Graham Everest, Gerard Mclaren, Tom Ward
Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited.
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Let $P$ be a non-torsion point on the elliptic curve $E_{a}: y^{2}=x^{3}+ax$. We show that if $a$ is fourth-power-free and either $n>2$ is even or $n>1$ is odd with $x(P)<0$ or $x(P)$ a perfect square, then the $n$-th element of the elliptic divisibility sequence generated by $P$ always has a primitive divisor.
We define the $\textit{Divisor Divisibility Sequence}$ associated to a Laurent polynomial $f\in\mathbb{Z}[X_1^{\pm1},\ldots,X_N^{\pm1}]$ to be the sequence $W_n(f)=\prod f(ζ_1,\ldots,ζ_N)$, where $ζ_1,\ldots,ζ_N$ range over all $n$'th roots of unity with $f(ζ_1,\ldots,ζ_N)\ne0$. More generally, we define $W_Λ(f)$ analogously for any finite subgroup $Λ\subset(\mathbb C^*)^N$. We investigate divisibility, factorization, and growth properties of $W_Λ(f)$ as a function of $Λ$. In particular, we give the complete factorization of $W_Λ(f)$ when $f$ has generic coefficients, and we prove an analytic …
Some effectivity results for primitive divisors of elliptic divisibility sequences
本文证明了椭圆曲线上非挠点的分母序列在项数足够大时每项均存在本原除子，并给出了仅依赖于曲线模型的可有效计算的常数。
Primitive divisors of elliptic divisibility sequences for elliptic curves with j=1728
本文研究 j=1728 有理椭圆曲线上的椭圆整除序列，证明 $B_{5m}$ 恒有素因子，并推广至 $B_{mp}$（$p\equiv1\pmod4$）。
We apply Vojta's conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta's conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.
Forbidden subgraphs in divisor graphs and an Erdős divisibility problem
本文研究了除数图中禁止子图的问题，证明了最大无元素相互整除的子集大小为c₂n+o(n)，并给出了此类子集数量的渐进行为。
We study the existence of primes and of primitive divisors in classical divisibility sequences defined over function fields. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.
It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with corresponding elliptic divisibility sequences having no perfect power terms. The proofs combine primitive divisor results with modular methods for Diophantine equations.
We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u^3+v^3=m, with m cube-free, all the terms beyond the first have a primitive divisor.
We study primitive divisors of terms of the sequence P_n=n^2+b, for a fixed integer b which is not a negative square. It seems likely that the number of terms with a primitive divisor has a natural density. This seems to be a difficult problem. We survey some results about divisors of this sequence as well as provide upper and lower growth estimates for the number of terms which have a primitive divisor.
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.
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.
We study the primitive divisors of the terms of $(Δ_n)_{n \geq 1}$, where $Δ_n=N_{K/ \mathbb{Q}}(u^n-1)$ for $K$ a real quadratic field, and $u>1$ a unit element of its ring of integers. The methods used allow us to find the terms of the sequence that do not have a primitive prime divisor.
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…
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。
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.
Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either divisible by $n$ or by a specific divisor of $n$. Applying this result, we construct an infinite family of certain tuples of imaginary quadratic fields of the form $$\left(\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1}), \mathbb{Q}(\sqrt{4d+1}), \mathbb{Q}(\sqrt{2d+4}), \mathbb{Q}(\sqrt{2d+16}), \cdots, \mathbb{Q}(\sqrt{2d+4^t}) \right)$$ with $d\in \mathbb{Z}$ and $1\leq 4^t\leq 2|d|…
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.
On big primitive divisors of Fibonacci numbers
本文证明 Fibonacci 数存在任意线性规模的大本原因子：对任意正整数 $κ$，充分大的第 $n$ 个 Fibonacci 数均含不小于 $(κ+1)n-1$ 的本原因子。
Primitive divisors of sequences associated to elliptic curves
本文研究椭圆曲线上点序列坐标分母的素因子存在性，证明素数阶挠点情形下大 $n$ 时必有素因子，并揭示其与 Lang-Trotter 猜想的关联。
Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R in E(k(T)), write x_R=A_R/D_R^2 with relatively prime polynomials A_R(T) and D_R(T) in k[T]. The sequence {D_{nR}) for n \ge 1 is called the ``elliptic divisibility sequence of R.'' Let P,Q in E(k(T)) be independent points. We conjecture that deg (gcd(D_{nP},D_{mQ})) is bounded for m,n \ge 1, and that gcd(D_{nP},D_{nQ}) = gcd(D_{P},D_{Q}) for infinitely many n \ge 1. We prove these conjectures in the case that j(E) is in k. More generally, we prove analo…
We provide an elementary proof to a conjecture by Robinson that multiples of (powers of) primes in the Somos-4 sequence are equally spaced. We also show, almost as a corollary, for the generalised Somos-4 sequence defined by $τ_{n+2}τ_{n-2}=ατ_{n+1}τ_{n-1}+βτ_n^2$ and initial values $τ_1=τ_2=τ_3=τ_4=1$, that the polynomial $τ_n(α,β)$ is a divisor of $τ_{n+k(2n-5)}(α,β)$ for all $n,k\in\mathbb Z$ and establish a similar result for the generalized Somos-5 sequence.
Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called $Φ^*_n(q)$, which is closely related to the cyclotomic polynomial $Φ_n(x)$ and to primitive prime divisors of $q^n-1$. Our definition of $Φ^*_n(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$, we give an algorithm for determining all pairs $(n,q)$ with $Φ^*_n(q)\le cn^k$. This algorithm is used to extend (and correct) a result of Hering which is useful for classifying certain families of subgr…
The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number
本文在奇完全数及更弱的奇本原非亏数假设下，证明了素因子倒数之和 T(n) 与乘积 H(n) 关于最小素因子的新上界。
We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of terms with a primitive divisor has a natural density. We discuss two heuristic arguments to suggest a value for that density, one using recent advances made about the distribution of roots of polynomial congruences.
In this note we compute a constant $N$ that bounds the number of non--primitive divisors in elliptic divisibility sequences over function fields of any characteristic. We improve a result of Ingram--Mah{é}--Silverman--Stange--Streng, 2012, and we show that the constant can be chosen independently of the specific point and to some extent of the specific curve, as predicted in loc. cit.
Let K be a number field, let f(x) in K(x) be a rational function of degree d> 1, and let z in K be a wandering point such that f^n(z) is nonzero for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that p | f^n(z) and p does not divide f^m(z) for all positive integers m < n. We prove the same result unconditionally for function fields of characteristic 0 when f is not isotrivial.
Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $ϕ(p-1) \geq ϕ(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that the reverse inequality holds for a small positive proportion of the twin primes. That is, $p$ tends to have more primitive roots than does $p+2$. We prove that Dickson's conjecture, which is much weaker than Bateman-Horn, implies that the quotients $\frac{ϕ(p+1)}{ϕ(p-1)}$, as $p,p+2$ range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of $\frac{…
We consider the family of polynomials $f_{d,c}(x)=x^d+c$ over the rational field $\Q$. Fixing integers $d, n\ge 2$, we show that the density of primes that can appear as primitive prime divisors of $f_{d,c}^n(0)$ for some $c\in\Q$ is positive. In fact, under certain assumptions, we explicitly calculate the latter density when $d=2$. Furthermore, fixing $d,n\ge 2$, we show that for a given integer $N>0$, there is $c\in \Q$ such that $\f^n(0)$ has at least $N$ primitive prime divisors each of which is appearing up to any predetermined power. This shows that there is no uniform upper bound on the…
Power Products in Elliptic Divisibility Sequences and Prime-Incidence Obstructions
研究椭圆可除序列中有限乘积为素数ρ次幂的条件，证明在D_1被2或3整除的假设下，索引的大素数因子必须满足ρ-平衡块约束。
Let D = (D_n)_{n\ge1} be an elliptic divisibility sequence. We study the set S(D) of indices n satisfying n | D_n. In particular, given an index n in S(D), we explain how to construct elements nd in S(D), where d is either a prime divisor of D_n, or d is the product of the primes in an aliquot cycle for D. We also give bounds for the exceptional indices that are not constructed in this way.
A divisibility sequence is a sequence of integers $\{d_n\}$ such that $d_m$ divides $d_n$ if $m$ divides $n$. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman conjectured that a similar behaviour should appear in many algebraic groups. We extend results by Ghioca-Hsia-Tucker and Silverman for elliptic curves and prove an analogue of Silverman's conjecture over function fields for abelian and split semiabelian varieties and some generalizations of …
Notes on the Divisibility of Divisor Sums of Catalan Numbers
本文研究卡塔兰数的约数和函数σ(C_n)的可除性，建立6k-1型素因子条件与可除性判据，并用de Bruijn定理给出渐近估计，揭示其算术结构。
Let $(b_n) = (b_1, b_2, ...)$ be a sequence of integers. A primitive prime divisor of a term $b_k$ is a prime which divides $b_k$ but does not divide any of the previous terms of the sequence. A zero orbit of a polynomial $f(z)$ is a sequence of integers $(c_n)$ where the $n$-th term is the $n$-th iterate of $f$ at 0. We consider primitive prime divisors of zero orbits of polynomials. In this note, we show that for integers $c$ and $d$, where $d > 1$ and $c \neq \pm 1$, every iterate in the zero orbit of $f(z) = z^d + c$ contains a primitive prime whenever zero has an infinite orbit. If $c = \…
Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = α^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this constant $α$. We also show that the method developed can be applied to many similar problems that can be stated in terms of the divisor graph, including other questions about primitive sets, geometric-progression-free sets, and the divisor graph path-cover problem.
