Related papers: Determinants concerning Legendre symbols

Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants involving $(\frac {j+k}p)\pm(\frac{j-k}p)$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. When $p\equiv1\pmod4$, we determine the characteristic polynomials…

Let $p>3$ be a prime. We investigate Legendre symbol of $\displaystyle \prod_{0<i<j<p/2 \atop p\nmid f(i,j) } f(i,j) \ $, where $i,j\in \Bbb Z, f(i,j)$ is a linear or quadratic form with integer coefficients. When $f=ai^2+bij+cj^2$ and…

Let $p$ be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for $\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we confirm several conjectures of Z.W. Sun. We also…

In 2019, Zhi-Wei Sun posed an interesting conjecture on certain determinants with Legendre symbol entries. In this paper, by using the arithmetic properties of $p$-th cyclotomic field and the finite field $\mathbb{F}_p$, we confirm this…

Let $m$ be any positive integer and let $\delta_1,\delta_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=\delta_1\…

Although squaring integers is deterministic, squares modulo a prime, $p$, appear to be random. First, because they are all generated by the multiplicative linear congruential equation, $x_{i+1} = g^2 x_i \mod p$, where $x_0 = 1$ and $g$ is…

For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k$. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related…

Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\det\left[\left(\frac{j^2-k^2}p\right)+\left(\frac{jk}p\right)w\right]_{\delta\le j,k\le (p-1)/2}$$ with $\delta\in\{0,1\}$.…

In this paper, we prove that for any $A>0$ there exist infinitely many primes $p$ for which sums of the Legendre symbol modulo $p$ over an interval of length $(\ln p)^A$ can take large values.

For $0\leq \alpha<1$ and prime number $p$ let $L(\alpha,p)$ be the sum of the first $[\alpha p]$ values of Legendre symbol modulo $p$. We study positivity of $L(\alpha,p)$ and prove that for $|\alpha-\frac13|<2\cdot 10^{-6}$ and for…

Motivated by the recent work of Zhi-Wei Sun on determinants involving the Legendre symbol, in this paper, we study some matrices concerning subgroups of finite fields. For example, let $q\equiv 3\pmod 4$ be an odd prime power and let $\phi$…

For p=1 (mod 4), we prove the formula (conjectured by R. Chapman) for the determinant of the matrix C with C(i,j)=LegendreSymbol(j-i,p), i,j=0,...,(p-1)/2.

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms $2^a3^b$ and $2^c5^d$.…

For each odd prime $p$, let $\zeta_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and $p>3$ the…

We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$,…

Given a squarefree positive integer $d$, we want to find integers (or rational numbers with denominators not divisible by large primes) $a_0,a_1,a_2,\ldots$ such that for sufficiently large primes $p$ we have $\sum_{k=0}^{p-1}a_k\equiv…

In this paper we study some determinants and permanents. In particular, we investigate the new type determinants $$\det[(i^2+cij+dj^2)^{p-2}]_{1\le i,j\le p-1}\ \text{and} \ \det[(i^2+cij+dj^2)^{p-2}]_{0\le i,j\le p-1}$$ modulo an odd prime…

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^3/m^k\pmod{p^2}$, $\sum_{k=0}^{p-1}\binom{2k}k\b{4k}{2k}/m^k\pmod p$ and…

In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$ with $x+3y$ a square. Meanwhile, he also…

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $b\in\mathbb Z$ and $\varepsilon\in\{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):\ 1<a<p\ \text{and}\ \left(\frac…