2020/07/08 by Darij Grinberg, Zhi‐Wei Sun, Grinberg, Darij +3
Mathematics · #11A07 #11A15 #11C20 #15A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2007.06453
openalex publication_date 2020/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer n>3 divides the determinant |(i2+dj2)(\fraci2+dj2n)|0≤ i,j≤ (n-1)/2, where d is any integer and (\frac⋅n) is the Jacobi symbol. Then we prove some divisibility results concerning |(i+dj)n|0≤ i,j≤ n-1 and |(i2+dj2)n|0≤ i,j≤ n-1, where d\not=0 and n>2 are integers. Finally, for any odd prime p and integers c and d with p\nmid cd, we determine completely the Legendre symbol (\fracSc(d,p)p), where Sc(d,p):=|(\fraci2+dj2+cp)|1≤ i,j≤(p-1)/2.