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For every positive integer $k$ such that $k>1,$ there are an infinity of oddintegers $h$ with $\omega(h) =k$ distinct prime divisors such that there do notexist a Circulant Hadamard matrix $H$ of order $n=4h^2.$ Moreover, our mainresult implies that for all of the odd numbers $h$, with $1< h < 10^{13}$ thereis no Circulant Hadamard matrix of order $n=4h^2.$

Quadratic residues and a new infinity of orders for which a conjecture of Ryser about Circulant Hadamard matrices holds