Divisibility Estimates for Gaussian Norms and Almost-Prime Orders of a CM Elliptic Curve
该研究针对椭圆曲线E:y²=x³-x，结合高斯CM筛框架与交叉项，证明了特定素数p下#E(F_p)/8的无平方因子性及素因子个数限制，为相关数论问题提供了可除性估计与分布结果。
Primitive divisors of sequences associated to elliptic curves with complex multiplication
本文研究复乘椭圆曲线上点序列的素因子，证明在特定条件下，几乎所有理想均有素因子，推广了椭圆整除序列的结果。
Primitive Divisors of Lucas Sequences in Polynomial Rings
本文研究多项式环中 Lucas 序列的本原除子，建立与经典正则 Lucas 序列相对应的结果，并纠正既有定理。
A primitive prime divisor of an element a_n of a sequence (a_1,a_2,a_3,...) is a prime P that divides a_n, but does not divide a_m for all m < n. The Zsigmondy set Z of the sequence is the set of n such that a_n has no primitive prime divisors. Let f : X --> X be a self-morphism of a variety, let D be an effective divisor on X, and let P be a point of X, all defined over the algebraic closure of Q. We consider the Zsigmondy set Z(X,f,P,D) of the sequence defined by the arithmetic intersection of the f-orbit of P with D. Under various assumptions on X, f, D, and P, we use Vojta's conjecture wit…
Let $\mathcal C_Q$ be the cyclic group of order $Q$, $n$ a divisor of $Q$ and $r$ a divisor of $Q/n$. We introduce the set of $(r,n)$-free elements of $\mathcal C_Q$ and derive a lower bound for the the number of elements $θ\in \mathbb F_q$ for which $f(θ)$ is $(r,n)$-free and $F(θ)$ is $(R,N)$-free, where $ f, F \in \mathbb F_q[x]$. As an application, we consider the existence of $\mathbb F_q$-primitive points on curves like $y^n=f(x)$ and find, in particular, all the odd prime powers $q$ for which the elliptic curves $y^2=x^3 \pm x$ contain an $\mathbb F_q$-primitive point.
Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F.
We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a prime and $r$ a positive integer, we prove $K(\mathcal{E}[p^r])=K(x_1,ζ_p,y_2)$, where $\{(x_1, y_1),(x_2,y_2)\}$ is a generating system of ${\mathcal{E}}[p^r]$ and $ζ_p$ is a primitive $p$-th root of the unity. If $\mathcal{E}$ has a $K$-rational point of order $p$, then $K(\mathcal{E}[p^r])=K(ζ_{p^r},\sqrt[m_1]{a})$, with $a\in K(ζ_{p^r})$ and $m_1|p^r$. In…
We prove the finiteness of the Zsigmondy set associated to the critical orbit of f(z) = z^d+c for rational values of c by finding an effective bound on the size of the set. For non-recurrent critical orbits, the Zsigmondy set is explicitly computed by utilizing effective dynamical height bounds. In the general case, we use Thue-style Diophantine approximation methods to bound the size of the Zsigmondy set when d >2, and complex-analytic methods when d=2.
In this work we discuss Dirichlet products evaluated at Fibonacci numbers. As first applications of the results we get a representation of Fibonacci numbers in terms of Euler's totient function, an upper bound on the number of primitive prime divisors and representations of some related Euler products. Moreover, we sum functions over all primitive divisors of a Fibonacci number and obtain a non--trivial fixed point of this operation.
We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are given, using a strong form of Siegel's Theorem and some congruence arguments. Our main result is applied to the study of prime divisors of Somos sequences.
