Related papers: Lucas' theorem modulo $p^2$
The classical $abc$ theorem for polynomials (often called Mason's theorem) deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound for the number of distinct zeros of the polynomial $abc$ in terms of…
Let $\{U_n\}_{n\geq 0}$ be a Lucas sequence. Then the equation $$|U_n|=m_1!m_2!\cdots m_k!$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in \{1,2, 3, 4, 6, 8, 12\}$. Further the equation $$|U_n|=D_{m_1}D_{m_2}\cdots D_{m_k}, \qquad…
In this paper, we introduce relations between binomial sums involving (generalized) Fibonacci and Lucas numbers, and different kinds of binomial coefficients. We also present some relations between sums with two and three binomial…
In this article we will be dedicated some algorithms of addition, subtraction, multiplication and division of two positive integers using Zeckendorf form. Such results find application in coding theory.
Repdigits are natural numbers formed by the repetition of a single digit. In this paper, we study the problem of writing repdigits as the difference of two balancing or Lucas-balancing numbers. The method of proof involves the application…
The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the…
The definition of the binomial coefficient in terms of gamma functions also allows non-integer arguments. For nonnegative integer arguments the gamma functions reduce to factorials, leading to the well-known Pascal triangle. Using a…
In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that $$…
In this paper, by presenting bi-periodic Lucas numbers as a binomial sum, we introduce the bi-periodic incomplete Lucas numbers. After that, by using the bi-periodic incomplete Lucas numbers, we derive the recurrence relation and the…
For a given prime $p$, we determine the limit, as $\lambda \to \infty$, of the density of residues modulo $p^\lambda$ attained by the Fibonacci sequence. In particular, we show that this limiting density is related to zeros in the sequence…
We derive a collection of identities for bivariate Fibonacci and Lucas polynomials using essentially a matrix approach as well as properties of such polynomials when the variables $x$ and $y$ are replaced by polynomials. A wealth of…
This note gives an elementary exposition of a variant of the spread polynomials in terms of Fibonacci and Lucas polynomials.
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…
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…
The generalized Lucas numbers are polynomials in two variables with nonnegative integer coefficients. Lucas versions of some combinatorial numbers with known formulas in terms of quotient and products of nonnegative integers have been…
In this note, we study the divisibility relation $U_m\mid U_{n+k}^s-U_n^s$, where ${\bf U}:=\{U_n\}_{n\ge 0}$ is the Lucas sequence of characteristic polynomial $x^2-ax\pm 1$ and $k,m,n,s$ are positive integers.
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…
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…
Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A…
Let p be a prime and let a be a positive integer. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p.…