2020/08/24 by Chao Huang, Huang, Chao
Mathematics · #11T24 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 11A07 #Secondary 11A15 #math.NT #msc:11A07 #msc:11A15 #msc:11T24
paper · pdf · doi:10.48550/arxiv.2008.10502
arxiv created 2020/08/24 · openalex publication_date 2020/08/24 · arxiv updated 2020/08/25 · openalex created_date 2020/09/01 · openalex updated_date 2026/07/28
Let p>3 be a prime. We investigate Legendre symbol of ∏0<i<j<p/2 \atop p\nmid f(i,j) f(i,j) , where i,j∈ \Bbb Z, f(i,j) is a linear or quadratic form with integer coefficients. When f=ai2+bij+cj2 and p\nmid c(a+b+c) , we prove that to evaluate the product is equivalent to determine ∑y=1p-1 ((y(y+1)(y+k))/(p)) \pmod16 , where 4c(a+b+c)k ≡ (4ac-b2)\pmodp. Parallel results are given for ∏i,j=1 \atop p\nmid f(i,j) (p-1)/2 (( f(i,j) )/(p)). Then we show that ∑y=1p-1 ((y(y+1)(y+k))/(p)) \pmod16 can be evaluated explicitly when k=2,4,5,9,10 or k is a square. And for several classes of f(i,j) these two kinds of products can be evaluated explicitly. Finally when f is a linear form we give unified identities for these products. Thus we prove these kind of problems raised in Sun's paper.