Paley's Theorem

Proved in 1933. If $q$ is an Odd Prime or $q = 0$ and $n$ is any Positive Integer, then there is a Hadamard Matrix of order

$$m = 2^e (q^n + 1),$$

where $e$ is any Positive Integer such that $m\equiv 0\ \left({{\rm mod\ } {4}}\right)$. If $m$ is of this form, the matrix can be constructed with a Paley Construction. If $m$ is divisible by 4 but not of the form (1), the Paley Class is undefined. However, Hadamard Matrices have been shown to exist for all $m\equiv 0\ \left({{\rm mod\ } {4}}\right)$ for $m<428$.

See also Hadamard Matrix, Paley Class, Paley Construction