On a problem suggested by Olga Taussky-Todd

The problem considered is to characterize those integers m such that m = det(C), C an integral n × n circulant. It is shown that if $(m,n) = 1$ then such circulants always exist, and if $(m,n)> 1$ and p is a prime dividing $(m,n)$ such that pᵗ||n, then pᵗ⁺¹|m. This implies for example, that n never occurs as the determinant of an integral n × n circulant, if $n > 1$.