Jiangshuai Yang, Yingpu Deng
In [16], we obtained some congruences for Lucas quotients of two infinite families of Lucas sequences by studying the combinatorial sum $$\sum_{k\equiv r(\mbox{mod}m)}{n\choose k}a^k.$$ In this paper, we show that the sum can be expressed in terms of some recurrent sequences with orders not exceeding $φ{(m)}$ and give some new congruences.
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In this paper, we study the combinatorial sum $$\sum_{k\equiv r(\mbox{mod }m)}{n\choose k}a^k.$$ By studying this sum, we obtain new congruences for Lucas quotients of two infinite families of Lucas sequences. Only for three Lucas sequences, there are such known results.
We obtain explicit factored closed-form expressions for Fibonacci and Lucas sums of the form \mbox{$\sum_{k = 1}^n {F_{2rk}^3 }$} and \mbox{$\sum_{k = 1}^n {L_{2rk}^3 }$}, where $r$~and~$n$ are integers.
We obtain closed-form expressions for all sums of the form \mbox{$\sum_{k = 1}^n {F_{mk}{}^4 }$} and \mbox{$\sum_{k = 1}^n {L_{mk}{}^4 }$} and their alternating versions, where $F_i$ and $L_i$ denote Fibonacci and Lucas numbers respectively. Our results complement those of Melham who studied the alternating sums.
We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In particular, we give precise values for the greatest common divisor (GCD) of all sums of $k$ consecutive terms of $\left(G_n\right)_{n \geq 0}$. When $G_0 = 0$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Fibonacci numbers, and when $G_0 = 2$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Lucas numbers. Denoting the GCD of all sum…
In 1878 É. Lucas proved a remarkable result which provides a simple way to compute the binomial coefficient ${n\choose m}$ modulo a prime $p$ in terms of the binomial coefficients of the base-$p$ digits of $n$ and $m$: {\it If $p$ is a prime, $n=n_0+n_1p+\cdots +n_sp^s$ and $m=m_0+m_1p+\cdots +m_sp^s$ are the $p$-adic expansions of nonnegative integers $n$ and $m$, then \begin{equation*} {n\choose m}\equiv \prod_{i=0}^{s}{n_i\choose m_i}\pmod{p}. \end{equation*}} The above congruence, the so-called {\it Lucas' theorem} (or {\it Theorem of Lucas}), plays an important role in Number Theory and C…
Balancing numbers $n$ are originally defined as the solution of the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+\cdots+(n+r)$, where $r$ is called the balancer corresponding to the balancing number $n$. By slightly modifying, $n$ is the cobalancing number with the cobalancer $r$ if $1+2+\cdots+n=(n+1)+\cdots+(n+r)$. Let $B_n$ denote the $n^{th}$ balancing number and $b_n$ denote the $n^{th}$ cobalancing number. Then $8B_n^2+1$ and $8b_n^2+8b_n+1$ are perfect squares. The $n^{th}$ Lucas-balancing number $C_n$ and the $n^{th}$ Lucas-cobalancing number $c_n$ are the positive roots of $8B_n^2+1$ …
Let $u_n$ be a nondegenerate Lucas sequence. We generalize the results of Bugeaud, Mignotte, and Siksek, 2006 to give a systematic approach towards the problem of determining all perfect powers in any particular Lucas sequence. We then prove a general bound on admissible prime powers in a Lucas sequence assuming the Frey-Mazur conjecture on isomorphic mod $p$ Galois representations of elliptic curves.
Let $w_n=w_n(P,Q)$ be numerical sequences which satisfy the recursion relation \begin{equation*} w_{n+2}=Pw_{n+1}-Qw_n. \end{equation*} We consider two special cases $(w_0,w_1)=(0,1)$ and $(w_0,w_1)=(2,P)$ and we denote them by $U_n$ and $V_n$ respectively. Vieta-Lucas polynomial $V_n(X,1)$ is the polynomial of degree $n$. We show that the congruence equation $V_n(X,1)\equiv C \mod p$ has a solution if and only if $U_{(p-ε)/d}(C+2,C+2)$ is divisible by $p$, where $ε\in\{\pm 1\}$ depends on $C$ and $p$, and $d=\gcd(n,p-ε)$.
