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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 105: Issue 4
DOI:
https://doi.org/10.1023/b:amhu.0000049279.34821.25
Author:
Xian-Meng Meng
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Acta Mathematica Hungarica
Volume/Issue:
Volume 99: Issue 1-2
DOI:
https://doi.org/10.1023/a:1024513613734
Author:
Li Hongze

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)<< X0.982.

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Acta Mathematica Hungarica
Volume/Issue:
Volume 97: Issue 4
DOI:
https://doi.org/10.1023/a:1021613315705
Author:
Jan-Christoph Puchta
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Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 58: Issue 1
DOI:
https://doi.org/10.1556/012.2021.58.1.1487
Authors:
Jinjiang Li
,
Min Zhang
, and
Haonan Zhao

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 p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 6 + p 6 6 ,

where p 1 , p 2 , p 3 , p 4 , p 5 , p 6 are prime numbers.

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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 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 135: Issue 3
DOI:
https://doi.org/10.1007/s10474-011-0193-0
Author:
Y. C. Wang

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.

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