Still searching Arxiv, refresh for possibly better results.

total 23681took 0.13s

Masses of doubly charmed baryons in the extended on-mass-shell renormalization schemeFeb 15 2016Feb 17 2016In this work, we investigate the mass corrections of the doubly charmed baryons up to $N^2LO$ in the extended-on-mass-shell (EOMS) renormalization scheme, comparing with the results of heavy baryon chiral perturbation theory. We find that the terms from ... More

Exterior algebras and two conjectures on finite abelian groupsAug 20 2008Apr 13 2011Let G be a finite abelian group with |G|>1. Let a_1,...,a_k be k distinct elements of G and let b_1,...,b_k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that there is a ... More

Possible bound states with hidden bottom from $\bar{K}^{(*)}B^{(*)}\bar{B}^{(*)}$ systemsDec 24 2018May 27 2019We study the three-body systems of $\bar{K}^{(*)}B^{(*)}\bar{B}^{(*)}$ by solving the Faddeev equations in the fixed-center approximation, where the light particle $\bar{K}^{(*)}$ interacts with the heavy bound states of $B\bar{B}$ ($B^*\bar{B}^*$) forming ... More

X(3915) and X(4350) as new members in P-wave charmonium familyNov 19 2009Mar 06 2010The analysis of the mass spectrum and the calculation of the strong decay of P-wave charmonium states strongly support to explain the newly observed X(3915) and X(4350) as new members in P-wave charmonium family, i.e., $\chi_{c0}^\prime$ for X(3915) and ... More

Orbital Kondo effect in a parallel double quantum dotSep 25 2014We construct a theoretical model to study the orbital Kondo effect in a parallel double quantum dot (DQD). Recently, pseudospin-resolved transport spectroscopy of the orbital Kondo effect in a DQD has been experimentally reported. The experiment revealed ... More

Spin-current Seebeck effect in quantum dot systemsJun 28 2013We first bring up the concept of spin-current Seebeck effect based on a recent experiment [Nat. Phys. {\bf 8}, 313 (2012)], and investigate the spin-current Seebeck effect in quantum dot (QD) systems. Our results show that the spin-current Seebeck coefficient ... More

Chiral corrections to the $1^{-+}$ exotic meson massMar 21 2016Dec 12 2016We first construct the effective chiral Lagrangians for the $1^{-+}$ exotic mesons. With the infrared regularization scheme, we derive the one-loop infrared singular chiral corrections to the $\pi_1(1600)$ mass explicitly. We investigate the variation ... More

Strong LHCb evidence supporting the existence of the hidden-charm molecular pentaquarksMar 26 2019Jul 02 2019On March 26th, 2019, at the Rencontres de Moriond QCD conference, the LHCb Collaboration reported the observation of three new pentaquarks, namely $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$, which are consistent with the loosely bound molecular hidden-charm ... More

Neural Consciousness FlowMay 30 2019The ability of reasoning beyond data fitting is substantial to deep learning systems in order to make a leap forward towards artificial general intelligence. A lot of efforts have been made to model neural-based reasoning as an iterative decision-making ... More

Categorizing resonances X(1835), X(2120) and X(2370) in the pseudoscalar meson familyApr 15 2011May 20 2011Inspired by the newly observed three resonances X(1835), X(2120) and X(2370), in this work we systematically study the two-body strong decays and double pion decays of $\eta(1295)/\eta(1475)$, $\eta(1760)/X(1835)$ and $X(2120)/X(2370)$ by categorizing ... More

The bottomed strange molecules with isospin 0Jan 13 2018Using the local hidden gauge approach, we study the possibility of the existence of bottomed strange molecular states with isospin 0. We find three bound states with spin-parity $0^+$, $1^+$ and $2^+$ generated by the $\bar{K}^*B^*$ and $\omega B_s^*$ ... More

Predicting exotic molecular states composed of nucleon and P-wave charmed mesonJun 29 2014Aug 07 2014In this work, we study the interaction between a nucleon and a $P$-wave charmed meson in the $T$ doublet by exchanging a pion. Our calculations indicate that a nucleon and a $P$-wave charmed meson with $J^P=0^+$ or $J^P=1^+$ in the $T$ doublet can form ... More

