| Combinatorial congruences and additive combinatorics are important parts of Combinatorial Number Theory.In this thesis we study combinatorial congruences of Ramanujan type and extensions of the solved Erdos-Heilbronn conjecture on sumset with distinct summands.Combinatorial congruences are related to many fields of mathematics,such as p-adic analysis and Ramanujan-type series for 1/π.In this thesis,by using tools of Wilf-Zeilberger pairs,combinatorial identities and Bernoulli numbers,we prove the following two congruences conjectured by Z.W.Sun:For any prime p>3 we have and We also make progress towards another congruence conjectured by Z.W.Sun.Additive Combinatorics is concerned with combinatorial properties related to ad-ditive structure.This field is quite active in recent years.In this thesis,we mainly apply the polynomial method based on Alon’s Combinatorial Nullstellenstaz to obtain a polynomial extension of the solved Erdos-Heilbronn conjecture.This thesis consists of five chapters.In the first chapter,we first present a survey of supercongruences related to Ramanujan-type series for 1/π,and introduce known extensions of the Erdos-Heilbronn conjecture,and then we state our main results.Chapters 2-5 are devoted to our proofs of the main results in this thesis. |
