Let D_p(c,d) denote the determinant associated with the quadratic form i^2+cij+dj^2 in the paper. Legendre-symbol formula. For any odd prime p equivpm2 pmod5, (fracD_p(3,1)p)= (frac6p)&if p equiv1 pmod4, 0&if p equiv3 pmod4. The supplied text gives no resolution of this proposed formula.

A Legendre-symbol formula for $D_p(3,1)$

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Let $D_p(c,d)$ denote the determinant associated with the quadratic form $i^2+cij+dj^2$ in the paper. Legendre-symbol formula. For any odd prime $p\equiv\pm2\pmod5$ ,

\left(\frac{D_p(3,1)}{p}\right)= \begin{cases} \left(\frac{6}{p}\right)&\text{if }p\equiv1\pmod4,\\\\ 0&\text{if }p\equiv3\pmod4. \end{cases}

The supplied text gives no resolution of this proposed formula.

References

Primary source

Zhi-Wei Sun, “On some determinants and permanents”, arXiv:2207.13039 (2024).

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