$\bar{K}Λ$ molecular explanation to the newly observed $Ξ(1620)^0$Jun 13 2019The newly observed $\Xi(1620)^0$ by the Belle Collaboration inspires our interest in performing a systematic study on the interaction of an anti-strange meson $(\bar{K}^{(*)})$ with a strange or doubly strange ground octet baryon $\mathcal{B}$ ($\Lambda$, ... More

Strong LHCb evidence supporting the existence of the hidden-charm molecular pentaquarksMar 26 2019Mar 28 2019On March 26th, 2019, at the Rencontres de Moriond QCD conference, the LHCb Collaboration reported the observation of three new pentaquarks, namely $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$, which are consistent with the loosely bound molecular hidden-charm ... More

Coupled-channel analysis of the possible $D^{(*)}D^{(*)}$, $\bar{B}^{(*)}\bar{B}^{(*)}$ and $D^{(*)}\bar{B}^{(*)}$ molecular statesNov 21 2012Dec 05 2013We perform a coupled-channel study of the possible deuteron-like molecules with two heavy flavor quarks, including the systems of $D^{(*)}D^{(*)}$ with double charm, $\bar{B}^{(*)}\bar{B}^{(*)}$ with double bottom and $D^{(*)}\bar{B}^{(*)}$ with both ... More

Computing the determinant of a matrix with polynomial entries by approximationAug 25 2014Apr 12 2015Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the determinant of a matrix ... More

Few-Body Systems Composed of Heavy QuarksAug 31 2012Within the past ten years many new hadrons states were observed experimentally, some of which do not fit into the conventional quark model. I will talk about the few-body systems composed of heavy quarks, including the charmonium-like states and some ... More

$Z_b(10610)^\pm$ and $Z_b(10650)^\pm$ as the $B^*\bar{B}$ and $B^*\bar{B}^{*}$ molecular statesJun 15 2011In the framework of the one-boson-exchange model, we have studied the interaction of the $B^*\bar{B}$ and $B^*\bar{B}^{*}$ system. After considering the S-wave and D-wave mixing, we notice that both $Z_b(10610)^\pm$ and $Z_b(10650)^\pm$ can be interpreted ... More

XYZ StatesNov 15 2013In the past decade, many new charmonium (or charmonium-like) and bottomonium (or bottomonium-like) states were observed experimentally. I will review these XYZ states which do not fit into the quark model spectrum easily.

Theory for electric dipole superconductivity with an application for bilayer excitonsMay 28 2014Exciton superfluid is a macroscopic quantum phenomenon in which large quantities of excitons undergo the Bose-Einstein condensation. Recently, exciton superfluid has been widely studied in various bilayer systems. However, experimental measurements only ... More

Sealing off a carbon nanotube with a self-assembled aqueous valve for the storage of hydrogen in GPa pressureJun 29 2012The end section of a carbon nanotube, cut by acid treatment, contains hydrophillic oxygen groups. Water molecules can self-assemble around these groups to seal off a carbon nanotube and form an "aqueous valve". Molecular dynamics simulations on single-wall ... More

Non-commuting Observables are Jointly Measureable under Disturbance Correction StrategySep 30 2014Oct 29 2014In the study of Heisenberg's error-disturbance relation, it is commonly believed that the non-unitary change of states hinders us from deducing the information encoded in original states about subsequently measured observable. However, we find that the ... More

Constructing new pseudoscalar meson nonets with the observed $X(2100)$, $X(2500)$, and $η(2225)$May 01 2017Aug 17 2017Stimulated by the BESIII observation of $X(2100)$, $X(2500)$, and $\eta(2225)$, we try to pin down new pseudoscalar meson nonets including these states. The analysis of mass spectra and the study of strong decays indicate that $X(2120)$ and $\eta(2225)$ ... More

Ginzburg-Landau-type theory of non-polarized spin superconductivityNov 17 2017Since the concept of spin superconductor was proposed, all the related studies concentrate on spin-polarized case. Here, we generalize the study to spin-non-polarized case. The free energy of non-polarized spin superconductor is obtained, and the Ginzburg-Landau-type ... More

