On R. Chapman's "evil determinant": case p=1 (mod 4)

TL;DR

This paper proves a conjectured formula for the determinant of a matrix defined by Legendre symbols for primes p congruent to 1 mod 4, advancing understanding of number-theoretic determinants.

Contribution

It provides a rigorous proof of R. Chapman's conjectured determinant formula for the specific case p=1 mod 4.

Findings

Confirmed the conjectured determinant formula for the matrix C.

Established a new explicit expression for the determinant involving Legendre symbols.

Enhanced understanding of determinants related to quadratic residues in number theory.

Abstract

For p=1 (mod 4), we prove the formula (conjectured by R. Chapman) for the determinant of the matrix C with C(i,j)=LegendreSymbol(j-i,p), i,j=0,...,(p-1)/2.