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A basis of ℤₘ, II

Min TangYong-Gao Chen — 2007

Colloquium Mathematicae

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 ( n ̅ ) 5120 for all n̅ ∈ ℤₘ.

On near-perfect numbers

Min TangXiaoyan MaMin Feng — 2016

Colloquium Mathematicae

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.

A basis of Zₘ

Min TangYong-Gao Chen — 2006

Colloquium Mathematicae

Let σ A ( n ) = | ( a , a ' ) A ² : 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 ( n ̅ ) 768 for all n̅ ∈ Zₘ.

Some results on Poincaré sets

Min-wei TangZhi-Yi Wu — 2020

Czechoslovak Mathematical Journal

It is known that a set H of positive integers is a Poincaré set (also called intersective set, see I. Ruzsa (1982)) if and only if dim ( X H ) = 0 , where X H : = x = n = 1 x n 2 n : x n { 0 , 1 } , x n x n + h = 0 for all n 1 , h H and dim denotes the Hausdorff dimension (see C. Bishop, Y. Peres (2017), Theorem 2.5.5). In this paper we study the set X H by replacing 2 with b > 2 . It is surprising that there are some new phenomena to be worthy of studying. We study them and give several examples to explain our results.

On near-perfect and deficient-perfect numbers

Min TangXiao-Zhi RenMeng Li — 2013

Colloquium Mathematicae

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.

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