MathDB

Let

f, g

be functions from the positive integers to the integers. Vlad the impala is jumping around the integer grid. His initial position is

x_0 = (0, 0)

, and for every

n \ge 1

, his jump is

x_n - x_{n - 1} = (\pm f(n), \pm g(n))

(\pm g(n), \pm f(n)),

with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location

(0, 0)

infinitely many times when (a)

f, g

are polynomials with integer coefficients? (b)

f, g

are any pair of functions from the positive integers to the integers?

Problem Statement