Let s_v denote the pure power sum \sum_{k=1}^n z_k^v. In a previous paper we
proved that \sqrt n <= \inf_{|z_k| => 1} \max_{v=1,…,n^2} |s_v| <= \sqrt{n+1}
when n+1 is prime. In this paper we prove that \inf_{|z_k| = 1}
\max_{v=1,…,n^2-n} |s_v| = \sqrt{n-1} when n-1 is a prime power, and if 2 <=
i <= n-1 and n => 3 is a prime power then \inf_{|z_k| => 1}
\max_{v=1,…,n^2-i} |s_v| =\sqrt n. We give explicit constructions of n-tuples
(z_1,…,z_n) which we prove are global minima for these problems. These are
two of the few times in Turan power sum theory where solutions in the inf max
problem can be explicitly calculated.