Min Sha
In this paper, we obtain analogues of Zsigmondy's theorem and the primitive divisor results for the Lucas and Lehmer sequences in polynomial rings of several variables.
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In this paper we prove that if $n > 30,030$ then the $n$-th element of any Lucas or Lehmer sequence has a primitive divisor.
A composite positive integer n is Lehmer if ϕ(n) divides n-1, where ϕ(n) is the Euler's totient function. No Lehmer number is known, nor has it been proved that they don't exist. In 2007, the second author [7] proved that there is no Lehmer number in the Fibonacci sequence. In this paper, we adapt the method from [7] to show that there is no Lehmer number in the companion Lucas sequence of the Fibonacci sequence $(L_n)_{n\geq 0}$ given by $L_0 = 2, L_1 = 1$ and $L_{n+2} = L_{n+1} + L_n$ for all $n\geq 0.$
On the Lucas and Lehmer sequences in Dedekind domains
该研究将 Lucas 与 Lehmer 序列的强整除性推广到 Dedekind 整环，并在函数域中证明相应的 Zsigmondy 型定理与本原素因子结果。
Stewart reduced the problem of determining all Lucas and Lehmer sequences whose $n$-th element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences for $n \leq 30$. Further computations lead us to conjecture that, for $n > 30$, the $n$-th element of such sequences always has a primitive divisor.
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.
Let u(n) be the n-th term of a Lucas sequence or a Lehmer sequence.In this article we shall establish an estimate from below for the greatest prime factor of u(n) which is of the form nexp(logn/104loglogn). In so doing we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdos from 1965.In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.
Let $\al$ and $\be$ be conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all $\al$ and $\be$ with $\hgt(\be/\al) \leq 4$, the $n$-th element of these sequences has a primitive divisor for $n > 30$. In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.
We extend the necessity part of Lucas Lehmer iteration for testing Mersenne prime to all base and uniformly for both generalized Mersenne and Wagstaff numbers(the later correspond to negative base). The role of the quadratic iteration $x \rightarrow x^2-2$ is extended by Chebyshev polynomial $T_n(x)$ with an implied iteration algorithm because of the compositional identity $T_n(T_m(x))=T_{nm}(x)$. This results from a Chebyshev polynomial primality test based essentially on the Lucas pair $(ω_a,\overlineω_a)$, $ω_a=a+\sqrt{a^2-1}$, where $a \neq 0 \pm 1$. It seems interesting that the arithmeti…
We describe a computationally efficient approach to resolving equations of the form $C_1x^2 + C_2 = y^n$ in coprime integers, for fixed values of $C_1$, $C_2$ subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.
In this paper we discuss some relations between zeros of Lucas-Lehmer polynomials and Gray code. We study nested square roots of 2 applying a "binary code" that associates bits $0$ and $1$ to $\oplus$ and $\ominus$ signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas-Lehmer polynomials, which assume the form of nested square roots of 2.
Jacobsthal and Jacobsthal-Lucas numbers with Lehmar property
本文证明了 Jacobsthal 和 Jacobsthal-Lucas 数列中不存在具有 Lehmer 性质的数。
On the Eight Levels theorem and applications towards Lucas-Lehmer primality test for Mersenne primes, I
本文证明了八级定理，并基于此提出了三种 Lucas-Lehmer Mersenne 素性检验新版本及一个 Mersenne 合数新判据，以提升算法效率并促进新 Mersenne 素数的发现。
The Lucas-Lehmer (LL) primality test for Mersenne numbers is the fastest known primality test. In 1969, Hans Riesel published a modification of LL to test numbers of the form $N = h \cdot 2^n - 1$, where $h < 2^n$ is an odd integer and $n \ge 2$ \cite{Riesel}. This test is now known as the Lucas-Lehmer-Riesel (LLR) primality test. In Algorithm \ref{PrimalityAlgorithm}, we present a modification of LLR which works for any odd integer $N$. A probabilistic version of our algorithm runs in expected time $\tilde{O}(\log^3 N)$, and a deterministic version in expected $\tilde{O}(\log^4 N)$. We conclu…
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…
On Pillai's Problem involving Lucas sequences of the second kind
本文研究第二类Lucas序列与完全幂之差的Pillai型丢番图方程，证明在至少存在三个解时，解的大小和特征多项式系数均有上界。
This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form $φ(|au_n |)=|bv_m|,$ where $φ$ is the Euler totient function, $\{u_n\}_{n\geq 0}$ and $\{v_m\}_{m\geq 0}$ ar…
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