Lucas congruences using modular forms
本文利用模形式证明众多 Apéry 型序列对任意素数满足 Lucas 同余，从而证实 S. Cooper 的四个相关猜想并重新解释已有结果。
Partial Lucas-type congruences
本文证明了一个一般性结果，表明所有可表示为多元Laurent多项式幂的常数项的序列均满足部分Lucas同余，推广了Chamberland和Dilcher关于特定二项和序列模素数同余的研究。
We develop a general framework for finding all perfect powers in sequences derived by shifting non-degenerate quadratic Lucas-Lehmer binary recurrence sequences by a fixed integer. By combining this setup with bounds for linear forms in logarithms and results based upon the modularity of elliptic curves defined over totally real fields, we are able to answer a question of Bugeaud, Luca, Mignotte and the third author by explicitly finding all perfect powers of the shape $F_k \pm 2 $ where $F_k$ is the $k$-th term in the Fibonacci sequence.
In this note we shall improve some congruences of D.F. Bailey [Two p^3 variations of Lucas' Theorem, JNT 35(1990), pp. 208-215] to higher prime power moduli, by studying the relation between irregular pairs of the form (p,p-3) and refined version of Wolstenholme's theorem.
We study Gibonacci sequences mod $m$, giving special attention to the Lucas numbers. It is known which $m$ have the property that the Fibonacci sequence contains all residues mod $m$. When $m$ has this property, we say that the Fibonacci sequence is complete mod $m$. We extend this work to all Gibonacci sequences, concluding by determining the set of $m$ in which a generic Gibonacci sequence containing the relatively prime consecutive terms $a, b$ is complete mod $m$.
For each of the $15$ known sporadic Apéry-like sequences, we prove congruences modulo $p^2$ that are natural extensions of the Lucas congruences modulo $p$. This extends a result of Gessel for the numbers used by Apéry in his proof of the irrationality of $ζ(3)$. Moreover, we show that each of these sequences satisfies two-term supercongruences modulo $p^{2r}$. Using special constant term representations recently discovered by Gorodetsky, we prove these supercongruences in the two cases that remained previously open.
We extend to Segal-Piatetski-Shapiro sequences previous results on the Luca-Schinzel question over integral valued polynomial sequences. Namely, we prove that for any real $c$ larger than $1$ the sequence $(\sum_{m\le n} φ(\lfloor m^c \rfloor) /\lfloor m^c \rfloor)_n$ is dense modulo $1$, where $φ$ denotes Euler's totient function. The main part of the proof consists in showing that when $R$ is a large integer, the sequence of the residues of $\lfloor m^c \rfloor$ modulo $R$ contains blocks of consecutive values which are in an arithmetic progression.
Let $p$ be a prime, and let $k,n,m,n_0$ and $m_0$ be nonnegative integers such that $k\ge 1$, and $_0$ and $m_0$ are both less than $p$. K. Davis and W. Webb established that for a prime $p\ge 5$ the following variation of Lucas' Theorem modulo prime powers holds $$ {np^k +n_0 \choose mp^k+m_0}\equiv{np^{\lfloor(k-1)/3\rfloor} \choose mp^{\lfloor(k-1)/3\rfloor}} {n_0 \choose m_0} \pmod{p^k}. $$ In the proof the authors used their earlier result that present a generalized version of Lucas' Theorem. In this paper we present a a simple inductive proof of the above congruence. Our proof is based o…
Lucas congruences for the Apéry numbers modulo $p^2$
本文给出 Apéry 数整函数插值的泰勒系数公式，并证明对特定基 $p$ 数字的整数 $n$，$A(n)$ 满足模 $p^2$ 的 Lucas 同余式。
In this note we prove that {equation*} {np^s\choose mp^s+r}\equiv (-1)^{r-1}r^{-1}(m+1){n\choose m+1}p^s \pmod{p^{s+1}} {equation*} where $p$ is any prime, $n$, $m$, $s$ and $r$ are nonnegative integers such that $n\ge m$, $s\ge 1$, $1\le r\le p^s-1$ and $r$ is not divisible by $p$. We derive a proof by induction using a multiple application of Lucas' theorem and two basic binomial coefficient identities. As an application, we prove that a similar congruence for a prime $p\ge 5$ established in 1992 by D. F. Bailey holds for each prime $p$.
