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

Abstract

For every positive integer $k$ such that $k>1,$ there are an infinity of odd integers $h$ with $\omega (h)=k$ distinct prime divisors such that there do not exist a Circulant Hadamard matrix $H$ of order $n=4h^2.$ Moreover, our main result implies that for all of the odd numbers $h$, with $1<h<10^{13}$ there is no Circulant Hadamard matrix of order $n=4h^2.$

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