Unfortunatly, there has been confusion about the status of this problem for decades.
All correct results on the circulant Hadamard matrix conjecture are contained in the following papers:
 Turyn, Richard J.: Character sums and difference sets. Pacific J. Math. 15 1965 319–346.
 Leung, Ka Hin; Schmidt, Bernhard: The field descent method. Des. Codes Cryptogr. 36 (2005), no. 2, 171–188.
 Leung, Ka Hin; Schmidt, Bernhard: New restrictions on possible orders of circulant Hadamard matrices. Des. Codes Cryptogr. 64 (2012), no. 1-2, 143–151.
Computational results and the latest status on the smallest open cases can be found in
 Peter Borwein, Michael J. Mossinghoff: Wieferich pairs and Barker sequences, II.