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Acta Mathematica Hungarica
Volume/Issue:
Volume 124: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-009-8163-5
Author:
K. Matomäki

Abstract  

We prove that whenever
\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} $$\mathcal{A}$$ \end{document}
and
\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} $$\mathcal{B}$$ \end{document}
are dense enough subsets of {1, ..., N}, there exist a ∈
\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} $$\mathcal{A}$$ \end{document}
and b ∈
\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} $$\mathcal{B}$$ \end{document}
such that the greatest prime factor of ab + 1 is at least
\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} $$N^{1 + |\mathcal{A}|/(9N)}$$ \end{document}
.
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Acta Mathematica Hungarica
Volume/Issue:
Volume 134: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-011-0161-8
Authors:
Yingchun Cai
and
Minggao Lu

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity, and let E 3(N) denote the number of natural numbers not exceeding N that are congruent to 4 modulo 24 yet cannot be represented as the sum of three squares of primes and the square of one P 5. Then we have E 3(N)≪log1053 N. This result constitutes an improvement upon that of D. I. Tolev, who obtained the same bound, but with P 11 in place of P 5.

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Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 50: Issue 4
DOI:
https://doi.org/10.1556/sscmath.50.2013.4.1255
Author:
Yingchun Cai
Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper we show that the inequality
\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} $$\left\{ {\sqrt p } \right\} < p^{ - \tfrac{1} {{15.5}}}$$ \end{document}
has infinitely many solutions in primes p such that p + 2 = P 4.
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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 116: Issue 4
DOI:
https://doi.org/10.1007/s10474-007-6063-0
Author:
J. Sivak-Fischler

Abstract  

Combining Goldston-Yildirim’s method on k-correlations of the truncated von Mangoldt function with Maier’s matrix method, we show that
\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} $$\Xi _r : = \lim \inf _{n \to \infty } \tfrac{{p_{n + r} - p_n }} {{\log p_n }} \leqq e^{ - \gamma } \left( {r - \tfrac{{\sqrt r }} {2}} \right)$$ \end{document}
for all r ≧ 1 where p n denotes the nth prime number and γ is Euler’s constant. This is the best known result for any r ≧ 11.
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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 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 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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Periodica Mathematica Hungarica
Volume/Issue:
Volume 58: Issue 1
DOI:
https://doi.org/10.1007/s10998-009-9001-9
Author:
Sylvain Col

Abstract  

Les propriétés multiplicatives des nombres ellipséphiques peuvent être obtenues à l’aide des moments de la série génératrice de cette suite. Nous donnons des estimations précises pour les grands moments par deux méthodes distinctes: l’une combinatoire fournit un résultat précis dans le cas réputé le plus difficile des nombres n’utilisant que les 0 et les 1; la seconde purement analytique fournit un résultat sans condition sur les chiffres.

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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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