Double Wieferich Pairs and Circulant Hadamard Matrices

We show that all but \(4489\) integers \(n\) with \(4 < n \leq 4 \cdot 10^{30}\) cannot occur as the order of a circulant Hadamard matrix. Our algorithm allows us to search \(10000\) times farther than prior efforts, while substantially reducing memory requirements. The principal improvement over prior methods involves the incorporation of a separate search for double Wieferich prime pairs \(\{p, q\}\), which have the property that \(p^{q-1} \equiv 1 \pmod{q^2}\) and \(q^{p-1} \equiv 1 \pmod{p^2}\).