Journal article
Proof of three conjectures on determinants related to quadratic residues
Linear & multilinear algebra, v ahead-of-print(ahead-of-print), pp 1-13
01 Dec 2020
Abstract
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 vertical bar(i(2) + dj(2)) (i(2) + dj(2)/n)vertical bar(0 <= i,j <=(n-1)/2), where d is any integer and (center dot/n) is the Jacobi symbol. Then we prove some divisibility results concerning
vertical bar(i + dj)(n)vertical bar(0 <= i,j <= n-1) and vertical bar(i(2) + dj(2))(n)vertical bar(0 <= i,j <= n-1),
where d not equal 0 and n> 2 are integers. Finally, for any odd prime p and integers c and d with p inverted iota cd, we determine completely the Legendre symbol (S-c(d,p)/p), where S-c(d, p) := vertical bar(i(2)+dj(2)+c/p)vertical bar(1 <= i,j <=(p-1)/2).
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Details
- Title
- Proof of three conjectures on determinants related to quadratic residues
- Creators
- Darij Grinberg - Drexel UniversityZhi-Wei Sun - Nanjing UniversityLilu Zhao - Shandong University
- Publication Details
- Linear & multilinear algebra, v ahead-of-print(ahead-of-print), pp 1-13
- Publisher
- Taylor & Francis
- Number of pages
- 13
- Grant note
- 11971222 / Natural Science Foundation of China; National Natural Science Foundation of China (NSFC)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000596641000001
- Scopus ID
- 2-s2.0-85097073392
- Other Identifier
- 991019168403404721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics