Hadamard Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A Hadamard matrix of order $m$ is a matrix $H=(h_{ij})\in\{-1,1\}^{m\times m}$ satisfying $HH^{\mathsf T}=mI_m$, where $H^{\mathsf T}$ is its transpose and $I_m$ is the identity matrix. Equivalently, its rows are pairwise orthogonal for the usual real dot product; each row automatically has squared length $m$.

Statement recorded in the supplied source.

For every integer $k\geq1$, does there exist a Hadamard matrix $H$ of order $4k$? Thus the required equality is $HH^{\mathsf T}=4kI_{4k}$ over the integers, rather than a congruence modulo a fixed modulus. [S2]

Short English statement

Does every positive order divisible by four admit a square sign matrix with mutually orthogonal rows?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1102 (ALG-002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Shalom Eliahou, A 64-modular Hadamard matrix of order 668, Australasian Journal of Combinatorics 93(2) (2025), 422--427, §1, p.422. https://ajc.maths.uq.edu.au/pdf/93/ajc_v93_p422.pdf.