Residues of Terms of Lucas Sequences Modulo $3^k$
本文将Fibonacci序列模$3^k$的剩余分布与稳定性结果推广至由参数$P$定义的两类Lucas序列，并证明这些序列均不满足模$3$稳定。
We consider the family of Lucas sequences uniquely determined by $U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$ with initial values $U_0(k)=0$ and $U_1(k)=1$ and $k\ge 1$ an arbitrary integer. For any integer $n\ge 1$ the discriminator function $\mathcal{D}_k(n)$ of $U_n(k)$ is defined as the smallest integer $m$ such that $U_0(k),U_1(k),\ldots,U_{n-1}(k)$ are pairwise incongruent modulo $m$. Numerical work of Shallit on $\mathcal{D}_k(n)$ suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every $k\ge 1$ there is a co…
We observe that a sequence satisfies Lucas congruences modulo $p$ if and only if its values modulo $p$ can be described by a linear $p$-scheme, as introduced by Rowland and Zeilberger, with a single state. This simple observation suggests natural generalizations of the notion of Lucas congruences. To illustrate this point, we prove explicit generalized Lucas congruences for integer sequences that can be represented as the constant terms of $P(x,y)^n Q(x,y)$ where $P$ and $Q$ are certain Laurent polynomials.
An extension of Lucas Theorem
本文给出计算模素数二项式系数的同余判据的初等证明，并据此推广Lucas定理，应用于推导二项式系数和的恒等式与同余式。
In 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 (mod 30), and p=7,23 (mod 30) respectively, where {F_n} is the Fibonacci sequence. We also raise several conjectures.
We develop a new method for proving algebraic independence of $G$-functions. Our approach rests on the following observation: $G$-functions do not always come with a single linear differential equation, but also sometimes with an infinite family of linear difference equations associated with the Frobenius that are obtained by reduction modulo prime ideals. When these linear difference equations have order one, the coefficients of the $G$-function satisfy congruences reminiscent of a classical theorem of Lucas on binomial coefficients. We use this to derive a Kolchin-like algebraic independence…
On the discriminator of Lucas sequences. II
本文研究Shallit序列（一类Lucas序列）的判别子，利用Matveev定理和Koksma-Erdős-Turán不等式给出判别子简单刻画成立的阈值上界，并对自然密度为68/75的整数k完全确定了所有判别子。
Harmonic numbers $H_k=\sum_{0<j\le k}1/j (k=0,1,2,...)$ arise naturally in many fields of mathematics. In this paper we initiate the study of congruences involving both harmonic numbers and Lucas sequences. One of our three theorems is as follows: Let u_0=0, u_1=1, and u_{n+1}=u_n-4u_{n-1} for n=1,2,3,.... Then, for any prime p>5 we have $$\sum_{k=0}^{p-1}u_{k+δ}H_k/2^k=0 (mod p),$$ where $δ=0$ if p=1,2,4,8 (mod 15), and $δ=1$ otherwise.
Let p be any prime and a be a positive integer. For nonnegative integers l,n and an integer r, the normalized cyclotomic $ψ$-coefficient $${n,r}_{l,p^a}:=p^{-[(n-p^{a-1}-lp^a)/(p^{a-1}(p-1))]} \sum_{k=r(mod p^a)}(-1)^k{n \choose k}{{(k-r)/p^a} \choose l}$$ is known to be an integer. In this paper, we show that this coefficient behaves like binomial coefficients and satisfies some Lucas-type congruences. This implies that a congruence of Wan is often optimal, and two conjectures of Sun and Davis are true.
Dwork-type $q$-congruences through the $q$-Lucas theorem
本文运用 $q$-Lucas 定理和已知 $q$-超同余，证明了若干 Dwork 型 $q$-同余，证实了三个先前提出的猜想，并验证了 Swisher (H.3) 猜想的较弱形式。
In this paper, we consider $D=\mathbb{Z}[θ]$, where $$θ= \sqrt{-k} \,\,\,\, \mbox{if}\;\;\;-k\not\equiv 1 \;(\mbox{mod}\;4)\,\,\,\,\mbox{or}\,\,\,\, θ=\frac{\sqrt{-k}+1}{2} \,\,\,\, \mbox{if}\;\;\;-k\equiv 1 \;(\mbox{mod}\;4),$$ $k\geq 2$ is a squarefree integer, and we proved that the number $R(y)$ of representations of a monic polynomial $f(x)\in \mathbb{Z}[θ][x]$, of degree $d\geq 1$, as a sum of two monic irreducible polynomials $g(x)$ and $h(x)$ in $\mathbb{Z}[θ][x]$, with the coefficients of $g(x)$ and $h(x)$ bounded in complex modulus by $y$, is asymptotic to $(4y)^{2d-2}$.
