A refinement of the Cameron-Erdős conjecture

AbstractErdős conjectured that for any set A ⊆ ℕ with positive lower asymptotic density, there are infinite sets B;C ⊆ ℕ such that B + C ⊆ A. We verify Erdős’ conjecture in the case where A has Banach density exceeding ½ . As a consequence, we prove that, for A ⊆ ℕ with positive Banach density (amuch weaker assumption than positive lower density), we can find infinite B;C ⊆ ℕ such that B+C is contained in the union of A and a translate of A. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.

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On a Conjecture of Livingston

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-065-1 ◽

2017 ◽

Vol 60 (1) ◽

pp. 184-195 ◽

Cited By ~ 1

Author(s):

Siddhi Pathak

Keyword(s):

Linear Independence ◽

Arithmetical Function ◽

Algebraic Numbers ◽

Erdös Conjecture

AbstractIn an attempt to resolve a folklore conjecture of Erdös regarding the non-vanishing at s = 1 of the L-series attached to a periodic arithmetical function with period q and values in {−1, 1}, Livingston conjectured the -linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston’s conjecture for composite q ≥ 4, highlighting that a newapproach is required to settle Erdös conjecture. We also prove that the conjecture is true for prime q ≥ 3, and indicate that more ingredients will be needed to settle Erdös conjecture for prime q.

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Erdös Conjecture on Connected Residual Graphs

Journal of Computers ◽

10.4304/jcp.7.6.1497-1502 ◽

2012 ◽

Vol 7 (6) ◽

Cited By ~ 1

Author(s):

Jiangdong Liao ◽

Gonglun Long ◽

Mingyong Li

Keyword(s):

Erdös Conjecture

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Fundamental units and consecutive squarefull numbers

International Journal of Number Theory ◽

10.1142/s1793042117500142 ◽

2016 ◽

Vol 13 (01) ◽

pp. 243-252

Author(s):

Kevser Aktaş ◽

M. Ram Murty

Keyword(s):

Abc Conjecture ◽

Elementary Methods ◽

Fundamental Units ◽

Erdös Conjecture

We connect several seemingly unrelated conjectures of Ankeny, Artin, Chowla and Mordell to a conjecture of Erdös on consecutive squarefull numbers. We then study the Erdös conjecture and relate it to the abc conjecture. We also derive by elementary methods several unconditional results pertaining to the Erdös conjecture.

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The degree of primitive sequences and Erdős conjecture

Mathematica Montisnigri ◽

10.20948/mathmontis-2021-52-4 ◽

2021 ◽

Vol 52 ◽

pp. 37-42

Author(s):

Ilias Laib

Keyword(s):

Prime Factors ◽

Primitive Sequences ◽

Positive Integers ◽

Erdös Conjecture

A sequence A of strictly positive integers is said to be primitive if none of its term divides another. Z. Zhang proved a result, conjectured by Erdős and Zhang in 1993, on the primitive sequences whose the number of the prime factors of its terms counted with multiplicity is at most 4. In this paper, we extend this result to the primitive sequences whose the number of the prime factors of its terms counted with multiplicity is at most 5.

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