  There is a standard "word length" metric canonically associated to any set of
generators for a group. In particular, for any integers a and b greater than 1,
the additive group of integers has generating sets {a^i}_{i=0}^{\infty} and
{b^j}_{j=0}^{\infty} with associated metrics d_A and d_B, respectively. It is
proved that these metrics are bi-Lipschitz equivalent if and only if there
exist positive integers m and n such that a^m = b^n.
