The Research On Some Congruence Identity Problems In Number Theory

The properties of identities have been studied in combinatorial mathematics and number theory.The identities involving the binomial coefficients and the identities involving special sequences have aroused the interest of many experts and scholars who from all over the world,so far,there have been many interesting conclusions.Using the method of elementary number theory and some techniques of combinatorial mathematics,some important identities are studied,and some research results of very theoretical value are obtained.Firstly,the main purpose is to discuss the properties of the Smarandache primitive function and combinatorial numbers.Combined combinatorial numbers C_n~m(n=2k-1,k?Z~+)with Smarandache primitive function,a new indeterminate equation are established and the integer solutions of the equation are gived.Secondly,the main purpose is to study a relation between binomial coefficients and cubes of Fibonacci numbers.By using the properties of Fibonacci numbers and combinatorial numbers,an interesting identity is obtained.Thirdly,based on the question two,the main purpose is to study the relationship between the combinatorial numbers and Fibonacci numbers.By using the same method,two identities concerning binomial coefficients and 12 and 16 powers of Fibonacci numbers are obtained,it is the generalization of the content of the question two.Fourthly,the main purpose is to study an identity in the form of (?) By using the method of classification discussion,the conclusion that this kind of integers are all anti-sociable numbers is proved.Finally,the problems of identities concerning binomial coefficients and Fibonacci numbers in this paper are concluded and some issues which need further research are pointed out.