Hello, need your help thanks

Prove that if then

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- Apr 24th 2009, 12:59 PMmaria18not divide
Hello, need your help thanks

Prove that if then - Apr 25th 2009, 11:57 PMOpalg
If then . Therefore , and

, 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.