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Acta Mathematica Hungarica
Volume/Issue:
Volume 137: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-012-0226-3
Author:
Kostadinka Lapkova

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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Acta Mathematica Hungarica
Volume/Issue:
Volume 116: Issue 4
DOI:
https://doi.org/10.1007/s10474-007-6041-6
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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Acta Mathematica Hungarica
Volume/Issue:
Volume 126: Issue 3
DOI:
https://doi.org/10.1007/s10474-009-9061-6
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 ≦ i ≦ j), for δ j = 1/45; 1/30; 1/25; 2/45, respectively.
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Acta Mathematica Hungarica
Volume/Issue:
Volume 129: Issue 3
DOI:
https://doi.org/10.1007/s10474-010-0028-4
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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Acta Mathematica Hungarica
Volume/Issue:
Volume 130: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-010-0002-1
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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Acta Mathematica Hungarica
Volume/Issue:
Volume 122: Issue 3
DOI:
https://doi.org/10.1007/s10474-008-8019-4
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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Acta Mathematica Hungarica
Volume/Issue:
Volume 120: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-007-7107-1
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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Acta Mathematica Hungarica
Volume/Issue:
Volume 120: Issue 4
DOI:
https://doi.org/10.1007/s10474-008-7068-z
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 3 ≦ Q 3, all reduced residues a 3 mod q 3, almost all q 2 ≦ Q 2, all admissible residues a 2 mod q 2, almost all q 1 ≦ Q 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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Acta Mathematica Hungarica
Volume/Issue:
Volume 132: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-010-0064-0
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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