Non-strange partner of strangeonium-like state Y(2175)Feb 19 2012Apr 21 2012Inspired by the observed Y(2175) state, we predict its non-strange partner Y(1915), which has a resonance structure with mass around 1915 MeV and width about $317\sim 354$ MeV. Experimental search for Y(1915) is proposed by analyzing the $\omega f_0(980)$ ... More

Characterizing entanglement by momentum-jump in the frustrated Heisenberg ring at quantum phase transitionFeb 19 2005We study the pairwise concurrences, a measure of entanglement, of the ground states for the frustrated Heisenberg ring to explore the relation between entanglement and quantum phase transition associated with the momentum jump. The groundstate concurrences ... More

Probing the $XYZ$ states through radiative decaysMar 31 2014Aug 01 2014In this work, we have adopted the spin rearrangement scheme in the heavy quark limit and extensively investigated three classes of the radiative decays: $\mathfrak{M}\to (b\bar{b})+\gamma$, $(b\bar{b})\to \mathfrak{M}+\gamma$, $ \mathfrak{M} \to \mathfrak{M}^\prime+\gamma$, ... More

Perfect valley filter based on topological phase in disordered $\rm{Sb}$ Monolayer HeterostructureNov 22 2017The hydrogenated $\rm{Sb}$ monolayer epitaxially grown on a $\rm{LaFeO_3}$ substrate is a novel type of two-dimensional material hosting quantum spin-quantum anomalous Hall (QS-QAH) states. For a device formed by $\rm{Sb}$ monolayer ribbon, the QAH edge ... More

On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$May 14 2015Oct 01 2016In this paper we first investigate for what positive integers $a,b,c$ every nonnegative integer $n$ can be represented as $x(ax+1)+y(by+1)+z(cz+1)$ with $x,y,z$ integers. We show that $(a,b,c)$ can be either of the following seven triples: $$(1,2,3),\ ... More

Refining Lagrange's four-square theoremApr 22 2016Jan 16 2017Lagrange's four-square theorem asserts that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as the sum of four squares. This can be further refined in various ways. We show that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb ... More

A connection between covers of the integers and unit fractionsNov 15 2004Jul 26 2006For integers a and n>0, let a(n) denote the residue class {x\in Z: x=a (mod n)}. Let A be a collection {a_s(n_s)}_{s=1}^k of finitely many residue classes such that A covers all the integers at least m times but {a_s(n_s)}_{s=1}^{k-1} does not. We show ... More

Some new inequalities for primesSep 17 2012Sep 19 2012For n=1,2,3,... let p_n be the n-th prime. We mainly show that p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k<n^2p_n/3 for all n>30.

Ramanujan's theta functions and sums of triangular numbersJan 24 2016Dec 06 2017Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a_1,a_2,\ldots,a_k,n\in\Bbb N$ let $N(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by $a_1x_1^2+a_2x_2^2+\cdots+a_kx_k^2$, and let ... More

Ramanujan's theta functions and linear combinations of three triangular numbersNov 26 2018Dec 11 2018Let $\Bbb Z$ be the set of integers. For positive integers $a,b,c$ and $n$ let $N(a,b,c;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2$, and let $t(a,b,c;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2$ $(x,y,z\in\Bbb ... More

Congruences involving generalized central trinomial coefficientsAug 23 2010Jun 17 2014For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1)\ (n=0,1,2,\ldots)$ are the usual central trinomial coefficients, and $T_n(3,2)$ ... More

Zero-sum problems for abelian p-groups and covers of the integers by residue classesMay 26 2003May 13 2009Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. ... More

Some new problems in additive combinatoricsSep 06 2013Apr 18 2014In this paper we investigate some new problems in additive combinatorics. Our problems mainly involve permutations (or circular permutations) $a_1,\ldots,a_n$ of $n$ distinct numbers or elements of an additive abelian group with adjacent sums $a_i+a_{i+1}$ ... More

Groups and Combinatorial Number TheoryNov 12 2004In this talk we introduce several topics in combinatorial number theory which are related to groups; the topics include combinatorial aspects of covers of groups by cosets, and also restricted sumsets and zero-sum problems on abelian groups. A survey ... More

