Yao-Qiang Li
We study some divisibility properties related to the factors of the discriminant of the characteristic polynomial of generalized Fibonacci sequences $(G_n)_{n\ge0}$ defined by $G_0=0$, $G_1=1$ and $G_n=pG_{n-1}+qG_{n-2}$ for $n\ge2$, where $p,q$ are given integers. As corollaries, we give some divisibility properties on some well known sequences.
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It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. It is called the strong divisibility property. However, this property does not hold for every second order sequence. In this paper we study the generalized Fibonacci polynomials and classify them in two types depending on their Binet formula. We give a complete characterization for those polynomials that satisfy the strong divisibility property. We also giveformulas to calculate the $\gcd$ of those polynomials that do not satisfy the strong divisibility property. Note. This paper is now published…
Fibonacci sums and divisibility properties
本文利用 Sury 多项式恒等式的变体推导有限斐波那契与卢卡斯和的新表达式，并将其推广到相关多项式及 Horadam 序列，进而用于发现和证明整除性质。
We give a survey of some known and some new results about factors of different sorts of $q-$Fibonacci numbers.
Let $F_n$ be the $n$th Fibonacci number. Let $m, n$ be positive integers. Define a sequence $(G(k,n,m))_{k\geq 1}$ by $G(1,n,m) = F^m_n$, and $G(k+1,n,m) = F_{nG(k,n,m)}$ for all $k\geq 1$. We show that $F_n^{k+m-1}\mid G(k,n,m)$ for all $k, m, n\in\mathbb N$. Then we calculate $\frac{G(k,n,m)}{F_n^{k+m-1}}\pmod{F_n}$.
Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.
For a prime $p$, let $Z(p)$ be the smallest positive integer $n$ so that $p$ divides $F_{n}$, the $n$th term in the Fibonacci sequence. Paul Bruckman and Peter Anderson conjectured a formula for $ζ(m)$, the density of primes $p$ for which $m | Z(p)$ on the basis of numerical evidence. We prove Bruckman and Anderson's conjecture by studying the algebraic group $G : x^{2} - 5y^{2} = 1$ and relating $Z(p)$ to the order of $α= (3/2,1/2) \in G(\F_{p})$. We are then able to use Galois theory and the Chebotarev density theorem to compute $ζ(m)$.
A second order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these type of sequences are: Fibonacci polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials. The \emph{resultant} of two polynomials is the determinant of the Sylvester matrix and the \emph{discriminant} of a polynomial $p$ is the resultant of $p$ and its derivative. We study …
Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials that divide their compositions with every positive integer power. We completely characterize such polynomials in terms of their factorizations into cyclotomic polynomials using labeled Hasse diagrams, and construct new integer divisibility sequences based on them. We also show that, unlike the Fibonacci numbers, these non-classical sequences do not have the p…
On the divisibility of sums of Fibonacci numbers
本文证明存在无穷多个奇数 $n$，使前 $n$ 个非零斐波那契数之和能被 $n$ 整除，从而解决 Fatehizadeh 与 Yaqubi 的猜想。
Let $n$ be a squarefree positive odd integer. We will show that there exist infinitely many imaginary quadratic number fields with discriminant divisible by $n$ and-at the same time-having an element of order $n$ in the class group. We then apply our result to prove that for a given squarefree positive odd integer $n$ there exist infinitely many $N$ such that $n$ divides both $N$ and $r(N)$, where $r(N)$ is the representation number of $N$ as sums of three squares.
The Discriminant of the Characteristic Polynomial of the $k$th Fibonacci sequence is not a member of the $k$th Lucas sequence
本文研究了$k$次Lucas序列的数论性质，证明了$k$次Fibonacci序列特征多项式的判别式不属于$k$次Lucas序列。
Consider tuples of separable algebras over a common local or global number field, related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants. We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.
Murty proved that for all sufficiently large $X$ there exist at least ${c(\ell,\eps) X^{1/{4\ell}-\eps}}$ real quadratic fields with class number divisible by $\ell$ and discriminant not exceeding $X$ in absolute value. We extend this this to number fields of arbitrary degree.
