The Hadamard conjecture
A Hadamard matrix of order is an matrix with entries whose rows are mutually orthogonal, so that . Such a matrix can exist only if is 1, 2 or a multiple of 4. Prove that a Hadamard matrix exists for every multiple of 4.
History and references
Still open as of 2026.
James Joseph Sylvester constructed examples of every order that is a power of 2 in 1867, and Jacques Hadamard found orders 12 and 20 in 1893. Raymond Paley's 1933 construction from finite fields produces many more orders, and Kronecker products of Hadamard matrices give still more. The smallest order not covered by the methods of Sylvester and Paley, 92, was found by computer in 1962. For many years 668 was the smallest order with no known example, and in 2026 explicit matrices were announced for 668 and the eleven other open orders below 2000. Finite constructions like these cannot settle the conjecture, which remains open.
References
Solutions 0
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