Let
be an even positive integer. Prove that there exists a positive integer
such that
for some polynomials
having integer coefficients. If
denotes the least such
determine
as a function of
i.e. show that
where
is the odd integer determined by
Note: This is variant A6' of the three variants given for this problem.
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![k = f(x) \cdot (x+1)^n + g(x) \cdot (x^n + 1)](/media/m/9/1/1/911858cc3eaee0466076c65c01b846c0.png)
for some polynomials
![f(x), g(x)](/media/m/b/9/e/b9eaa8b2336d491901ae09d6b6d0a1c3.png)
![k_0](/media/m/0/5/c/05cdd33bbb072691972a3f85b661072b.png)
![k,](/media/m/3/b/3/3b3f59143c4e876b4adc54b666df1462.png)
![k_0](/media/m/0/5/c/05cdd33bbb072691972a3f85b661072b.png)
![n,](/media/m/5/f/2/5f26ebab144fee216fbc733cb1fa2f2b.png)
![k_0 = 2^q](/media/m/a/c/2/ac202764662ac83b3e968981f91c0b79.png)
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
![n = q \cdot 2^r, r \in \mathbb{N}.](/media/m/e/a/a/eaaf3ccfc17c3c449f15d1233f18b76f.png)
Note: This is variant A6' of the three variants given for this problem.