Typical primitive polynomials over integer residue rings

Let N be a product of distinct prime numbers and Z/(N) be the integer residue ring modulo N. In this paper, a primitive polynomial f(x) over Z/(N) such that f(x) divides x^s−c for some positive integer s and some primitive element c in Z/(N) is called a typical primitive polynomial. Recently typical primitive polynomials over Z/(N) were shown to be very useful, but the existence of typical primitive polynomials has not been fully studied. In this paper, for any integer m1, a necessary and sufficient condition for the existence of typical primitive polynomials of degree m over Z/(N) is proved.     

Polynomial Cunningham chains

A sequence of prime numbers p₁,p₂,p₃,…, such that pi=2pi−1+? for all i, is called a Cunningham chain of the first or second kind, depending on whether ?=1 or −1 respectively. If k is the smallest positive integer such that 2pk+? is composite, then we say the chain has length k. It is conjectured that there are infinitely many Cunningham chains of length k for every positive integer k. A sequence of polynomials f₁(x),f₂(x),… in Z[x], such that f₁(x) has positive leading coefficient, each fi(x) is irreducible in Q[x] and fi(x)=xfi−1(x)+? for all i, is defined to be a polynomial Cunningham chain of the first or second kind, depending on whether ?=1 or −1 respectively. If k is the least positive integer such that fk+1(x) is reducible in Q[x], then we say the chain has length k. In this article, for polynomial Cunningham chains of both kinds, we prove that there are infinitely many chains of length k and, unlike the situation in the integers, that there are infinitely many chains of infinite length, by explicitly giving infinitely many polynomials f₁(x), such that fk+1(x) is the only term in the sequence that is reducible.     

Values of coefficients of cyclotomic polynomials

Let a(k,n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that . In this paper, we improve this result and prove that for any prime p and any integer l≥1, we have

{a(k,p^ln)∣n,k∈N}=Z.     

A weighted Turán sieve method

We develop a weighted Turán sieve method and applied it to study the number of distinct prime divisors of f(p) where p is a prime and f(x) a polynomial with integer coefficients.     

Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part

Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.     

On multipliers of polynomial addition sets

Let G be a group of order v, and f(x) be a nonzero integral polynomial. A (v, k, f(x))-polynomial addition set in G is a subset D of G with k distinct elements such that f(Σ_d?Dd) = λΣ_g?Gg for some integer λ. We discuss the multipliers of polynomial addition sets. The structure of some polynomial addition sets is studied, and in particular, we give a complete characterization of the case where G is cyclic and f(x) is irreducible.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

Polynomials That Force a Unital Ring to be Commutative

We characterize polynomials f with integer coefficients such that a ring with unity R is necessarily commutative if f(R) = 0, in the sense that f(x) = 0 for all ${x \in R}$ . Such a polynomial must be primitive, and for primitive polynomials the condition f(R) = 0 forces R to have nonzero characteristic. The task is then reduced to considering rings of prime power characteristic and the main step towards the full characterization is a characterization of polynomials f such that R is necessarily commutative if f(R) = 0 and R is a unital ring of characteristic some power of a fixed prime p.     

Periodic decomposition of measurable integer valued functions

We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough. We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so-called decomposition property. We characterize those periods a₁,…,ak for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a₁,…,ak)-periodic decomposition if and only if Δa₁?Δakf=0, where Δaf(x)=f(x+a)−f(x).     

A complete determination of Rabinowitsch polynomials

Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x²+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers x=x₀,x₀+1,…,x₀+s−1, |fm(x)| is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials fm(x).     

The p-Regular Partition Function Modulo p

Let b_?(n) denote the number of ?-regular partitions of n, where ? is a positive power of a prime p. We study in this paper the behavior of b_?(n) modulo powers of p. In particular, we prove that for every positive integer j, b_?(n) lies in each residue class modulo p^j for infinitely many values of n.     

