Sequences with bounded l.c.m. of each pair of terms
On addition of two sets of integers
On a problem of Sierpiński
On a conjecture of Sárközy and Szemerédi
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(minA(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,...
On the Siegel-Tatuzawa-Hoffstein theorem
On disjoint arithmetic progressions
On subset sums of a fixed set
Remarks on systems of congruence classes
On a question regarding visibility of lattice points, II
Prime values of reducible polynomials, I
Sequences with bounded l.c.m. of each pair of terms, III
A quantitative Erdös-Fuchs theorem and its generalization
A generalization of the classical circle problem
Sequences with bounded l.c.m. of each pair of terms II
Visibility of lattice points
A basis of ℤₘ, II
Given a set A ⊂ ℕ let 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, 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 for all n̅ ∈ ℤₘ.
On the average of the sum-of-a-divisors function
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.
On the index of an odd perfect number
Suppose that N is an odd perfect number and is a prime power with . Define the index . We prove that m cannot take the form , 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,...
A basis of Zₘ
Let , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which 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 for all n̅ ∈ Zₘ.
On the subset sums of exponential type sequences
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