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On a determinant involving linear combinations of Legendre symbols
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On a determinant involving linear combinations of Legendre symbols
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In this paper, we prove a conjecture of the second author by evaluating the determinant $$\det\left[x + \left(\frac{i-j}p\right) + \left(\frac ip\right)y + \left(\frac jp\right)z + \left(\frac{ij}p\right)w\right]_{0\le i,j\le(p-3)/2}$$ for any odd prime $p$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. In particular, the determinant is equal to $x$ when $p\equiv 3\pmod4$.
Forward citations
Cited by 2 Pith papers
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Two determinant evaluations in Sun's conjectures involving Legendre symbols
Proves two determinant evaluations for Legendre symbol matrices, resolving the p≡1 mod 4 cases of Sun's Conjectures 4.8(i) and 4.10(i) via matrix factorizations.
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Two determinant evaluations in Sun's conjectures involving Legendre symbols
Proves explicit determinant formulas and a uniform congruence modulo p for Legendre symbol matrices, resolving parts of Sun's conjectures 4.8(i) and 4.10(i).
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