Congruence Properties For The Sum Of Binomial Coefficients

Congruence properties for the sum of binomialcoefficientsBinomial coefficients have been studied for about 700 years. People find that the sum of binomial coefficients have many good properties which are related to many mathematical problems. In 1978, Apery proved the irrationality of (?)(2) and (?)(3) using the recurrence for the sum of the binomial coefficients. Since then, many people pay attention to the sum of binomial coefficients, so a lot of new results are obtained: On the other hands, with the development of Cryptology, it is a hot topic that how to find a big prime effectively in recent years. In 2004, a polynomial time primality test was given by Agral, Kayal and Saxeha, which is related to the congruence properties of a kind of sum of binomial coefficients. In this paper, we mainly study several kinds of sum of binomial coefficents via the method of divisibility and congruence.In this paper, we mainly discuss the following problems.Firstly, we study the sum of binomial coefficients d(n) = (?) and obtain the congruence properties of d(p), d(p - 1), d(p + 1), d(2p) modulo p³,p².Secondly, two kinds of sum of binomial coefficients are consideredAs a result, we prove that:1. The congruence properties of (?) modulo p and somedeductions.2. The congruence properties of u_a,b,c^ε(2), u_a,b,c^ε(3), u_a,b,c^ε(2^r) respectively modulo 2³, 3³, 2^3r.These results will give people more effective information about these kinds of sumof binomial coefficients, which is useful for the further study of related problems.