Hadamard conjecture

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 gives all orders $2^m$. Paley's construction uses quadratic residues in a finite field $\F_q$ 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 $668$.