MathDB

The following problem is open in the sense that the answer to part (b) is not currently known.
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_n(x,y,z) h_n(x,y,z) \pmod{2},

for some integer polynomials

g_n(x,y,z)

and

h_n(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_n(x,y,z) h_n(x,y,z) \pmod{2},

for some integer polynomials

g_n(x,y,z)

and

h_n(x,y,z)

, neither of which is constant modulo 2.
(Two integer polynomials are \emph{congruent modulo 2} if every coefficient of their difference is even. A polynomial is \emph{constant modulo 2} if it is congruent to a constant polynomial modulo 2.)

Problem Statement