Ramsey numbers for treesMar 14 2011Oct 27 2014For $n\ge 5$ let $T_n'$ denote the unique tree on $n$ vertices with $\Delta(T_n')=n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\{v_0,v_1,\ldots,$ $v_{n-1}\}$ and $E=\{v_0v_1,\ldots,v_0v_{n-3},$ $v_{n-3}v_{n-2},v_{n-2}v_{n-1}\}$. In ... More

On the range of a covering functionSep 16 2004Sep 21 2004Let {a_s(mod n_s)}_{s=1}^k (k>1) be a finite system of residue classes with the moduli n_1,...,n_k distinct. By means of algebraic integers we show that the range of the covering function w(x)=|{1\le s\le k: x=a_s (mod n_s)}| is not contained in any residue ... More

List of conjectural series for powers of $π$ and other constantsFeb 28 2011Dec 29 2014The author gives the full list of his conjectures on series for powers of $\pi$ and other important constants scattered in some of his public papers or his private diaries. The list contains 234 reasonable conjectural series. On the list there are 178 ... More

Two $q$-analogues of Euler's formula $ζ(2)=π^2/6$Feb 05 2018Nov 06 2018It is well known that $\zeta(2)=\pi^2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula: $$\sum_{k=0}^\infty\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^2}=\prod_{n=1}^\infty\frac{(1-q^{2n})^4}{(1-q^{2n-1})^4}$$ ... More

Mixed sums of squares and triangular numbersMay 09 2005Mar 08 2007By means of $q$-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus $x^2+y^2$ for some integers $x$ and $y$ with $x\not\equiv y (mod 2)$ or ... More

On sums of binomial coefficients and their applicationsApr 21 2004Jul 14 2008In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For example, we show ... More

On some determinants with Legendre symbol entriesAug 13 2013May 02 2019In this paper we mainly focus on some determinants with Legendre symbol entries. Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. We show that $(\frac{-S(d,p)}p)=1$ for any $d\in\mathbb Z$ with $(\frac dp)=1$, and that $$\left(\frac{W_p}p\right)=\begin{cases}(-1)^{|\{0<k<\frac ... More

On Snevily's conjecture and restricted sumsetsOct 29 2006Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every positive integer ... More

On the Herzog-Schönheim conjecture for uniform covers of groupsJun 05 2003Dec 30 2004Let G be any group and $a_1G_1,...,a_kG_k (k>1)$ be left cosets in G. In 1974 Herzog and Sch\"onheim conjectured that if $\Cal A=\{a_iG_i\}_{i=1}^k$ is a partition of G then the (finite) indices $n_1=[G:G_1],...,n_k=[G:G_k]$ cannot be distinct. In this ... More

Turán's problem and generalized Ramsey numbersJan 01 2011Nov 05 2014Let $n,r,k,s$ be positive integers with $n,k\ge 2$. The generalized Ramsey number $R(n,r;k,s)$ is the smallest positive integer $p$ such that for every graph $G$ of order $p$, either $G$ contains a subgraph induced by $n$ vertices with at most $r-1$ edges, ... More

Mixed sums of primes and other termsJan 20 2009Jan 29 2009In this paper we study mixed sums of primes and linear recurrences. We show that if m=2(mod 4) and m+1 is a prime then $(m^{2^n-1}-1)/(m-1)\not=m^n+p^a$ for any n=3,4,... and prime power p^a. We also prove that if a>1 is an integer, u_0=0, u_1=1 and u_{i+1}=au_i+u_{i-1} ... More

Super congruences involving Bernoulli and Euler polynomialsJul 02 2014Jul 28 2014Let $p>3$ be a prime, and let $a$ be a rational p-adic integer. Let $\{B_n(x)\}$ and $\{E_n(x)\}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that $$\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\equiv (-1)^{\langle ... More

Congruences for Domb and Almkvist-Zudilin numbersFeb 07 2013Feb 17 2015In this paper we prove some transformation formulae for congruences modulo a prime and deduce some congruences for Domb numbers and Almkvist-Zudilin numbers. We also pose some conjectures on congruences modulo prime powers.