A Hardy field extension of Szemerédi's theorem

In 1975 Szemerédi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form p(n) where p(n) is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference of the progression in Szemerédi's theorem can be of the form [nδ] where δ is any positive real number and [x] denotes the integer part of x. More generally, the common difference can be of the form [a(n)] where a(x) is any function that is a member of a Hardy field and satisfies a(x)/xk→∞ and a(x)/xk+1→0 for some non-negative integer k. The proof combines a new structural result for Hardy sequences, techniques from ergodic theory, and some recent equidistribution results of sequences on nilmanifolds.     

On multiplicative functions which are additive on sums of primes

Let ${f : \mathbb{N} \to \mathbb{C}}$ be a multiplicative function satisfying f(p ₀) ≠ 0 for at least one prime number p ₀, and let k ≥ 2 be an integer. We show that if the function f satisfies f(p ₁ + p ₂ + . . . + p _k ) = f(p ₁) + f(p ₂) + . . . + f(p _k ) for any prime numbers p ₁, p ₂, . . . ,p _k then f must be the identity f(n) = n for each ${n \in \mathbb{N}}$ . This result for k = 2 was established earlier by Spiro, whereas the case k = 3 was recently proved by Fang. In the proof of this result for k ≥ 6 we use a recent result of Tao asserting that every odd number greater than 1 is the sum of at most five primes.     

Periods of polynomials over a Galois ring

The period of a monic polynomial over an arbitrary Galois ring GR(pe,d) is theoretically determined by using its classical factorization and Galois extensions of rings. For a polynomial f(x) the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois ring, the period of f(x) is characterized by the periods of the irreducible polynomial and an associated polynomial of the form (x-1)m+pg(x). Further results on the periods of such associated polynomials are obtained, in particular, their periods are proved to achieve an upper bound value in most cases. As a consequence, the period of a monic polynomial over GR(pe,d) is equal to pe-1 times the period of its modulo p remainder polynomial with a probability close to 1, and an expression of this probability is given.     

Effective construction of an algebraic variety nonsingular in codimension one over a ground field of zero characteristic

Let k be a field of zero characteristic finitely generated over a primitive subfield. Let f be a polynomial of degree at most d in n variables, with coefficients from k, irreducible over an algebraic closure [`(k)] \bar{k} . Then we construct an algebraic variety V nonsingular in codimension one and a finite birational isomorphism V → Z(f), where Z(f) is the hypersurface of all common zeros of the polynomial f in the affine space. The running time of the algorithm for constructing V is polynomial in the size of the input. Bibliography: 8 titles.     

On the finiteness of certain Rabinowitsch polynomials II

Let m be a positive integer and f_m(x) be a polynomial of the form f_m(x)=x²+x−m. We call a polynomial f_m(x) a Rabinowitsch polynomial if for and consecutive integers is either 1 or prime. In Byeon (J. Number Theory 94 (2002) 177), we showed that there are only finitely many Rabinowitsch polynomials f_m(x) such that 1+4m is square free. In this note, we shall remove the condition that 1+4m is square free.     

A note on the least prime in an arithmetic progression

Let k, l denote positive integers with (k, l) = 1. Denote by p(k, l) the least prime p ≡ l(mod k). Let P(k) be the maximum value of p(k, l) for all l. We show $\text{lim inf}\text{P(k)}\text{(?(k)}\text{log}\text{k)}\text{≥ e}{}^{\mathrm{\gamma }}\text{= 1.78107…}$, where γ is Euler's constant and ? is Euler's function. We also show $\text{P(k)}\text{(?(k)}\text{log}\text{k)}\text{→ ∞}$ for almost all k.     

Uniqueness of entire functions

Let f be a nonconstant entire function and let a be a meromorphic function satisfying T(r,a)=S(r,f) and a?a′. If f(z)=a(z)⇔f′(z)=a(z) and f(z)=a(z)⇒f″(z)=a(z), then f≡f′, and a?a′ is necessary. This extended a result due to Jank, Mues and Volkmann.     

On the 2<Emphasis Type="Italic">m</Emphasis>-th power mean of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions with the weight of trigonometric sums

Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a ₀ + a ₁ x + ... + a _k x ^k is a k-degree polynomial with integral coefficients such that (p, a ₀, a ₁, ..., a _k ) = 1, for any integer m, we study the asymptotic property of