If k is a positive integer and p is
a prime with p ≡ 1 (mod2k), then 2(p−1)∕2k is a 2k-th root of unity modulo p. We
consider the problem of determining 2(p−1)∕2k modulo p. This has been done for
k = 1,2,3 and the present paper treats k = 4 and 5, extending the work of
Cunningham, Aigner, Hasse, and Evans.