On some determinants and permanents

TL;DR

This paper explores specific determinants and permanents involving quadratic forms modulo prime numbers, providing new formulas and conjectures in the area of algebraic combinatorics.

Contribution

It introduces new types of determinants involving quadratic forms and investigates their properties modulo primes, along with proposing conjectures for future research.

Findings

Derived formulas for determinants involving quadratic forms modulo primes

Identified patterns and properties of these determinants

Proposed conjectures for further mathematical exploration

Abstract

In this paper we study some determinants and permanents. In particular, we investigate the new type determinants $$\det[(i^2+cij+dj^2)^{p-2}]_{1\le i,j\le p-1}\ \text{and} \ \det[(i^2+cij+dj^2)^{p-2}]_{0\le i,j\le p-1}$$ modulo an odd prime $p$, where $c$ and $d$ are integers. We also pose some conjectures for further research.