Question

If `alpha, beta` are the roots of `x^2+px+q=0` and also of `x^(2n) + p^nx^n + q^n` = 0 and if `alpha/beta, beta/alpha` are the roots of `x^n + 1 + (x+1)^n = 0 ,` then prove that n must be an even integer?

`alpha`

Answer 1

See below

Explanation:

From `x^n + 1 + (x+1)^n = 0` we have

`2x^n+(n-1)x^{n-1}+cdots+2=0` and also
`x^n+(n-1)/2x^{n-1}+cdots+1=0`. This polynomial obeys

`(alpha/beta)^{n_1}(beta/alpha)^{n_2} = 1` with
`n_1` and `n_2` integers such that `n_1+n_2=n`
Calling now `y = (alpha/beta)` we have
`y^{n_1-n_2} = y^{2n_1-n} = 1->2n_1-n=0->n=2r_1`

so `n` is an even integer.

Note.

`alpha ne beta` because `x=pm1` is not root for

`x^{2n} + 1 + (x+1)^{2n} = 0`