Related papers: On some determinants involving Jacobi symbols

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 $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any…

In this paper, we partly prove a supercongruence conjectured by Z.-W. Sun in 2013. Let $p$ be an odd prime and let $a\in\mathbb{Z}^{+}$. Then if $p\equiv1\pmod3$, we have \begin{align*}…

The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some…

Let $p$ be a prime and $c,d\in\mathbb{Z}$. Sun introduced the determinant $D_p^-(c,d):=\det[(i^2+cij+dj^2)^{p-2}]_{1<i,j<p-1}$ for $p>3$. In this paper, we confirm three conjectures on $D_p^-(c,d)$ proposed by Zhi-Wei Sun.

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2}…

In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Ap\'ery numbers. For any nonnegative integer $n$, define…

Let $p>3$ be a prime and let $a$ be a positive integer. We show that if $p\equiv1\pmod 4$ or $a>1$ then $$\sum_{k=0}^{\lfloor\frac34p^a\rfloor}\frac{\binom{2k}k^2}{16^k}\equiv\l(\frac{-1}{p^a}\r)\pmod{p^3}$$ with $(-)$ the Jacobi symbol,…

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

Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures…

In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det […

Let n>=3 be an odd integer. For any integer a prime to n, define the permutation gamma_{a,n} of {1,...,(n-1)/2} by gamma_{a,n}(x)=n-\dec{ax}_n if {ax}_n>=(n+1)/2, and {ax}_n if {ax}_n<=(n-1)/2, where {x}_n denotes the least nonnegative…

The Sun polynomials $g_n(x)$ are defined by \begin{align*} g_n(x)=\sum_{k=0}^n{n\choose k}^2{2k\choose k}x^k. \end{align*} We prove that, for any positive integer $n$, there hold \begin{align*} &\frac{1}{n}\sum_{k=0}^{n-1}(4k+3)g_k(x)…

In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. We deduce some new results in this direction; for example, we prove that for any prime $p\equiv3\pmod4$ we have…

Let $p$ be an odd prime. For $b,c\in\mathbb Z$, Sun introduced the determinant $$D_p(b,c)=\left|(i^2+bij+cj^2)^{p-2}\right|_{1\leqslant i,j \leqslant p-1},$$ and investigated the Legendre symbol $(\frac{D_p(b,c)}p)$. Recently Wu, She and Ni…

Let $p$ be an odd prime. For any $b,c\in\mathbb{Z}$, Z.-W. Sun introduced the new-type determinant $$D_p(b,c)=|(i^2+bij+cj^2)^{p-2}|_{1\leqslant i,j\leqslant p-1},$$ and studied its arithmetic properties. In this paper we mainly prove that…

Let $p$ be an odd prime and $x$ be an indeterminate. Recently, Z.-W. Sun proposed the following conjecture: $$\det\left[x+\left(\frac{j-i}{p}\right)\right]_{0\le i,j\le \frac{p-1}{2}}=\begin{cases} (\frac{2}{p})pb_px-a_p & \mbox{if}\…

Let $p$ be an odd prime and let $a,b\in\mathbb Z$ with $p\nmid ab$. In this paper we mainly evaluate $$T_p^{(\delta)}(a,b,x):=\det\left[x+\tan\pi\frac{aj^2+bk^2}p\right]_{\delta\le j,k\le (p-1)/2}\ \ (\delta=0,1).$$ For example, in the case…

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…

For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\sum_{k=0}^n {n\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0 \pmod{2n^2},…

The numbers $R_n$ and $W_n$ are defined as \begin{align*} R_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{1}{2k-1},\ \text{and}\ W_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{3}{2k-3}. \end{align*} We prove that, for any positive…