The Hadamard conjecture is an open problem in combinatorics concerning Hadamard matrices, square matrices named after the French mathematician Jacques Hadamard whose entries are either +1 or -1 and whose rows are mutually orthogonal. The conjecture proposes that a Hadamard matrix of order 4k exists for every positive integer k; it has also been attributed to Raymond Paley, since the claim was considered implicitly by other mathematicians before Paley's own work on the subject. Hadamard matrices have been constructed for a great many orders, but no general proof covering every multiple of four has yet been found.
Facts
StatementA Hadamard matrix of order 4k exists for every positive integer k. 1 Progress Toward ResolutionHadamard matrices have been constructed for a great many orders: Sylvester's 1867 construction yields every power of two, Hadamard himself constructed orders 12 and 20 in 1893, order 92 was found computationally in 1962, and order 428 was constructed in 2005. As of 2014 twelve multiples of four below 2000 remained without a known construction. In August 2026 Levent Alpoge, Philippe Voinov and Saul Reynolds-Haertle, assisted by Anthropic's AI model Claude, announced explicit constructions for all twelve of those previously unknown orders below 2000, including order 668; a general proof covering every multiple of four has still not been found. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
Sources
1. Hadamard Matrix (Wikipedia)
Hadamard conjecture section
Is there a Hadamard matrix of order 4k for every positive integer k?
Status as of 2026 section
In August 2026, Levent Alpöge, Philippe Voinov, and Saul Reynolds-Haertle, assisted by Anthropic's AI model Claude, announced explicit constructions of Hadamard matrices for all twelve previously unknown orders below 2000, including order 668.
- Lead section
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