tailieunhanh - Đề tài " On a coloring conjecture about unit fractions "

We prove an old conjecture of Erd˝s and Graham on sums of unit fractions: o There exists a constant b 0 such that if we r-color the integers in [2, br ], then there exists a monochromatic set S such that n∈S 1/n = 1. 1. Introduction We will prove a result on unit fractions which has the following corollary. Corollary. There exists a constant b so that for every partition of the integers in [2, br ] into r classes, there is always one class containing a subset S with the property n∈S 1/n = 1 | Annals of Mathematics On a coloring conjecture about unit fractions By Ernest S. Croot III Annals of Mathematics 157 2003 545 556 On a coloring conjecture about unit fractions By Ernest S. Croot III Abstract We prove an old conjecture of Erdos and Graham on sums of unit fractions There exists a constant b 0 such that if we r-color the integers in 2 br then there exists a monochromatic set S such that 2nes 1 n 1. 1. Introduction We will prove a result on unit fractions which has the following corollary- CoROLLARy. There exists a constant b so that for every partition of the integers in 2 br into r classes there is always one class containing a subset S with the property nes 1 n 1. In fact we will show that b may be taken to be e167000 if r is sufficiently large though we believe that b may be taken to be much smaller also note that b cannot be taken to be smaller than e since the integers in 2 er-o r can be placed into r classes in such a way that the sum of reciprocals in each class is just under 1. This corollary implies the result mentioned in the abstract and so resolves an unsolved problem of Erdos and Graham which appears in 2 3 and 5 . We will need to introduce some notation and definitions in order to state the Main Theorem as well as the propositions and lemmas in later sections For a given set of integers C let Qc denote the set of all the prime power divisors of elements of C and let S C 22q Q 1 q. Define C X Y 0 to be the integers in X Y all of whose prime power divisors are Xe and let C X Y ff be those integers n G C X Y ff such that w n Cl n log log n where w n and Q n denote the number of prime divisors and the number of prime power divisors of n respectively. Our Main Theorem then is as follows. 546 ERNEST S. CROOT III Main Theorem. Suppose C c C N N1 0 0 where 0 ỗ 0 and Ỗ 0 1 4. If N Pop 1 and E 1 6 nec n then there exists a subset S c C for which neS 1 n 1. To prove the corollary we will show in the next section that for r sufficiently large 6r

• Thạc sĩ - Tiến sĩ - Cao học
• unit fractions
• subjects of mathematics
• Annals of Mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Research methods
• Data collection methods
• Understanding scientific research
• Ensuring research validity
• Measurement techniques
• Sampling methods
• Scientific research capacity
• Educational institutions
• Policy and management studies
• Vietnam Academy for Ethnic minorities
• Scientific bases
• earth sciences
• Research
• VNU Journal of Science
• scientific reports
• Action research
• linguistics
• research projects
• Foreign Languages
• cooperative research
• Renewable energy
• Introduction to research
• Scientific investigation
• Scientific investigation research
• Experimental designs
• Measurement of variables
• Reading research
• Conducting and reading research
• Research process
• Research report
• Purpose of research
• Scientific method
• Conducting research
• The effect of university autonomy on training and scientific research
• Vietnam national university of agriculture
• Training and scientific research
• Vascular plant
• Flora
• Yen Tu
• Quang Ninh Research Forest
• statements and documents scientific reports
• medical reports
• medical knowledge
• medical research
• clinical trials
• medical management
• health care
• Methods in behavioral research
• Behavioral research
• Scientific understanding
• Fundamental research issues
• Measurement concepts
• Observational methods
• reading comprehension
• education
• optimal picket
• model creation
• urban expansion
• organizing spatial
• Journal of Science
• Journal of Technology
• Technology Management
• Research quality
• Strategy and policy
• Summary of Doctoral thesis
• Summary of Doctoral thesis in public Management
• State management
• Female scientific research human
• Research human resource development
• Scientific research and teaching
• Defense and security education
• Current situation and solutions
• Security education
• Structural biochemistry
• epigenetics
• MicroRNAs
• Tạp chí khoa học
• Tạp chí Công nghệ
• Quản lý công nghệ
• Human resources
• Science management
• Poliomyelitis vaccine Research
• Non Ri Maprec
• Production center