2). It follows from the previous paragraph that is within distance less than 1 from an odd multiple of 3. So its integer part cannot be a multiple of 6.