Related papers: Legendre symbols related to certain determinants

Let $p$ be an odd prime, and define $$G_p(x)=\prod_{k=1}^{(p-1)/2}\left(x-e^{2\pi i k^2/p}\right).$$ In this paper we study values of $G_p(x)$ at roots of unity via Galois theory, and confirm some previous conjectures. For example, for any…

Let $f(n)$ be a random completely multiplicative function such that $f(p) = \pm 1$ with probabilities $1/2$ independently at each prime. We study the conditional probability, given that $f(p) = 1$ for all $p < y$, that all partial sums of…

In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let $p$ be a prime and let $a$ be any positive integer. We determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}$ mod $p^2$ for…

The main purpose of this paper is using a very simple constructive method to study an old number theory problem related to the Legendre symbol modulo p, and completely solved it. The proving method of the result is purely elementary and has…

In this paper we prove the supercongruence $$\sum_{n=0}^{(p-1)/2}\frac{6n+1}{256^n}\binom{2n}n^3\equiv p(-1)^{(p-1)/2}+(-1)^{(p-1)/2}\frac{7}{24}p^4B_{p-3}\pmod{p^5}$$ for any prime $p>3$, which was conjectured by Sun in 2019.

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…

We consider the operator of taking the $2p$th derivative of a function with zero boundary conditions for the function and its first $p-1$ derivatives at two distinct points. Our main result provides an asymptotic formula for the eigenvalues…

In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom…

The harmonic numbers $H_n=\sum_{0<k\ls n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. With helps of some combinatorial identities, we establish the following two congruences:…

Let $p>3$ be a prime, and let $q_p(2)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base 2. Recently, Z. H. Sun proved that \sum_{k=1}^{p-1}\frac{1}{k\cdot 2^k}\equiv q_p(2)-\frac{p}{2}q_p(2)^2 \pmod{p^2} which is a generalization of a…

We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes iff n is composite. If the dimension is a prime p, then the…

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is…

In this paper, motivated by the work of Chapman, Vsemirnov and Sun et al., we investigate some arithmetic properties of the generalized Legendre matrices over finite fields. For example, letting $a_1,\cdots,a_{(q-1)/2}$ be all non-zero…

Let p be a prime, and let n>0 and r be integers. In 1913 Fleck showed that $$F_p(n,r)=(-p)^{-[(n-1)/(p-1)]}\sum_{k=r(mod p)}\binom{n}{k}(-1)^k\in\Z.$$ Nowadays this result plays important roles in many aspects. Recently Sun and Wan…

We investigate the Lebesgue--Nagell equation \begin{align*} x^2-2=y^p \end{align*} in integers $x,y,p$ with $p\geq 3$ an odd prime. A longstanding folklore conjecture asserts that the only solutions are the ``trivial'' ones with $y=-1$. We…

Recently, Z.-W. Sun made the following conjecture: for any odd prime $p$ and odd integer $m$, $$ \frac{1}{m^2{m-1\choose (m-1)/2}}\Bigg(\sum_{k=0}^{(pm-1)/2}\frac{{2k\choose k}}{8^k}…

The Franel numbers are defined by $ f_n=\sum_{k=0}^n {n\choose k}^3. $ Motivated by the recent work of Z.-W. Sun on Franel numbers, we prove that \begin{align*} \sum_{k=0}^{n-1}(3k+1)(-16)^{n-k-1} {2k\choose k} f_k &\equiv…

We show that if p is an odd prime then $$\sum_{k=0}^{p-1}E_kE_{p-1-k}=1 (mod p)$$ and $$\sum_{k=0}^{p-3}E_kE_{p-3-k}=(-1)^{(p-1)/2}2E_{p-3} (mod p),$$ where E_0,E_1,E_2,... are Euler numbers. Moreover, we prove that for any positive integer…

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.

Let $p$ be an odd prime and let $n$ be a positive integer. For any positive integer $\alpha$ and $m\in\{1,2,3\}$, we have \begin{align*}…