Modulo P~2 Congruence With Harmonic Number

This thesis mainly studies the properties of(?)km Hkn(mod p2).According to the different pairs of(m,n),the problem is classified and studied,and the general research methods are given.In addition,some congruences involving central binomial coefficients and Catalan numbers are proved by using congruences of harmonic numbers,some of which are generalizations of known results.This dissertation is divided into seven chapters,the specific research contents are as follows1.In the first chapter,we briefly introduce the research history and development status of identities and congruences involving harmonic numbers.We also show the research motivation and structure of this thesis.2.In the second chapter,we study the properties of(?)kmHk(n)(mod p2)and prove the congruences for m??p-2,p-3,p-4,p-5},n ?{1,2,3,4}.3.In the third chapter,we prove two Euler-type congruences by using Kummer's congruence and Euler's theorem.4.In the forth chapter,we study the properties of(?)km Hk2(mod p2)and give some results and a recurrence formula of(?)kmHk3(mod p2).5.In the fifth chapter,we derive a formula which converts the calculations of(?)kmHk4(mod p2)to the calculations of modulus p congruences.We also establish the congruences for m=3,5 and discuss the relation between this kind of congruences and the Euler-type.6.In the sixth chapter,we prove some congruences involving central binomial coeffi-cients and Catalan numbers and generalize some known results.7.In the seventh chapter,we summarize the work of this paper and look forward to the future.