Double Wieferich Pairs and Circulant Hadamard Matrices

We show that all but 4489 integers n with 4 < n ≤ 4 ·1030 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 pq−1 ≡ 1 mod q2 and qp−1 ≡ 1 mod p2.