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A basis of ℤₘ, II

100%

Tang M. , Chen Y.-G.

Colloquium Mathematicum

2007

tom 108

nr 1

141-145

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.

A quantitative Erdös-Fuchs theorem and its generalization

100%

Chen Y.-G. , Tang M.

Acta Arithmetica

2011

tom 149

nr 2

171-180

A basis of Zₘ

100%

Tang M. , Chen Y.-G.

Colloquium Mathematicum

2006

tom 104

nr 1

99-103

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.

The Diophantine equation $(bn)^{x} + (2n)^{y} = ((b+2)n)^{z}$

100%

Tang M. , Yang Q.-H.

Colloquium Mathematicum

2013

tom 132

nr 1

95-100

Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b ≠ 89, the Diophantine equation $b^{x} + 2^{y} = (b+2)^{z}$ has only the solution (x,y,z) = (1,1,1). We give an extension of this result.

On near-perfect and deficient-perfect numbers

80%

Tang M. , Ren X.-Z. , Li M.

Colloquium Mathematicum

2013

tom 133

nr 2

221-226

For a positive integer n, let σ(n) denote the sum of the positive divisors of n. Let d be a proper divisor of n. We call n a near-perfect number if σ(n) = 2n + d, and a deficient-perfect number if σ(n) = 2n - d. We show that there is no odd near-perfect number with three distinct prime divisors and determine all deficient-perfect numbers with at most two distinct prime factors.

On near-perfect numbers

80%

Tang M. , Ma X. , Feng M.

nr 2

157-188

For a positive integer n, let σ(n) denote the sum of the positive divisors of n. We call n a near-perfect number if σ(n) = 2n + d where d is a proper divisor of n. We show that the only odd near-perfect number with four distinct prime divisors is 3⁴·7²·11²·19².