Open problem, Combinatorics and discrete mathematics, Algebra, posed 1893

Hadamard conjecture

Open

For every positive integer $k \ge 1$ , there exists a Hadamard matrix of order $n = 4k$ : an $n \times n$ matrix $H \in \{-1, +1\}^{n \times n}$ with entries in $\{-1, +1\}$ whose rows are pairwise orthogonal, satisfying $H H^{\mathsf{T}} = n I_n$ .

Research frontier as of 2026

As of 2026, the general Hadamard conjecture remains open: no proof is known that a Hadamard matrix exists for every $n = 4k$ , nor even that the set of multiples of $4$ admitting a Hadamard matrix has positive natural density. However, combining Paley matrices, Williamson and Goethals–Seidel circulant block arrays, Turyn Base Sequences, and 2026 AI-assisted block searches has produced explicit Hadamard matrices for every multiple of $4$ up to $n = 2000$ . Asymptotically, de Launey and Gordon (2001) proved under the Extended Riemann Hypothesis that the number of $m \le x$ for which a Hadamard matrix of order $4m$ exists is at least $\frac{x}{\log x} e^{(C + o(1))(\log\log\log x)^2}$ .

Best known results

Infinite families of Hadamard matrices are proved for $n = 2^a 12^b 20^c \prod (q_i + 1) \prod 2(r_j + 1)$ where $q_i \equiv 3 \pmod 4$ and $r_j \equiv 1 \pmod 4$ are prime powers (Sylvester 1867; Paley 1933).
Explicit Hadamard matrices have been constructed for every multiple of $4$ up to $n = 2000$ (including order $428$ by Kharaghani–Tayfeh-Rezaie 2005 and order $668$ in 2026).

Tools and where they stop

Tool	Achieved	Where it stops
Paley quadratic-residue difference sets over $\mathbb{F}_q$	Uses the character-sum orthogonality of the Jacobsthal matrix over $\mathbb{F}_q$ to construct Hadamard matrices of orders $q + 1$ ( $q \equiv 3 \pmod 4$ ) and $2(q + 1)$ ( $q \equiv 1 \pmod 4$ ).	Prime powers have zero asymptotic density $\Theta(x / \log x)$ among all integers, leaving infinitely many composite orders $n = 4k$ untouched even after Kronecker multiplication.
Goethals–Seidel arrays, $T$ -sequences, and circulant block decompositions	Reduces an order- $4m$ Hadamard matrix to four $m \times m$ circulant $\pm 1$ matrices $A, B, C, D$ whose autocorrelation sequences sum to zero ( $AA^{\mathsf{T}} + BB^{\mathsf{T}} + CC^{\mathsf{T}} + DD^{\mathsf{T}} = 4m I_m$ ), resolving individual sporadic orders up to $2000$ .	Requires a separate finite combinatorial search for each length $m$ rather than providing a uniform algebraic proof for all integers $m \ge 1$ .

Open questions

Does the set of positive integers $k$ for which a Hadamard matrix of order $4k$ exists have positive lower asymptotic density?
Does every multiple of $4$ admit a skew-Hadamard matrix $H = I_n + S$ with $S^{\mathsf{T}} = -S$ ?

References

Jacques Hadamard (1893). Résolution d'une question relative aux déterminants
Raymond E. A. C. Paley (1933). On orthogonal matrices · DOI:10.1002/sapm1933121311
Hadi Kharaghani, Behruz Tayfeh-Rezaie (2005). A Hadamard matrix of order 428 · DOI:10.1002/jcd.20043
Epoch AI (2026). Hadamard Matrix of Order 668 — FrontierMath: Open Problems