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1
Content available remote On a problem of Sierpiński
100%
Chen Y.-G.
Acta Arithmetica
|
2012
|
tom 156
|
nr 4
373-382
2
Content available remote On a conjecture of Sárközy and Szemerédi
100%
Chen Y.-G.
|
|
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.
3
Content available remote On disjoint arithmetic progressions
88%
Chen Y.-G.
Acta Arithmetica
|
2005
|
tom 118
|
nr 2
143-148
4
Content available remote On subset sums of a fixed set
75%
Chen Y.-G.
Acta Arithmetica
|
2003
|
tom 106
|
nr 3
207-211
5
Content available remote On the average of the sum-of-a-divisors function
63%
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.
6
Content available remote A basis of ℤₘ, II
63%
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.
7
Content available remote On the index of an odd perfect number
63%
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.
8
Content available remote Sequences with bounded l.c.m. of each pair of terms II
63%
Dai L.-X. , Chen Y.-G.
Acta Arithmetica
|
2006
|
tom 124
|
nr 4
315-326
9
Content available remote A quantitative Erdös-Fuchs theorem and its generalization
63%
Chen Y.-G. , Tang M.
Acta Arithmetica
|
2011
|
tom 149
|
nr 2
171-180
10
Content available remote Remarks on systems of congruence classes
63%
Chen Y.-G. , Porubský Š.
|
|
nr 1
1-10
11
Content available remote A basis of Zₘ
63%
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.
12
Content available remote Visibility of lattice points
63%
Chen Y.-G. , Cheng L.-F.
Acta Arithmetica
|
2003
|
tom 107
|
nr 3
203-207
13
Content available remote Prime values of reducible polynomials, I
63%
Chen Y.-G. , Ruzsa I. Z.
Acta Arithmetica
|
2000
|
tom 95
|
nr 2
185-193
14
Content available remote Sequences with bounded l.c.m. of each pair of terms, III
63%
Chen Y.-G. , Dai L.-X.
Acta Arithmetica
|
2007
|
tom 128
|
nr 2
125-133
15
Content available remote Egyptian fractions with restrictions
51%
Chen Y.-G. , Elsholtz C. , Jiang L.-L.
Acta Arithmetica
|
2012
|
tom 154
|
nr 2
109-123
16
Content available remote On the subset sums of exponential type sequences
51%
Chen Y.-G. , Fang J.-H. , Hegyvári N.
|
|
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 $n = p^2 F_{k+2p-1}^2$.
17
Content available remote Prime values of reducible polynomials, II
45%
Chen Y.-G. , Kun G. , Pete G. , Ruzsa I. , Timar A.
|
|
tom 104
|
nr 2
117-127
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