## You are here

HomeHadamard conjecture

## Primary tabs

# Hadamard conjecture

There exists a Hadamard matrix of order $n$ = $4m$, for all $m\in\mathbb{Z}^{+}.$

A Hadamard matrix of order 428 (m=107) has been recently constructed [1].

A Hadamard matrix of order 764 has also recently been constructed [2].

Also, Paley’s theorem guarantees that there always exists a Hadamard matrix $H_{n}$ when $n$ is divisible by 4 and of the form $2^{e}(p^{m}+1)$, for some positive integers e and m, and p an odd prime and the matrices can be found using Paley construction.

This leaves the order of the lowest unknown Hadamard matrix as 668. There are 13 integers $m$ less than 500 for which no Hadamard matrix of order $4m$ is known:

$167,179,223,251,283,311,347,359,419,443,479,487,491$ |

and all of them are primes congruent to 3 mod 4.

# References

- 1
H. Kharaghani, B. Tayfeh-Rezaie,
*A Hadamard matrix of order 428*, J. Comb. Designs 13, (2005), 435-440. - 2
D.Z. Doković,
*Hadamard matrices of order 764 exist*, preprint.

Related:

HadamardMatrix

Synonym:

Hadamard's conjecture

Type of Math Object:

Conjecture

Major Section:

Reference

## Mathematics Subject Classification

15-00*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections