MathDB

Let

P(x,y)=x^2y+xy^2

and

Q(x,y)=x^2+xy+y^2.

For

n=1,2,3,\dots,

let \begin{align*}F_n(x,y)=(x+y)^n-x^n-y^n \text{ and,}\\ G_n(x,y)=(x+y)^n+x^n+y^n. \end{align*} One observes that

G_2=2Q, F_3=3P, G_4=2Q^2, F_5=5PQ, G_6=2Q^3+3P^2.

Prove that, in fact, for each

n

either

F_n

G_n

is expressible as a polynomial in

P

and

Q

with integer coefficients.

Problem Statement