Period patterns, entry points, and orders in the Lucas sequences: theory and applications
本文将模 m 的 Fibonacci 和 Lucas 序列统计理论扩展至 Lucas 序列 U 和 V，基于阶、周期与进入点等概念推广了统计恒等式，并应用于新颖的图解方法以揭示序列项的周期模式。
In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that $$ \sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k^2\frac{P_k}{8^k}\equiv0\pmod{p^2} $$ for any prime $p\equiv 7\mod{8}$, where $P_k$ is the $k$-th Pell number. Further, we also propose three new congruences of the same type.
In 1876, Edouard Lucas showed that if an integer $b$ exists such that $b^{n-1} \equiv 1 (\mathrm{mod} \ n)$ and $b^{(n-1)/p} \not\equiv 1( \mathrm{mod} \ n)$ for all prime divisors $p$ of $n-1$ , then $n$ is prime, a result known as Lucas's converse of Fermat's little theorem. This result was considerably improved by Henry Pocklington in 1914 when he showed that it's not necessary to know all the prime factors of $n-1$ in order to determine if $n$ is prime. In this paper we optimize Pocklington's primality test for integers of the form $ap^{k}+1$ where $p$ is prime, $a<p$, $k\ge 1$. An extensi…
Let $f(z)$ be in $1+z\mathbb{Q}[[z]]$ and $\mathcal{S}$ be an infinite set of prime numbers such that, for all $p\in\mathcal{S}$, we can reduce $f(z)$ modulo $p$. We let $f(z)_{\mid p}$ denote the reduction of $f(z)$ modulo $p$. Generally, when $f(z)$ is D-finite, $f(z)_{\mid p}$ is algebraic over $\mathbb{F}_p(z)$. It turns out that if $f(z)$ is a solution of a polynomial of the form $X-A_p(z)X^{p^l}$, we can use this type of equations to obtain results of transcendence and algebraic independence over $\mathbb{Q}(z)$. In the present paper, we look for conditions on the differential operators …
For any polynomial $P(x)\in\mathbb{Z}[x],$ we study arithmetic dynamical systems generated by $\displaystyle{F_P(n)=\prod_{k\le n}}P(n)(\text{mod}\ p),$ $n\ge 1.$ We apply this to improve the lower bound on the number of distinct quadratic fields of the form $\mathbb{Q}(\sqrt{F_P(n)})$ in short intervals $M\le n\le M+H$ previously due to Cilleruelo, Luca, Quirós and Shparlinski. As a second application, we estimate the average number of missing values of $F_P(n)(\text{mod}\ p)$ for special families of polynomials, generalizing previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and oth…
Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \not\equiv0\pmod p$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ mod $p^2$ for $d=0,1$; for example, $$\sum_{k=0}^{p^a-1}\frac{\binom{2k}k}{m^k}\equiv\left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}\pmod{p^2},$$ where $(-)$ is the Jacobi symbol, and $\{u_n\}_{n\geqslant0}$ is the Lucas sequence given by $u_0=0$, $u_1=1$ and $u_{n+1}=(m-2)u_n-u_{n-1}$ for $n=1,2,3,\ldots$. As an application, we determine $\sum_{0<k<p^a,\, k\equiv r\pmod{p-1}}C_k$ modulo $p^2$ for…
A novel public key cryptography based on generalized Lucas matrices
本文提出基于广义卢卡斯矩阵的新型公钥密码方案，通过减少密钥传输复杂度和扩大密钥空间实现高效安全的加密解密
The 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 mainly establish for any prime $p>3$ the following congruences: \begin{align*}\sum_{k=0}^{p-1}(-1)^kf_k&\equiv\left(\frac p3\right)\ \ (\mbox{mod}\ p^2), \\ \sum_{k=0}^{p-1}(-1)^k\,kf_k&\equiv-\frac 23\left(\frac p3\right)\ \ (\mbox{mod}\ p^2), \\ \sum_{k=1}^{p-1}\frac{(-1)^k}kf_k &\equiv0\ \ (\mbox{mod}\ p^2), \\ \sum_{k=1}^{p-1}\frac{(-1)^k}{k^2…
$p$-Linear schemes for sequences modulo $p^r$
本文研究模素数幂 $p^r$ 下组合序列的复杂行为，构造了描述该行为的 $p$-线性方案，并证明了对于固定 $r$，状态数上界不依赖于素数 $p$。
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