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We proved recently (see \cite{lhgarasu}) the result on the title for oddprime divisors of such an $n.$ The result implies for many $n's$, moreprecisely, for an infinity of $n$'s with an arbitrary fixed number of primedivisors, the inexistence of circulant Hadamard matrices, and the inexistenceof Barker sequences of length $n >13$. The proof used a result of Arasu. Itturns out that there is another, shorter proof, of the more general result thatincludes the prime $p=2.$ This new proof is based on a result of Brock (see\cite[Theorem 3.1]{brock}), and besides that, requires just the definition ofthe Fourier transform. I noticed Brock's result in a preprint (see\cite{winterhofetal}) of Winterhof et al. where it is used to study theinexistence of related Butson-Hadamard matrices.

If $p^a \vert \vert n$ where $n >4$ is the order of a Circulant Hadamard matrix, then the order of $p$ modulo $n/p^a$ is odd