Primitive Prime Divisors In Orbits Related To Arithmetic Dynamics

The study of primes appearing in sequences is a recurring theme in number theory,ranging from Dirichlet s theorem on primes in arithmetic progressions to the conjectural existence of infinitely many primes in some special sequences.Thus it is natural to look for primes appearing in orbits related to arithmetic dynamics.In this theme,there are three different topics:the density of prime,primitive prime divisors,and Iwasawa sequences.The main topic of the paper is the primitive prime divisors in orbits related to arithmetic dynamics.We will also consider two other topics:integral points of elliptic curves and Lehmer’s problem.This paper is divided into four chapters.In Chapter 1,we give some main results and preliminaries.In Chapter 2,let(?) denote the absolute logarithmic height.Lehmer’s conjecture asserts that there is an absolute positive constant k such that if φ(z)∈Z[z]is a monic polynomial of degree d>1 whose roots are not roots of unity,then(?)This problem has a long history.Many mathematicians have considered Lehmer’sproblem for restricted values of a.In this chapter,we prove a similar result for a family of polynomials related to weighted homogeneous polynomials.In Chapter 3,based on a well-known theorem of Siegel,which states that anelliptic curve contains only finitely many integral points,there has been much interest in the problem of determining the integral points on elliptic curves,and many advanced methods have been developed to solve such problems.Moreover,V.Mahe explained how the primality conjecture for magnified elliptic curve divisibility sequences is linked to two classical problems in diophantine geometry:solving Thue equations and finding integer points on elliptic curves.In this chapter,for appropriate prime numbers p,q,we find all integral points for elliptic curves Epq:y2=x3(pq-12)x-2(pq-8).In Chapter 4,let φ(z)∈Q(z)be a rational function of degree d>2.Let φn denote the nth iterate of φ,for a given point α∈Q,the orbit of a is the set Oφ(α)={φn(α):n≥0}.Then an important problem in number theory is to prove the existence of primitive divisors of an arithmetically defined sequence.Let A=(An)n≥1 be an integer sequence.A prime p dividing a term An is called a primitive prime divisor of An if p does not divide any term Am,1≤m≤n.Let A=(An)n≥1 be an integer sequence.The set z(A)= {n:An does not have a primitive prime divisor}is called the Zsigmondy set of the integer sequence A.In this chapter,for the zero orbit of weighted homogeneous polynomial ft(x)and a special subsequence of the zero orbit of rational function,we prove that the Zsimondy set is finite.