Maximal Height Of Divisors Of X^（pq）b-1

The height H(f(x)) of a polynomial f(x) is the largest coefficient of f(x) in absolute value. Let n-th cyclotmic polynomial A(n):=H(Φn(z)) is called the height of polynomialΦn(x). A(n) has been much studied a lot. On this basis, Pomerance and Ryan ([25])introduced the concept of B(n), such as B(n):=max{H(f(x)):f(x)(?)xn-1and f(x)€Z[x]}. The problem of determining B(n) has been studied extensively. Ryan ([27]) gave some conjectures on the maximal height of divisors of xn-1.Conjecture1. Let p<q be odd primes. Suppose b>2, Then B(pqb)> p..Conjecture2. For a fixed odd prime p and fixed positive integer b, the finite list of values B(pqb) as p<q varies are all divisible by p.Conjecture3. Let p<q be primes. ThenIn this paper we investigate the maximal height of divisors of xpaqb-1, obtain some conclusions about Conjecture1and Conjecture2.In chapter1, we introduce the background of this article, research and develop-ment profiles.In chapter2, we give some element lemmas and some calculation results which are needed in the prove of our main result. In chapter3, we investigate the maximal height of divisors of xpq3-1and get the following main resultTheorem3.1. Let p, q be distinct primes and ρ,σ the unique positive integers such that pp+aq=(p-1)(q-1). ThenIn chapter4, we investigate the maximal height of divisors of xpq4-1and get the following three resultTheorem4.1. Let p≠q be primes. Then B(pq4)<p(p-1).Theorem4.4. Let p<q be.primes. ThenTheorem4.6. Let p<q<r be primes and q=±r(mod p). If b≤4, then B(pqb)=B(prb).In chapter5, we investigate Conjecture2and Conjecture3, prove that Conjecture2is true and give a result about Conjecture3.