Prime power order circulant determinants

Newman showed that for primes p≥5, an integral circulant determinant of prime power order pt cannot take the value pt+1 once t≥2. We show that many other values are also excluded. In particular, we show that p2t is the smallest power of p attained for any t≥3, p≥3. We demonstrate the complexity involved by giving a complete description of the 25×25 and 27×27 integral circulant determinants. The former case involves a partition of the primes that are 1mod5 into two sets, Tanner’s artiads and perissads, which were later characterized by E. Lehmer.