Journal ArticleDOI
The solution of Graham's greatest common divisor problem
TLDR
The following conjecture of R. L. Graham is verified: ifn≧n0, wheren0 is an explicitly computable constant, then for anyn distinct positive integersa1,a2, ...,an the authors have ai/(ai,aj) ≧ ≧n, and equality holds only in two trivial cases.Abstract:
The following conjecture of R. L. Graham is verified: Ifn≧n 0, wheren 0 is an explicitly computable constant, then for anyn distinct positive integersa 1,a 2, ...,a n we have\(\mathop {\max }\limits_{i,j} \) a i /(a i ,a j ) ≧ ≧n, and equality holds only in two trivial cases. Here (a i ,a j ) stands for the greatest cnmmon divisor ofa i anda j .read more
Citations
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Journal ArticleDOI
Covering the integers by arithmetic sequences. II
Zhi-Wei Sun,Zhi-Wei Sun +1 more
TL;DR: In this article, the authors reveal further connections between the common differences in an (exact) m-cover of Z and Egyptian fractions, and show that m-covers A of Z can be defined as systems where every integer is covered by A at least (exactly) m times.
Journal ArticleDOI
Covering the integers by arithmetic sequences
TL;DR: In this article, the authors reveal connections between the common dierences in an (exact) m-cover of Z and Egyptian fractions, and show that m-covers A of Z can be defined as systems where every integer is covered by A at least (exactly) m times.
Journal ArticleDOI
Some Problems and Results on Combinatorial Number Theory
Paul Erdös,Paul Erdös +1 more
TL;DR: In this article, the authors try to write this paper in such a way that it will not entirely be contained in the union of the set of my previous papers and that a t least some of the open problems I state will not be entirely hopeless.
ReportDOI
Towards a Katona Type Proof for the 2-intersecting Erdős-Ko-Rado Theorem
TL;DR: The possibility of a Katona type proof for the Erdős-Ko-Rado theorem for 2-and 3-intersecting families of sets was studied in this paper.
References
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Journal ArticleDOI
On the Difference Between Consecutive Primes.
D. R. Heath-Brown,Henryk Iwaniec +1 more
TL;DR: In this article, it was shown that (2) holds with & = 13/23 and (3) is admissible with and = 0 • 5652 and that (4) can be relaxed to # > 5/9 by an elaboration of the argument.
Book ChapterDOI
A Survey of Problems in Combinatorial Number Theory
TL;DR: A survey of problems in combinatorial number theory can be found in this paper, where the authors discuss problems connected with Van der Waerden's and Szemeredi's theorem.