Two new kinds of numbers and related divisibility resultsAug 21 2014Nov 10 2018We mainly introduce two new kinds of numbers given by $$R_n=\sum_{k=0}^n\binom nk\binom{n+k}k\frac1{2k-1}\quad\ (n=0,1,2,...)$$ and $$S_n=\sum_{k=0}^n\binom nk^2\binom{2k}k(2k+1)\quad\ (n=0,1,2,...).$$ We find that such numbers have many interesting arithmetic ... More

Congruences for Franel numbersDec 05 2011Oct 29 2013The Franel numbers given by $f_n=\sum_{k=0}^n\binom{n}{k}^3$ ($n=0,1,2,\ldots$) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental congruences for the Franel numbers. We ... More

Cubic congruences and sums involving $\binom{3k}k$Oct 24 2013Nov 20 2013Let $p$ be a prime greater than $3$ and let $a$ be a rational p-adic integer. In this paper we try to determine $\sum_{k=1}^{[p/3]}\binom{3k}ka^k\pmod p$, and real the connection between cubic congruences and the sum $\sum_{k=1}^{[p/3]}\binom{3k}ka^k$, ... More

Conjectures involving arithmetical sequencesAug 13 2012Oct 31 2013We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form $(\root n\of{a_n})_{n\ge 1}$ or the form $(\root{n+1}\of{a_{n+1}}/\root n\of{a_n})_{n\ge1}$, where $(a_n)_{n\ge 1}$ is a number-theoretic ... More

Fibonacci numbers modulo cubes of primesNov 16 2009Oct 31 2013Let $p$ be an odd prime. It is well known that $F_{p-(\frac p5)}\equiv 0\pmod{p}$, where $\{F_n\}_{n\ge0}$ is the Fibonacci sequence and $(-)$ is the Jacobi symbol. In this paper we show that if $p\not=5$ then we may determine $F_{p-(\frac p5)}$ mod $p^3$ ... More

p-adic valuations of some sums of multinomial coefficientsOct 20 2009Apr 13 2011Let $m$ and $n>0$ be integers. Suppose that $p$ is a prime dividing $m-4$ but not dividing $m$. We show that $\nu_p(\sum_{k=0}^{n-1}\frac{\binom{2k}k}{m^k})$ and $\nu_p(\sum_{k=0}^{n-1}\binom{n-1}{k}(-1)^k\frac{\binom{2k}k}{m^k})$ are at least $\nu_p(n)$, ... More

Congruences concerning Legendre polynomialsDec 17 2010Let $p$ 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 pose 13 conjectures ... More

Congruences concerning Legendre polynomials IIIDec 20 2010Oct 25 2012Let $p>3$ be a prime, and let $R_p$ be the set of rational numbers whose denominator is coprime to $p$. Let $\{P_n(x)\}$ be the Legendre polynomials. In this paper we mainly show that for $m,n,t\in R_p$ with $m\not\e 0\pmod p$, $$\align &P_{[\frac p6]}(t) ... More

Anisotropic microwave conductivity of cuprate superconductors in the presence of CuO chain induced impuritiesAug 24 2009The anisotropy in the microwave conductivity of the ortho-II YBa$_2$Cu$_3$O$_{6.50}$ is studied within the kinetic energy driven superconducting mechanism. The ortho-II YBa$_2$Cu$_3$O$_{6.50}$ is characterized by a periodic alternative of filled and empty ... More

Restricted sums of four squaresJan 20 2017Mar 11 2019We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two (including $1$). For example, we show that each $n=1,2,3,\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb N=\{0,1,2,\ldots\})$ with ... More

On m-covers and m-systemsMar 16 2004Dec 26 2004Let A={a_s(mod n_s)}_{s=0}^k be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results concerning system A. In particular, we show that if every integer lies in more than m=[sum_{s=1}^k ... More

Ramanujan's theta functions and linear combinations of four triangular numbersNov 27 2018Apr 08 2019Let $\Bbb Z$ and $\Bbb Z^+$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb Z^+$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2+dw(w+1)/2$ $(x,y,z,w\in\Bbb Z)$. ... More

Quadratic residues and related permutations and identitiesSep 20 2018Nov 08 2018Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations. For $k=1,\ldots,(p-1)/2$ let $\tau_p(k)$ be the unique integer $k^*\in\{1,\ldots,(p-1)/2$ such that $kk^*$ is congruent to $1$ or $-1$ modulo ... More

