We prove that the lattice points on the circles x2 + y2 = n are well distributed for most circles containing lattice points.

A partition of the positive integer n into distinct parts is a decreasing sequence of positive integers whose sum is n, and the number of such partitions is denoted by Q(n). If we adopt the convention that Q(0) = 1, then we have the generating function

Improved upper and lower bounds of the counting functions of the conceivable additive decomposition sets of the set of primes are established. Suppose that where, ℝ′ differs from the set of primes in finitely many elements only and .

It is shown that the counting functions A(x) of ℐ and B(x) of ℬ for sufficiently large x, satisfy

The purpose of this paper is to show how a sieve method which has had many applications to problems involving rational primes can be modified to derive new results on Gaussian primes (or, more generally, prime ideals in algebraic number fields). One consequence of our main theorem (Theorem 2 below) is the following result on rational primes.

Let (bn) be a sequence of integers, obtained by traversing the rows of Pascal's triangle, as follows: start from the element at the top of the triangle, and at each stage continue from the current element to one of the elements at the next row, either the one immediately to the left of the current element or the one immediately to its right. Consider the distribution of the sequence (bnα) modulo 1 for an irrational α. The main results show that this sequence “often” fails to be uniformly distributed modulo 1, and provide answers to some questions raised by Adams and Petersen.

In this paper we continue the investigation begun in [11]. Let λ1…., λs and μ1, …, μs be real numbers, and define the forms

Further, let τ be a positive real number. Our goal is to determine conditions under which the system of inequalities

has a non-trivial integral solution. As has frequently been the case in work on systems of diophantine inequalities (see, for example, Brüdern and Cook [6] and Cook [7]), we were forced in [11] to impose a condition requiring certain coefficient ratios to be algebraic. A recent paper of Bentkus and Gotze [4] introduced a method for avoiding such a restriction in the study of positivede finite quadratic forms, and these ideas are in fact flexible enough to be applied to other problems. In particular, Freeman [10] was able to adapt the method to obtain an asymptotic lower bound for the number of solutions of a single diophantine inequality, thus finally providing the expected strengthening of a classical theorem of Davenport and Heilbronn [9]. The purpose of the present note is to apply these new ideas to the system of inequalities (1.1).

We consider ineducible Goppa codes of length qm over Fq defined by polynomials of degree r, where q = pt and p, m, r are distinct primes. The number of such codes, inequivalent under coordinate permutations and field automorphisms, is determined.

In this paper we derive a relation between character sums and partial sums of Dirichlet series.

Using a result on arithmetic progressions, we describe a method for finding the rational h–tuples ρ = (ρl,…,ρh) such that all the multiples mρ (for m coprime to a denominator of ρ) lie in a linear variety modulo Z. We give an application to hypergeometric functions.

We find a basis for the universal punctured even distribution and then a basis for the cyclotomic units over function fields.

The set of integers represented as the sum of three cubes of natural numbers is widely expected to have positive density (see Hooley [7] for a discussion of this topic). Over the past six decades or so, the pursuit of an acceptable approximation to the latter statement has spawned much of the progress achieved in the theory of the Hardy-Littlewood method, so far as its application to Waring's problem for smaller exponents is concerned. Write R(N) for the number of positive integers not exceeding N which are the sum of three cubes of natural numbers.

§1. Introduction. In 1946, Davenport and Heilbronn [9] proved a result which opened up the study of Diophantine inequalities. Suppose that Q(x) is a diagonal quadratic form with non-zero real coefficients in s variables. We write

In this paper we prove that every positive definite n-ary integral quadratic form with 12 < n < 13 (respectively 14 ≦ n ≤ 20) that can be represented by a sum of squares of integral linear forms is represented by a sum of 2 · 3n + n + 6 (respectively 3 · 4n + n + 3) squares. We also prove that every positive definite 7-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 25 squares.

An original linear algebraic approach to the basic notion of Freiman's isomorphism is developed and used in conjunction with a combinatorial argument to answer two questions, posed by Freiman about 35 years ago.

First, the order of growth is established of t(n), the number of classes isomorphic n-element sets of integers: t(n) = n(2 + σ(1))n. Second, it is proved linear Roth sets (sets of integers free of arithmetic progressions and having Freiman rank 1) exist and, moreover, the number of classes of such cardinality n is amazingly large; in fact, it is “the same as above”: .

Let φ be the Euler function, and consider the error term H in the asymptotic formula

A new criterion on Catalan's equation is proved by elementary means

This shows, without appealing either to the theory of linear forms in logarithms, or to any computation, that (C) has no solution (x, y, p, q) with min {p, q}≤41, except (3,2, 2, 3).

We consider the problem of finding, for each even number m, a basis of orthogonal vectors of length in the Leech lattice. We give such a construction by means of double circulant codes whenever m = 2p and p is a prime not equal to 11. From this one can derive a construction for all even m not of the form 2· 11r.

Let ϑ be an integer of multiplicative order t≥1 modulo a prime p. Sums of the form

are introduced and estimated, with a sequence such that kz1, …, kzT is a permutation of z1, …, zT, both sequences taken modulo t, for sufficiently many distinct modulo t values of k. Such sequences include

xn for x = 1 ,…,t with an integer n≥1;

xn for x = 1 ,…,t and gcd (x, t) = 1 with an integer n≥1;

ex for x = 1 ,…,T with an integer e, where T is the period of the sequence ex modulo t.

Some of the results can be extended to composite moduli and to sums of multiplicative characters as well. Character sums with the above sequences have some cryptographic motivation and applications and have been considered in several papers by J. B. Friedlander, D. Lieman and I. E. Shparlinski. In particular several previous bounds are generalized and improved.

Let q be a prime number and let a = (a1, …, as) be an s-tuple of distinct integers modulo q. For any x coprime with q, let be such that . For fixed s and q→∞ an asymptotic formula is given for the number of residue classes x modulo q for which

The more general case, when q is not necessarily prime and x is restricted to lie in a given subinterval of [1, q], is also treated.

New upper and lower bounds are given for Fh(g, N), the maximum size of a Bh[g] sequence contained in [1, N]. It is proved that and that

and that, for any ε > 0 and g > g(ε, h),