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On some determinants and permanents

Zhi-Wei Sun

TL;DR

This paper investigates the new-type determinants det[( i 2 + cij + dj 2 ) p − 2 ] 1 ≤ i,j ≤ p − 1 modulo an odd prime p, where c and d are integers.

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.

On some determinants and permanents

TL;DR

This paper investigates the new-type determinants det[( i 2 + cij + dj 2 ) p − 2 ] 1 ≤ i,j ≤ p − 1 modulo an odd prime p, where c and d are integers.

Abstract

In this paper we study some determinants and permanents. In particular, we investigate the new type determinants modulo an odd prime , where and are integers. We also pose some conjectures for further research.
Paper Structure (1 section, 3 theorems, 12 equations)

This paper contains 1 section, 3 theorems, 12 equations.

Table of Contents

  1. Introduction

Key Result

Theorem 1.1

Let $A=[a_{i,j}]_{1\leqslant i,j\leqslant n}$ be an matrix over a commutative ring. Suppose that $a_{i,j}=0$ whenever $i+j$ is an even number greater than two. (i) If $n=2m$ for some $m\in\Bbb Z^+$, then and (ii) If $n=2m+1$ for some $m\in\Bbb Z^+$, then and

Theorems & Definitions (5)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5