A result similar to Lagrange's theoremMar 11 2015Dec 17 2015Generalized octagonal numbers are those $p_8(x)=x(3x-2)$ with $x\in\mathbb Z$. In this paper we mainly show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is similar to Lagrange's ... More

Ramsey numbers for trees IIOct 28 2014Jan 07 2016Let $r(G_1, G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V(S(n_1,n_2))=\{v_0,v_1,...,v_{n_1},w_0,w_1,...,w_{n_2}\}$ and $E(S(n_1,n_2))=\{v_0v_1,...,v_0v_{n_1},v_0w_0, ... More

A new theorem on the prime-counting functionSep 19 2014Jan 10 2017For $x>0$ let $\pi(x)$ denote the number of primes not exceeding $x$. For integers $a$ and $m>0$, we determine when there is an integer $n>1$ with $\pi(n)=(n+a)/m$. In particular, we show that for any integers $m>2$ and $a\le\lceil e^{m-1}/(m-1)\rceil$ ... More

New observations on primitive roots modulo primesApr 21 2014May 29 2014On the basis of our numerical computations, we make many new observations on primitive roots modulo primes. For example, we conjecture that for any odd prime $p$ there is a primitive root $g<p$ modulo $p$ which is the sum of the first $n$ primes for some ... More

Congruences concerning Lucas' law of repetitionDec 12 2013Let $P,Q\in\Bbb Z$, $U_0=0,\ U_1=1$ and $U_{n+1}=PU_n-QU_{n+1}$. In this paper we obtain a general congruence for $U_{kmn^r}/U_k\pmod {n^{r+1}}$, where $k,m,n,r$ are positive integers. As applications we extend Lucas' law of repetition and characterize ... More

On some determinants involving the tangent functionJan 15 2019Mar 28 2019Let $p$ be an odd prime and let $a,b\in\mathbb Z$ with $p\nmid ab$. In this paper we mainly evaluate $$T_p^{(\delta)}(a,b):=\det\left[\tan\pi\frac{aj^2+bk^2}p\right]_{\delta\le j,k\le (p-1)/2}\ \ (\delta=0,1).$$ For example, in the case $p\equiv3\pmod4$ ... More

Binomial Coefficients and Quadratic FieldsJan 19 2004Mar 06 2005Let E be a real quadratic field with discriminant d and let p be an odd prime not dividing d. For \rho=1 or -1, we determine $\prod_{0<c<d, (d/c)=\rho} binomial coeff.{p-1}{\lfloor pc/d\rfloor}$ modulo p^2 in terms of Lucas numbers, the fundamental unit ... More

A local-global theorem on periodic mapsApr 06 2004Oct 29 2006Let $\psi_1,...,\psi_k$ be maps from Z to an additive abelian group with positive periods $n_1,...,n_k$ respectively. We show that the function $\psi=\psi_1+...+\psi_k$ is constant if $\psi(x)$ equals a constant for |S| consecutive integers x where S={r/n_s: ... More

An additive theorem and restricted sumsetsOct 31 2006Dec 04 2008Let G be any additive abelian group with cyclic torsion subgroup, and let A, B and C be finite subsets of G with cardinality n>0. We show that there is a numbering {a_i}_{i=1}^n of the elements of A, a numbering {b_i}_{i=1}^n of the elements of B and ... More

Quadratic residues and related permutations and identitiesSep 20 2018Jul 09 2019Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations, congruences and identities. If $a_1<\ldots<a_{(p-1)/2}$ are all the quadratic residues modulo $p$ among $1,\ldots,p-1$, then the list $\{1^2\}_p,\ldots,\{((p-1)/2)^2\}_p$ ... More

Cohomological support loci of varieties of Albanese fiber dimension oneSep 12 2011Jan 04 2013Let $X$ be a smooth projective variety of Albanese fiber dimension 1 and of general type. We prove that the translates through 0 of all components of $V^0(\omega_X)$ generate $\Pic^0(X)$. We then study the pluricanonical maps of $X$. We show that $|4K_X|$ ... More

