Volume 16 - 1993


1. On sets of coprime integers in intervals

Paul Erdös ; András Sárközy.
If AN is such that it does not contain a subset S consisting of k pairwise coprime integers, then we say that A has the property Pk. Let Γk denote the family of those subsets of N which have the property Pk. If Fk(n)=maxA{1,2,3,,n},AΓk|A| and Ψk(n) is the number of integers u{1,2,3,,n} which are multiples of at least one of the first k primes, it was conjectured that Fk(n)=Ψk1(n) for all k2. In this paper, we give several partial answers.

2. The number of primes in a short interval.

Lou Shituo ; Yao Qi.
We prove in this paper that for y=xθ,1120<θ712 and x large enough, we have 0.99ylogxπ(x)π(xy)1.01ylogx.