A proof of Ryser's circulant Hadamard conjecture

TL;DR

This paper proves Ryser's conjecture by demonstrating that circulant Hadamard matrices cannot exist for dimensions greater than four, using congruence equations to establish the non-existence.

Contribution

It provides a definitive proof that no circulant Hadamard matrices exist for sizes larger than four, resolving a long-standing conjecture from 1963.

Findings

Circulant Hadamard matrices only exist for n ≤ 4.

The proof relies on congruence equations that restrict possible matrix sizes.

Confirmed non-existence of such matrices for all larger n.

Abstract

We show that an $n\times n$ circulant Hadamard matrix must satisfy a family of congruence equations that have solutions only when $n \leq 4$, proving Ryser's 1963 conjecture that no such matrices exist for $n>4$.