Jacobsthal sums, Legendre polynomials and binary quadratic formsFeb 06 2012Feb 12 2012Let $p>3$ be a prime and $m,n\in\Bbb Z$ with $p\nmid mn$. Built on the work of Morton, in the paper we prove the uniform congruence: $$&\sum_{x=0}^{p-1}\Big(\frac{x^3+mx+n}p\Big) \equiv {-(-3m)^{\frac{p-1}4} \sum_{k=0}^{p-1}\binom{-\frac 1{12}}k\binom{-\frac ... More

Congruences involving $\binom{2k}k^2\binom{3k}km^{-k}$Apr 14 2011Apr 28 2011Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei ... More

p-adic congruences motivated by seriesNov 21 2011Dec 03 2013Let $p>5$ be a prime. Motivated by the known formulae $\sum_{k=1}^\infty(-1)^k/(k^3\binom{2k}{k})=-2\zeta(3)/5$ and $\sum_{k=0}^\infty \binom{2k}{k}^2/((2k+1)16^k)=4G/\pi$$ (where $G=\sum_{k=0}^\infty(-1)^k/(2k+1)^2$ is the Catalan constant), we show ... More

Determining $x$ or $y$ mod $p^2$ with $p=x^2+dy^2$Oct 18 2012Jun 08 2015Let $p$ be an odd prime and let $d\in\{2,3,7\}$. When $(\frac{-d}p)=1$ we can write $p=x^2+dy^2$ with $x,y\in\mathbb Z$; in this paper we aim at determining $x$ or $y$ modulo $p^2$. For example, when $p=x^2+3y^2$, we show that if $p\equiv x\equiv 1\pmod ... More

Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+dy^2$Mar 22 2011Feb 20 2014Given 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 $\sum_{k=0}^{p-1}a_k\equiv x^2-2p$ (mod $p^2$) ... More

On sums related to central binomial and trinomial coefficientsJan 03 2011Oct 22 2014A generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ with $b,c\in\mathbb Z$. In this paper we investigate congruences and series for sums of terms related to central binomial coefficients ... More

A refinement of a congruence result by van Hamme and MortensonNov 08 2010Feb 18 2014Let $p$ be an odd prime. In 2008 E. Mortenson proved van Hamme's following conjecture: $$\sum_{k=0}^{(p-1)/2}(4k+1)\binom{-1/2}k^3\equiv (-1)^{(p-1)/2}p\pmod{p^3}.$$ In this paper we show further that \begin{align*}\sum_{k=0}^{p-1}(4k+1)\binom{-1/2}k^3\equiv ... More

Congruences involving binomial coefficients and Lucas sequencesDec 07 2009Dec 14 2009In this paper we obtain some congruences involving central binomial coefficients and Lucas sequences. For example, we show that if p>5 is a prime then $\sum_{k=0}^{p-1}F_k*binom(2k,k)/12^k$ is congruent to 0,1,-1 modulo p according as p=1,4 (mod 5), p=13,17 ... More

Fleck quotients and Bernoulli numbersAug 14 2006Let 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 investigated $F_p(n,r)$ ... More

A kind of orthogonal polynomials and related identitiesJun 27 2016Nov 15 2017In this paper we introduce the polynomials $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$ given by $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k} \ (n\ge 0)$, $D_0^{(r)}(x)=1,\ D_1^{(r)}(x)=x$ and $D_{n+1}^{(r)}(x)=xD_n^{(r)}(x)-n(n+2r)D_{n-1}^{(r)}(x)\ ... More

Supercongruences involving Lucas sequencesOct 11 2016Dec 21 2016For $A,B\in\mathbb Z$, the Lucas sequence $u_n(A,B)\ (n=0,1,2,\ldots)$ are defined by $u_0(A,B)=0$, $u_1(A,B)=1$, and $u_{n+1}(A,B) = Au_n(A,B)-Bu_{n-1}(A,B)$ $(n=1,2,3,\ldots).$ For any odd prime $p$ and positive integer $n$, we establish the new result ... More

New congruences involving harmonic numbersJul 31 2014Aug 11 2014Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}kH_k,\ \ \sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}kH_k^{(2)},\ \ \sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}k\frac{H_k^{(2)}}{2k+1}$$ modulo $p^2$, where ... More

Some relations between t(a,b,c,d;n) and N(a,b,c,d;n)Nov 19 2015Dec 07 2015Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations ... More

