Hadamard's conjecture on the existence of Hadamard matrices

Hadamard’s conjecture, attributed to Paley, asserts that a suitable matrix exists for every positive order divisible by four. The general assertion remains open, with $668$ historically the smallest unresolved order.

Known results

• Paley’s constructions give matrices of order $q+1$ for primes $q\equiv 3 \pmod 4$, and order $2(q+1)$ for primes $q\equiv 1 \pmod 4$ (Paley).
• Kharaghani and Tayfeh-Rezaie constructed order $428$ in $2004$/$2005$, removing the previous smallest unknown order.
• Computational databases reported known constructions through order $1208$ and updated tables through order $2999$, while $668$ remained unknown in November $2024$.

2026 claimed constructions

Levent Alpöge reported examples for all previously unknown orders below $2000$, including $668$, attributing the work to Philippe Voinov, Saul Reynolds-Haertle, Claude, and himself. The report supplies no construction details or independent verification, so this is progress on individual orders, not a proof of Hadamard’s conjecture.

Current status (as of August 2026): The conjecture is not proved or disproved; a reported construction covering $668$ and other previously unknown orders below $2000$ remains unverified, and existence for every order divisible by $4$ is open.