scholarly journals The Erdős conjecture for primitive sets

2019 ◽  
Vol 6 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Jared Duker Lichtman ◽  
Carl Pomerance
Keyword(s):  
Primitive Sets ◽  
Erdös Conjecture

scholarly journals The number of maximum primitive sets of integers

2021 ◽  
pp. 1-15
Author(s):  
Hong Liu ◽  
Péter Pál Pach ◽  
Richárd Palincza
Keyword(s):  
Related Problem ◽  
Maximum Size ◽  
Multiplicative Error ◽  
Coprime Integers ◽  
Primitive Sets

Abstract A set of integers is primitive if it does not contain an element dividing another. Let f(n) denote the number of maximum-size primitive subsets of {1,…,2n}. We prove that the limit α = lim n→∞ f(n)1/n exists. Furthermore, we present an algorithm approximating α with (1 + ε) multiplicative error in N(ε) steps, showing in particular that α ≈ 1.318. Our algorithm can be adapted to estimate the number of all primitive sets in {1,…,n} as well. We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in {1,…n} is between ${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$ and ${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$ . We show that neither of these bounds is tight: there are in fact ${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$ such sets.

Erdös Conjecture on Sums of Distinct Numbers

1980 ◽  
Vol 63 (1) ◽  
pp. 87-92 ◽  
Author(s):  
G. W. Peck
Keyword(s):  
Erdös Conjecture

scholarly journals The probability of choosing primitive sets

2007 ◽  
Vol 125 (1) ◽  
pp. 39-49 ◽  
Author(s):  
Sergi Elizalde ◽  
Kevin Woods
Keyword(s):  
Primitive Sets

scholarly journals On a Sumset Conjecture of Erdős

2015 ◽  
Vol 67 (4) ◽  
pp. 795-809 ◽  
Author(s):  
Mauro Di Nasso ◽  
Isaac Goldbring ◽  
Renling Jin ◽  
Steven Leth ◽  
Martino Lupini ◽  
...  
Keyword(s):  
Positive Solution ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Amenable Groups ◽  
Banach Density ◽  
Infinite Sets ◽  
Erdös 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.

Application of primitive sets to multi-criteria optimization problems

1980 ◽  
Vol 7 (1) ◽  
pp. 77-90 ◽  
Author(s):  
Walter O. Rom ◽  
Ming S. Hung
Keyword(s):  
Optimization Problems ◽  
Primitive Sets

scholarly journals On a Conjecture of Livingston

2017 ◽  
Vol 60 (1) ◽  
pp. 184-195 ◽  
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.

scholarly journals Identifying complex Hadamard submatrices of the Fourier matrices via primitive sets

2021 ◽  
Vol 617 ◽  
pp. 121-150
Author(s):  
John E. Herr ◽  
Troy M. Wiegand
Keyword(s):  
Primitive Sets

Erdös Conjecture on Connected Residual Graphs

2012 ◽  
Vol 7 (6) ◽  
Author(s):  
Jiangdong Liao ◽  
Gonglun Long ◽  
Mingyong Li
Keyword(s):  
Erdös Conjecture

On properties of primitive sets of digraphs with common cycles

2019 ◽  
pp. 101-114
Author(s):  
Y. E. Avezova ◽  
Keyword(s):  
Common Cycles ◽  
Primitive Sets

Fundamental units and consecutive squarefull numbers

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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