SIMON-MARAIS 2024 B4

The following problem is open in the sense that the answer to part (b) is not currently known. A solution to part (a) will be awarded 7 points. Up to 7 additional points may be awarded for progress on part (b).

Let $n$ be an odd positive integer and let

$$f_n(x, y, z) = x^n + y^n + z^n + (x + y + z)^n.$$

(a) Prove that there exist infinitely many values of $n$ such that

$$f_n(x, y, z) \equiv (x + y)(y + z)(z + x)g(x, y, z)h(x, y, z) \pmod{2},$$

for some integer polynomials $g(x, y, z)$ and $h(x, y, z)$, neither of which is constant modulo 2.

(b) Determine all values of $n$ such that

$$f_n(x, y, z) \equiv (x + y)(y + z)(z + x)g(x, y, z)h(x, y, z) \pmod{2},$$

for some integer polynomials $g(x, y, z)$ and $h(x, y, z)$, neither of which is constant modulo 2.

(Two integer polynomials are congruent modulo 2 if every coefficient of their difference is even. A polynomial is constant modulo 2 if it is congruent to a constant polynomial modulo 2.)