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## Divisibility of class numbers of imaginary quadratic fields whose discriminant has only three prime factors

Acta Mathematica Hungarica
Author:

## 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.

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## Hua’s theorem with nine almost equal prime variables

Acta Mathematica Hungarica
Authors:
G. Lü
and
Y. Xu

## Abstract

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

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$N = p_1^3 + \cdots + p_9^3 with \left| {p_j - \sqrt[3]{{N/9}}} \right| \leqq U = N^{\tfrac{1} {3} - \tfrac{1} {{198}} + \varepsilon }$$ \end{document}
, where p j are primes. This result is as good as what was previously derived from the Generalized Riemann Hypothesis.

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## Sums of almost equal prime cubes

Acta Mathematica Hungarica
Author:
F. Zhao

## Abstract

We prove that almost all integers 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| ≦
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$N^{1/3 - \delta _j + \varepsilon }$$ \end{document}
(1 ≦ ij), for δ j = 1/45; 1/30; 1/25; 2/45, respectively.
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## Additive bases arising from functions in a Hardy field

Acta Mathematica Hungarica
Authors:
T. H. Chan
,
A. V. Kumchev
, and
M. Wierdl

## Abstract

A classical additive basis question is Waring’s problem. It has been extended to integer polynomial and non-integer power sequences. In this paper, we will consider a wider class of functions, namely functions from a Hardy field, and show that they are asymptotic bases.

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## An improvement on Waring–Goldbach problem for unlike powers

Acta Mathematica Hungarica
Author:
Zhixin Liu

## 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.

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## On a generalization of Hua’s theorem with five squares of primes

Acta Mathematica Hungarica
Authors:
G. Lü
and
H. Sun

## Abstract

We sharpen Hua’s theorem with five squares of primes by proving that every sufficiently large integer N congruent to 5 modulo 24 can be written in the form
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + p_5^2$$ \end{document}
with p 1
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$N^{\tfrac{{49}} {{288}}}$$ \end{document}
.
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## Difference sets and shifted primes

Acta Mathematica Hungarica
Author:
J. Lucier

## Abstract

We show that if A is a subset of {1, …, n} which has no pair of elements whose difference is equal to p − 1 with p a prime number, then the size of A is O(n(log log n)c(log log log log log n)) for some absolute c > 0.

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## On the ternary goldbach problem with primes in independent arithmetic progressions

Acta Mathematica Hungarica
Author:
K. Halupczok

## Abstract

We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q 1 = Q 2:= n 1/2(log n)ϑ and Q 3:= (log n) θ . Then for all q 3Q 3, all reduced residues a 3 mod q 3, almost all q 2Q 2, all admissible residues a 2 mod q 2, almost all q 1Q 1 and all admissible residues a 1 mod q 1, there exists a representation n = p 1 + p 2 + p 3 with primes p i a i (q i ), i = 1, 2, 3.

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## On unlike powers of primes and powers of 2

Acta Mathematica Hungarica
Authors:
Zhixin Liu
and
Guangshi Lü

## Abstract

It is proved that every sufficiently large even integer is a sum of one prime, one square of prime, two cubes of primes and 161 powers of 2.

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