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1
Content available remote

On a conjecture of Sárközy and Szemerédi

100%
Chen Y.-G.
Acta Arithmetica
|
2015
|
tom 169
|
nr 1
47-58
EN
Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. In 1994, Sárközy and Szemerédi conjectured that there exist infinite additive complements A and B with lim sup A(x)B(x)/x ≤ 1 and A(x)B(x)-x = O(min{A(x),B(x)}), where A(x) and B(x) are the counting functions of A and B, respectively. We prove that, for infinite additive complements A and B, if lim sup A(x)B(x)/x ≤ 1, then, for any given M > 1, we have $A(x)B(x) - x ≥ (min{A(x), B(x)})^M$ for all sufficiently large integers x. This disproves the above Sárközy-Szemerédi conjecture. We also pose several problems for further research.
2
Content available remote

Sequences with bounded l.c.m. of each pair of terms

100%
Chen Y.-G.
Acta Arithmetica
|
1998
|
tom 84
|
nr 1
71-95
3
Content available remote

On a problem of Sierpiński

100%
Chen Y.-G.
Acta Arithmetica
|
2012
|
tom 156
|
nr 4
373-382
4
Content available remote

On disjoint arithmetic progressions

88%
Chen Y.-G.
Acta Arithmetica
|
2005
|
tom 118
|
nr 2
143-148
5
Content available remote

On the Siegel-Tatuzawa-Hoffstein theorem

88%
Chen Y.-G.
Acta Arithmetica
|
2007
|
tom 130
|
nr 4
361-367
6
Content available remote

On subset sums of a fixed set

75%
Chen Y.-G.
Acta Arithmetica
|
2003
|
tom 106
|
nr 3
207-211
7
Content available remote

Prime values of reducible polynomials, I

64%
Chen Y.-G., Ruzsa I. Z.
Acta Arithmetica
|
2000
|
tom 95
|
nr 2
185-193
8
Content available remote

A basis of Zₘ

64%
Tang M., Chen Y.-G.
Colloquium Mathematicum
|
2006
|
tom 104
|
nr 1
99-103
EN
Let $σ_{A}(n) = |{(a,a') ∈ A²: a + a' = n}|$, where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, $σ_{A}(n)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which $σ_{A}(n)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and $σ_{A}(n̅) ≤ 768$ for all n̅ ∈ Zₘ.
9
Content available remote

On the average of the sum-of-a-divisors function

64%
Chen S.-C., Chen Y.-G.
Colloquium Mathematicum
|
2004
|
tom 100
|
nr 1
103-109
EN
We prove an Ω result on the average of the sum of the divisors of n which are relatively coprime to any given integer a. This generalizes the earlier result for a prime proved by Adhikari, Coppola and Mukhopadhyay.
10
Content available remote

On the index of an odd perfect number

64%
Chen F.-J., Chen Y.-G.
Colloquium Mathematicum
|
2014
|
tom 136
|
nr 1
41-49
EN
Suppose that N is an odd perfect number and $q^{α}$ is a prime power with $q^{α} || N$. Define the index $m = σ(N/q^{α})/q^{α}$. We prove that m cannot take the form $p^{2u}$, where u is a positive integer and 2u+1 is composite. We also prove that, if q is the Euler prime, then m cannot take any of the 30 forms q₁, q₁², q₁³, q₁⁴, q₁⁵, q₁⁶, q₁⁷, q₁⁸, q₁q₂, q₁²q₂, q₁³q₂, q₁⁴ q₂, q₁⁵q₂, q₁²q₂², q₁³q₂², q₁⁴q₂², q₁q₂q₃, q₁²q₂q₃, q₁³q₂q₃, q₁⁴q₂q₃, q₁²q₂²q₃, q₁²q₂²q₃², q₁q₂q₃q₄, q₁²q₂q₃q₄, q₁³q₂q₃q₄, q₁²q₂²q₃q₄, q₁q₂q₃q₄q₅, q₁²q₂q₃q₄q₅, q₁q₂q₃q₄q₅q₆, q₁q₂q₃q₄q₅q₆q₇, where q₁, q₂, q₃, q₄, q₅, q₆, q₇ are distinct odd primes. A similar result is proved if q is not the Euler prime. These extend recent results of Broughan, Delbourgo, and Zhou. We also pose a related problem.
11
Content available remote

A quantitative Erdös-Fuchs theorem and its generalization

64%
Chen Y.-G., Tang M.
Acta Arithmetica
|
2011
|
tom 149
|
nr 2
171-180
12
Content available remote

Sequences with bounded l.c.m. of each pair of terms II

64%
Dai L.-X., Chen Y.-G.
Acta Arithmetica
|
2006
|
tom 124
|
nr 4
315-326
13
Content available remote

A generalization of the classical circle problem

64%
Chen Y.-G., Tang M.
Acta Arithmetica
|
2012
|
tom 152
|
nr 3
279-290
14
Content available remote

Sequences with bounded l.c.m. of each pair of terms, III

64%
Chen Y.-G., Dai L.-X.
Acta Arithmetica
|
2007
|
tom 128
|
nr 2
125-133
15
Content available remote

A basis of ℤₘ, II

64%
Tang M., Chen Y.-G.
Colloquium Mathematicum
|
2007
|
tom 108
|
nr 1
141-145
EN
Given a set A ⊂ ℕ let $σ_A(n)$ denote the number of ordered pairs (a,a') ∈ A × A such that a + a' = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, $σ_A(n)$ is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and $σ_A(n̅) ≤ 5120$ for all n̅ ∈ ℤₘ.
16
Content available remote

Visibility of lattice points

64%
Chen Y.-G., Cheng L.-F.
Acta Arithmetica
|
2003
|
tom 107
|
nr 3
203-207
17
Content available remote

On a question regarding visibility of lattice points, II

64%
Adhikari S., Chen Y.-G.
Acta Arithmetica
|
1999
|
tom 89
|
nr 3
279-282
18
Content available remote

Remarks on systems of congruence classes

64%
Chen Y.-G., Porubský Š.
Acta Arithmetica
|
1995
|
tom 71
|
nr 1
1-10
19
Content available remote

On the subset sums of exponential type sequences

52%
Chen Y.-G., Fang J.-H., Hegyvári N.
Acta Arithmetica
|
2016
|
tom 173
|
nr 2
141-150
EN
For a sequence A ⊆ ℕ, let P(A) be the set of all sums of distinct terms taken from A. The sequence A is said to be complete if P(A) contains all sufficiently large integers. Let p > 1 be an integer. The following main results are proved: (a) Let $A_t = {a_1 ≤ ... ≤ a_t}$ be any sequence of positive integers (not necessarily distinct), $S_p = {p^i : i = 0, 1, ... }$ and $S_p A_t = {p^i a_j : i = 0, 1, ...; j = 1, ..., t}$. When t ≥ p-1, the sequence P(S_pA_t)$ has positive lower asymptotic density not less than $1/a_{p-1}$. The lower bounds p-1 and $1/a_{p-1}$ are both the best possible. (b) For any positive integer k, the sequence ${ p^i F_j : i = 0, 1, ... ; j = k, k+1, ..., n}$ is complete, where $F_j$ is the jth Fibonacci number and $n = p^2 F_{k+2p-1}^2$.
20
Content available remote

Fermat numbers and integers of the form $a^k+a^l+p^{α}$

52%
Chen Y.-G., Feng R., Templier N.
Acta Arithmetica
|
2008
|
tom 135
|
nr 1
51-61
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