Assuming Vojta's conjecture, we give a sufficient condition for the limit \[ \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x))} \] is equal to zero, where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$, $h_{H}$ is an ample height function on $X$, and $h_{Y}$ is a global height function associated with a closed subscheme $Y \subset X$ of codimension at least two. Based on this, we propose a conjecture on a sufficient condition for the limit to be zero. We point out that our conjecture implies Dynamical Mordell-Lang conjecture for endomor…
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。
We study the existence of a Zsigmondy bound for a sequence of divisors associated to points on an elliptic curve over a function field. More precisely, let $k$ be an algebraically closed field, let $\mathcal{C}$ be a nonsingular projective curve over $k$, and let $K$ denote the function field of $\mathcal{C}$. Suppose $E$ is an ordinary elliptic curve over $K$ and suppose there does not exist an elliptic curve $E_0$ defined over $k$ that is isomorphic to $E$ over $K$. Suppose $P\in E(K)$ is a non-torsion point and $Q\in E(K)$ is a torsion point of order $r$. The sequence of points $\{nP+Q\}\su…
Tamagawa number divisibility of central $L$-values of twists of the Fermat elliptic curve
研究Fermat椭圆曲线扭曲的中心L值代数部分的3-adic赋值，揭示Tate-Shafarevich群3-部分与惰性素因子个数的关系，并证明在分裂素数乘积情形下BSD猜想预测的阶为完全平方。
An Infinite Family of Primitive Heron Triangles with Two Sides as Perfect Squares
该研究通过椭圆曲线理论证明存在无限多个原始海伦三角形，其中两边为完全平方数，并暗示可能存在三边均为完全平方数的原始海伦三角形。
Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements and $r$ be a positive divisor of $q^n-1$. An element $α\in \mathbb{F}_{q^n}^*$ is called $r$-primitive if its multiplicative order is $(q^n-1)/r$. Also, $α\in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_α(x) = αx^{n-1}+ α^q x^{n-2} + \ldots + α^{q^{n-2}}x + α^{q^{n-1}}$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$. These concepts generalize the ideas of primitive and normal elements, respectively. In this paper, we consider non-negative integers $m_1,m_2,k_1,k_2$, po…
Motivated by a question of Venkataramana, we consider the greatest common divisor of $ϕ(f(n))$ where $f$ is a primitive polynomial with integer coefficients, and $n$ ranges over all natural numbers. Assuming Schinzel's hypothesis, we establish that this gcd may be bounded just in terms of the degree of the polynomial $f$. Unconditionally we establish such a bound for quadratic polynomials, as well as polynomials that split completely into linear factors. The paper also addresses a question of Calegari, and establishes that there are infinitely many $n$ such that $n^2+1$ is not divisible by any…
We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 \le r \le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0) where the \forall-quantifiers run over a total of 8 variables, and where F is a polynomial. This implies that the Σ_5-theory of Q is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of Z in Q with quantifier complexity \forall \exists, involving only one universally quantified variable. This improves the complexity of defining Z in Q in two ways, and implies that the Σ_3-theory, and even the Π_2-theory…
We define a primitive index of an integer in a sequence to be the index of the term with the integer as a primitive divisor. For the sequences $k^u+h^u$ and $k^u-h^u$, we discern a formula to find the primitive indexes of any composite number given the primitive indexes of its prime factors. We show how this formula reduces to a formula relating the multiplicative order of $k$ modulo $N$ to that of its prime factors. We then introduce immediate consequences of the formula: certain sequences which yield the same primitive indexes for numbers with the same unique prime factors, an expansion of t…
We find analogues of the primitive divisor results of Zsigmondy, Bang, Bilu-Hanrot-Voutier, and Carmichael in polynomial rings, following the methods of Carmichael.
The moments of split greatest common divisors
研究分裂最大公因子序列的矩，刻画了gcd(n,u_n)^λ的渐近行为，解决了代数群G_a×G_m上的矩问题，给出无条件和条件结果。
Let K be a number field or a function field of characteristic 0, let f be a K-rational function of degree greater than 1, and let a be an element of K. Let S be a finite set of places of K containing all the archimedean ones and the primes where f has bad reduction. After excluding all the natural counter-examples, we define a subset A(f,a) of pairs of integers (m,n) with m nonnegative and n positive, and show that for all but finitely many (m,n) in A(f,a) there is a prime p of K which is not in S such that the p-adic valuation of f^{m+n}(a)-f^m(a) is precisely equal to 1, and moreover a has p…
Large degree primitive points on curves
本文证明：若曲线 $X$ 上存在次数 $d>2g$ 的因子，则 $X$ 有无穷多个次数为 $d$ 的本原点，补充了低次数点非本原的已知结果。
For an integer $a$ consider the sequence $(T_{n}(a)-1)_{n=1}^{\infty}$ defined by the Chebyshev polynomials $T_{n}$. We list all pairs $(n,a)$ for which the term $T_{n}(a)-1$ has no primitive prime divisor.
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