On the Density of the Set of Known Hadamard Orders

Understand

Let $S(x)$ be the number of $n \leq x$ for which a Hadamard matrix of order $n$ exists.

• Hadamard's conjecture states that $S(x)$ is about $x/4$.
• From Paley's constructions of Hadamard matrices, we have that \[ S(x) = \Omega(x/\log x).
• \] In a recent paper, the first author suggested that counting the products of orders of Paley matrices would result in a greater density.