On Sequences Containing at Most 4 Pairwise Coprime Integers

Details On Sequences Containing at Most 4 Pairwise Coprime Integers On Sequences Containing at Most 4 Pairwise Coprime Integers

2010-12-30

arxiv.org/abs/1101.0050v2

Mathematics

Number Theory

17 pages

Scientific paper

Let $f(n,k)$ be the largest number of positive integers not exceeding $n$ from which one cannot select $k+1$ pairwise coprime integers, and let $E(n,k)$ be the set of positive integers which do not exceed $n$ and can be divided by at least one of $p_1, p_2,..., p_k$, where $p_i$ is the $i$-th prime. In 1962, P. Erd\H os conjectured that $f(n,k)=|E(n,k)|$ for all $n\ge p_k$. In 1973, S. L. G. Choi proved that the conjecture is true for $k=3$. In 1994, Ahlswede and Kachatrian disproved the conjecture for $k=212$. In this paper we prove that, for $n\ge 49$, if A(n,4) is a set of positive integers not exceeding $n$ from which one cannot select 5 pairwise coprime integers and $|A(n,4)|\ge |E(n,4)|$, then $A(n,4)=E(n,4)$. In particular, the conjecture is true for k=4. Several open problems and conjectures are posed for further research.

Affiliated with

Also associated with

No associations

Rating

On Sequences Containing at Most 4 Pairwise Coprime Integers does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On Sequences Containing at Most 4 Pairwise Coprime Integers, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Sequences Containing at Most 4 Pairwise Coprime Integers will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-400605