Strong divisibility sequences and sieve methods
本文利用筛法及除数闭集工具，研究强整除序列中含大素因子整数与素数项的密度界，并以 Fibonacci 数和椭圆整除序列为例分析方法局限。
We study p-divisibility of discriminants of Hecke algebras associated to spaces of cusp forms of prime level. We make a precise conjecture about the indexes of Hecke algebras in their normalisation which implies (if true) the conjecture that there are no mod p congruences between non-conjugate newforms of weight 2 and level Gamma_0(p).
Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers
将Midy定理推广到非整数基的计数系统，研究分数在基β>1下的展开性质，并刻画了基为黄金比例时满足Midy性质的素数分母q。
The symmetric numerical semigroups S(F_a,F_b,F_c) and S(L_k,L_m,L_n) generated by three Fibonacci (F_a,F_b,F_c) and Lucas (L_k,L_m,L_n) numbers are considered. Based on divisibility properties of the Fibonacci and Lucas numbers we establish necessary and sufficient conditions for both semigroups to be symmetric and calculate their Hilbert generating series, Frobenius numbers and genera.
For any integer $l\geq 1$, let $p_1, p_2, \ldots, p_{l+2}$ be distinct prime numbers $\geq 5.$ For all real numbers $X>1,$ we let $N_{3,l}(X)$ denote the number of real quadratic fields $K$ whose absolute discriminant $d_K\leq X$ and $d_K$ is divisible by $(p_1\ldots p_{l+2})$ together with the class number $h_K$ of $K$ divisible by $2^{l}\cdot 3.$ Then, in this short note, by following the method in \cite{Byeonkoh}, we prove that $N_{3,l}(X) \gg X^\frac{7}{8}$ for all large enough $X$'s.
In this paper, we first prove that for any strong divisibility sequences $\boldsymbol{a} = \left(a_n\right)_{n\geq 1}$, we have the identity: $\mathrm{lcm} \left\lbrace \binom{n}{0}_{\bf{a}}, \binom{n}{1}_{\bf{a}},\dots, \binom{n}{n}_{\bf{a}} \right\rbrace = \frac{\mathrm{lcm} \left(a_1,\dots , a_n , a_{n+1}\right)}{a_{n+1}}$ $\left(\forall n \geq 1\right)$, generalizing the identity of Farhi (obtained in 2009 for $a_n=n$). Then, we derive from this one some other interesting identities. Finally, we apply those identities to estimate the least common multiple of the consecutive terms of some L…
Perfect powers in elliptic divisibility sequences
本文研究椭圆可除序列中完全幂的有限性问题，通过构造全实域上的Frey曲线并结合模性与降阶论证，在三个附加假设下给出肯定回答，且在Langlands纲领标准猜想下可放宽假设得到相同结论。
The Fibonacci numbers are not an additive uniqueness set for multiplicative functions
本文通过构造反例，否证了 Spiro 关于斐波那契数构成乘法函数加法唯一性集合的猜想，并给出 AI 辅助搜索与验证方法。
Fibonacci Partial Sums Tricks
本文研究类斐波那契数列部分和的可除性，揭示前十项和等于第7项乘以11的魔术原理，并推广到其他二阶递推，证明类Pell数列存在类似技巧而类Jacobsthal数列不存在。
A sequence of nonzero integers $f = (f_1, f_2, \dots)$ is ``binomid'' if every $f$-binomid coefficient $\left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f$ is an integer. Those terms are the generalized binomial coefficients: \[ \left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f \ = \ \frac{ f_nf_{n-1}\cdots f_{n-k+1} }{ f_kf_{k-1}\cdots f_1 }. \] Let $Δ(f)$ be the infinite triangle with those numbers as entries. When $I = (1, 2, 3, \dots)$ then $Δ(I)$ is Pascal's Triangle so that $I$ is binomid. Surprisingly, every row and column of Pascal's Triangle is also binomid. For any $f$, …
We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the sequence $D_1$, $D_2, \ldots$ of denominators of multiples $P$, $2P,\ldots$ of $P$ is a strong divisibility sequence in the sense that $\gcd(D_m, D_n) = D_{\gcd(m,n)}$. This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences. A number $N$ is called a Zsigmondy bound of the sequence if each term $D_{n}$ with $n>N$ presents a new p…
