# Search Results

## Abstract

We prove the existence of infinitely many imaginary quadratic fields whose discriminant has exactly three distinct prime factors and whose class group has an element of a fixed large order. The main tool we use is solving an additive problem via the circle method.

## Abstract

Let *E(X)* denote the number of natural numbers not exceeding *X* which cannot be written as a sum of a prime and a square. In this paper we show that for sufficiently large *X* we have *E(X)<<* X^{0.982}.

Let *N* be a sufficiently large integer. In this paper, it is proved that, with at most *O*(*N*
^{119/270+}
*
^{s}
*) exceptions, all even positive integers up to

*N*can be represented in the form

where *p*
_{1}
*, p*
_{2}
*, p*
_{3}
*, p*
_{4}
*, p*
_{5}
*, p*
_{6} are prime numbers.

## Abstract

We sharpen Hua’s result by proving that each sufficiently large odd integer *N* can be written as

*p*

_{j}are primes. This result is as good as what was previously derived from the Generalized Riemann Hypothesis.

## Abstract

*N*satisfying some necessary congruence conditions are the sum of

*j*almost equal prime cubes with

*j*= 5; 6; 7; 8, i.e.,

*N*=

*p*

_{1}

^{3}+ ... +

*p*

_{ j }

^{3}with |

*p*

_{ i }− (

*N/j*)

^{1/3}| ≦

*i*≦

*j*), for

*δ*

_{ j }= 1/45; 1/30; 1/25; 2/45, respectively.

## Abstract

Let *p*
_{
i
} be prime numbers. In this paper, it is proved that for any integer *k*≧5, with at most exceptions, all positive even integers up to *N* can be expressed in the form . This improves the result for some *c*>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1–4] for the problem in the form . This method can also be used for some other similar forms.

## Abstract

*N*congruent to 5 modulo 24 can be written in the form

*p*

_{1}≦

## Abstract

Let denote the set {*n*∣2|*n*, ∀ *p*>2 with *p*−1|*k*}. We prove that when , almost all integers can be represented as the sum of a prime and a *k*-th power of prime for *k*≧3. Moreover, when , almost all integers *n*∊(*X*,*X*+*H*] can be represented as the sum of a prime and a *k*-th power of integer for *k*≧3.