Hadamard conjecture

Prove that for every positive integer n divisible by 4 there exists an n×n matrix H with entries in {±1} such that HHᵀ = nI, i.e., establish the existence of Hadamard matrices for all orders n that are multiples of four.

Background

A Hadamard matrix is an n×n matrix with entries in {±1} whose rows are mutually orthogonal. The Hadamard conjecture asserts existence for all n divisible by 4 and is a central open problem in combinatorial design theory and matrix theory. The paper studies counts of partial Hadamard matrices (rectangular ±1 matrices with pairwise orthogonal rows) and connects counting questions to a Fourier-analytic random walk framework, noting that the classical conjecture remains unresolved.

The authors also note an equivalence: for n divisible by 4, positivity of a certain Fourier integral at t = n is equivalent to the existence of an n×n Hadamard matrix, underscoring how counting/analytic perspectives interface with the conjecture.

References

The Hadamard conjecture asks whether, for every positive integer n divisible by~4, there exists an n\times n matrix H with entries in {\pm1} satisfying HH\top = nI. Despite sustained effort since Sylvester's recursive construction and Paley's finite-field families , the conjecture remains open; we refer to Horadam and Seberry--Yamada for background on Hadamard matrices, to Tressler and Browne et al. for surveys, and to Cati--Pasechnik for a recent computational database.

— Counting partial Hadamard matrices in the cubic regime  (2603.30013 - Davis, 31 Mar 2026) in Section 1 (Introduction)

The long-standing Hadamard Conjecture posits that this necessary condition is also sufficient—namely, that a Hadamard matrix exists for every positive integer multiple of 4. While constructive methods such as Sylvester's Kronecker-product technique, Paley's quadratic residue constructions, and Williamson array-based approaches have proven existence for vast families of orders, finding general construction methods for arbitrary multiples of 4 remains an active open pursuit.

— TSS Graphs for Hadamard Matrices: Real vs Complex  (2609.16813 - Lewis et al., 15 Sep 2026) in Section 1, Introduction