---
tag: 6X9S10E3M2
kind: conjecture
name: Hadamard conjecture
msc: [05B20]
uses: [DM59GHSMAJ]
about: [J0N4J606X0]
sources:
  - url: https://en.wikipedia.org/wiki/Hadamard_matrix
history:
  posed_by: "Raymond Paley"
  first_stated: "1933"
---

## Statement

For each positive integer $k$, there is a [[Hadamard matrix|DM59GHSMAJ]] of order $4k$.

## Remarks

A Hadamard matrix of order $n > 2$ can exist only when $n$ is a multiple of $4$, so the conjecture says that this necessary condition is also sufficient.
The conjecture is open.
Sylvester's construction by [[Kronecker products|0JK6M18QZH]] gives all orders $2^m$.
Paley's construction uses [[quadratic residues|0SQDB87Z3T]] in a [[finite field|08RTDHX1SK]] $\F_q$ and gives order $q + 1$ when $q \equiv 3 \pmod 4$ is a prime power.
Combining such constructions covers many multiples of $4$; the smallest multiple of $4$ for which no Hadamard matrix is known is $668$.

## Explanations

### L1 human

A Hadamard matrix is a square table of plus and minus signs in which any two rows agree in exactly half of the places.
Simple counting shows that the size must be a multiple of $4$, apart from sizes $1$ and $2$.
The conjecture says that every multiple of $4$ really is possible.