Chen-kai Ren

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20242026

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math.NT2026

On the natural density of integers $n$ for which $σ (k n + r_{1}) > σ (k n + r_{2})$

Xin-qi Luo, Chen-kai Ren

For any positive integer $n$ , let $σ (n) = \sum_{d ∣ n} d$ . In 2020, M. Kobayashi and T. Trudgian showed that the natural density of positive integers n with $σ(kn+r_1) \geq σ(kn+r_…

math.NT2025

Evaluation of a determinant involving Legendre symbols

Chen-Kai Ren, Zhi-Wei Sun

Let $p > 3$ be a prime, and let $(\frac{\cdot}{p})$ be the Legendre symbol. Let $A_{p} (x)$ denote the matrix $[x + a_{ij}]_{1 ⩽ i, j ⩽ (p - 1) /2}$ , where $$ a_{ij}=\begin{cases}…

math.NT2024

On some determinant conjectures

Ze-Hua Zhu, Chen-Kai Ren

Let $p$ be a prime and $c, d \in Z$ . Sun introduced the determinant $D_{p}^{-} (c, d) := det [(i^{2} + c ij + d j^{2})^{p - 2}]_{1 < i, j < p - 1}$ for $p > 3$ . In this paper, we confirm three conjecture…

math.NT2024

On certain determinants and the square root of some determinants involving Legendre Symbols

Chen-kai Ren, Xin-qi Luo

Let $p > 3$ be a prime and $(\frac{.}{p})$ be the Legendre symbol. For any integer $d$ with $p ∤ d$ and any positive integer $m$ , Sun introduced the determinants $$T_m(d,p)=\det\l…

math.NT2024

On some determinants arising from quadratic residues

Chen-Kai Ren, Zhi-Wei Sun

Let $p > 3$ be a prime, and let $d \in Z$ with $p ∤ d$ . For the determinants $$S_m(d,p)=\det\left[(i^2+dj^2)^{m}\right]_{1\leqslant i,j \leqslant (p-1)/2}\ \ \left(\frac{p-1…

On the natural density of integers n for which σ(kn+r1​)>σ(kn+r2​)

Evaluation of a determinant involving Legendre symbols

On some determinant conjectures

On certain determinants and the square root of some determinants involving Legendre Symbols

On some determinants arising from quadratic residues

On the natural density of integers $n$ for which $σ (k n + r_{1}) > σ (k n + r_{2})$