Quadratic residues and a new infinity of orders for which a conjecture of Ryser about Circulant Hadamard matrices holds

TL;DR

This paper proves that for infinitely many odd integers with a fixed number of prime factors, Circulant Hadamard matrices of order 4h^2 do not exist, extending Ryser's conjecture to a broad class of cases.

Contribution

It establishes an infinite family of odd integers h for which no Circulant Hadamard matrix of order 4h^2 exists, advancing understanding of Ryser's conjecture.

Findings

No Circulant Hadamard matrices of order 4h^2 for all odd h < 10^{13}

Existence of infinitely many odd h with a fixed number of prime divisors where such matrices do not exist

Extension of non-existence results to an infinite set of cases based on prime factorization

Abstract

For every positive integer $k$ such that $k>1,$ there are an infinity of odd integers $h$ with $\omega(h) =k$ distinct prime divisors such that there do not exist a Circulant Hadamard matrix $H$ of order $n=4h^2.$ Moreover, our main result implies that for all of the odd numbers $h$, with $1< h < 10^{13}$ there is no Circulant Hadamard matrix of order $n=4h^2.$