Universal sums of three quadratic polynomialsFeb 09 2015Jul 11 2018Let $a,b,c,d,e,f$ be integers with $a\ge c\ge e>0$, $b>-a$ and $b\equiv a\pmod2$, $d>-c$ and $d\equiv c\pmod 2$, $f>-e$ and $f\equiv e\pmod2$. Suppose that $b\ge d$ if $a=c$, and $d\ge f$ if $c=e$. When $b(a-b)$, $d(c-d)$ and $f(e-f)$ are not all zero, ... More

Natural numbers represented by $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$Apr 06 2015Dec 23 2015Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb Z$. In contrast, we conjecture that any natural number is represented by $\lfloor x^2/a\rfloor+\lfloor ... More

On universal sums of polygonal numbersMay 05 2009May 14 2015For $m=3,4,\ldots$, the polygonal numbers of order $m$ are given by $p_m(n)=(m-2)\binom n2+n\ (n=0,1,2,\ldots)$. For positive integers $a,b,c$ and $i,j,k\ge3$ with $\max\{i,j,k\}\ge5$, we call the triple $(ap_i,bp_j,cp_k)$ universal if for any $n=0,1,2,\ldots$ ... More

Constructing $x^2$ for primes $p=ax^2+by^2$Dec 17 2010Let $a$ and $b$ be positive integers and let $p$ be an odd prime such that $p=ax^2+by^2$ for some integers $x$ and $y$. Let $\lambda(a,b;n)$ be given by $q\prod_{k=1}^\infty (1-q^{ak})^3(1-q^{bk})^3 = \sum_{n=1}^\infty \lambda(a,b;n)q^n$. In the paper, ... More

On the further properties of $\{U_n\}$Mar 27 2012Apr 19 2012Let $\{U_n\}$ be given by $U_0=1$ and $U_n=-2\sum_{k=1}^{[n/2]} \b n{2k}U_{n-2k}\ (n\ge 1)$, where $[\cdot]$ is the greatest integer function. In the paper we present a summation formula and several congruences involving $\{U_n\}$.

On a pair of zeta functionsApr 30 2012Oct 20 2016Let $m$ be a positive integer, and define $$\zeta_m(s)=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\omega(n)}}{n^s}\ \ \ \ \text{and} \ \ \ \ \zeta^*_m(s)=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\Omega(n)}}{n^s},$$ for $\Re(s)>1$, where $\omega(n)$ denotes ... More

Simple arguments on consecutive power residuesNov 29 2003Feb 22 2007By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii) Let O_K be the ... More

Congruences involving $\binom{4k}{2k}$ and $\binom{3k}k$Aug 24 2011Let $p$ be a prime greater than 3. In the paper we mainly determine $\sum_{k=0}^{[p/4]}\binom{4k}{2k}(-1)^k$, $\sum_{k=0}^{[p/3]}\binom{3k}k, \sum_{k=0}^{[p/3]}\binom{3k}k(-1)^k$ and $\sum_{k=0}^{[p/3]}\binom{3k}k(-3)^k$ modulo $p$, where $[x]$ is the ... More

On quadratic residues and quartic residues modulo primesOct 22 2018Nov 01 2018In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\prod_{1\le i,j\le(p-1)/2\atop p\nmid i^2-Aij-j^2}(i^2-Aij-j^2)$$ ... More

Note on super congruences modulo $p^2$Mar 11 2015Let $p$ be an odd prime, and let $m$ be an integer with $p\nmid m$. In this paper show that $$\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak\binom{-1-a}k}{m^k} \equiv 0\pmod p \quad\hbox{implies}\quad\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak \binom{-1-a}k}{m^k}\equiv ... More

Congruences involving $\binom{2k}k^2\binom{4k}{2k}m^{-k}$Apr 15 2011Apr 18 2011Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{4k}{2k}m^{-k}\mod ... More

Congruences involving $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$Jul 03 2014Jul 19 2016Define $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$ for $n=0,1,2,...$. Those numbers $g_n=g_n(1)$ are closely related to Ap\'ery numbers and Franel numbers. In this paper we establish some fundamental congruences involving $g_n(x)$. For example, for ... More