Fibonacci along even powers is (almost) realizable
本文研究了Fibonacci序列沿特定幂次采样的可实现性，证明了沿奇数幂采样的序列不可实现，而乘以5后的偶数幂采样序列可实现，揭示了与判别式相关的算术现象。
In this paper, we look at numbers of the form $H_{r,k}:=F_{k-1}F_{r-k+2}+F_{k}F_{r-k}$. These numbers are the entries of a triangular array called the \emph{determinant Hosoya triangle} which we denote by ${\mathcal H}$. We discuss the divisibility properties of the above numbers and their primality. We give a small sieve of primes to illustrate the density of prime numbers in ${\mathcal H}$. Since the Fibonacci and Lucas numbers appear as entries in ${\mathcal H}$, our research is an extension of the classical questions concerning whether there are infinitely many Fibonacci or Lucas primes. W…
On the properties of fibotomic polynomials
本文定义了fibotomic多项式并研究了其性质，包括判别式、结果式及在素数域中的因式分解，并推广至二元齐次情形。
We describe how to compute the intersection of two Lucas sequences of the forms $\{U_n(P,\pm 1) \}_{n=0}^{\infty}$ or $\{V_n(P,\pm 1) \}_{n=0}^{\infty}$ with $P\in\mathbb{Z}$ that includes sequences of Fibonacci, Pell, Lucas, and Lucas-Pell numbers. We prove that such an intersection is finite except for the case $U_n(1,-1)$ and $U_n(3,1)$ and the case of two $V$-sequences when the product of their discriminants is a perfect square. Moreover, the intersection in these cases also forms a Lucas sequence. Our approach relies on solving homogeneous quadratic Diophantine equations and Thue equation…
Sophie Germain Primes and the Totient of Fibonacci Numbers
研究了在模Pisano周期π(q)下，使得q整除φ(F_m)对所有m≡r mod π(q)的残差类集合S(q)，证明若q为赛普蒂默· Germain素数且z(2q+1)整除π(q)，则S(q)是非空的等差数列，且对于q>5，其基数为奇数且q≡8 mod 15。
On the Diophantine Equation $F_n = F_l^k (F_l^m-1)$
本文研究丢番图方程 $F_n = F_l^k (F_l^m-1)$，利用 Matveev 定理、改进的 Baker-Davenport 约化方法及斐波那契数整除性质，证明唯一正整数解为 $(6,3,3,1)$。
This thesis is devoted to studying estimates of the least common multiple of some integer sequences. Our study focuses on effective bounding of the $\mathrm{lcm}$ of some class of quadratic sequences, as well as arithmetic progressions and strong divisibility sequences. First, we have used methods of commutative algebra and complex analysis to establish new nontrivial lower bounds for the $\mathrm{lcm}$ of some quadratic sequences. Next, a more profound study of the arithmetic properties of strong divisibility sequences allowed us to obtain three interesting identities involving the $\mathrm{l…
On the power of the discriminant of a univariate polynomial as a certain determinant in positive characteristic
研究正特征下，由多项式f(x)的e次幂的系数构造的d×d矩阵的行列式被判别式Δ(f(x))的幂次整除的条件，并给出相关公式和实验等式。
Monogenic trinomials and class numbers of related quadratic fields
本文研究形如 $x^N+Ax+B$ 的单基因三项式，通过其判别式的无平方因子约数 $δ$ 考察相关二次域 ${\mathbb Q}(\sqrtδ)$ 的类数可除性。
On certain $D(9)$ and $D(64)$ Diophantine triples
研究关于$D(9)$和$D(64)$丢番图三元组，推广刻画特定$D(l^2)$-丢番图三元组对应佩尔方程解的结果，利用该结果及代数数对数线性形式的界，对特定形式的$D(9)$和$D(64)$-丢番图三元组进行分类。
An application of Birch-Tate formula to tame kernels of real quadratic number fields
本文利用非本原Dirichlet L值的归纳论证得到实二次数域中$L(χ_F,-1)$的2-可整除性结果，并借助Birch-Tate公式确定$K_2\mathcal{O}_F$的2-准素部分。
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