Some Results On Congruences

In 1914,the Indian mathematician S.Ramanujan discovered a number of Ramanujan-type series for 1/? for example,#12In 2011,Zhi-wei Sun observed that some congruences,corresponding to Ramanujan-type series for 1/?,modulo fourth powers of primes involves Euler numbers.The Euler numbers E0,E1,E2,...are given by(?)In this thesis,we mainly use the Wilf-Zeilberger method to prove two congruences involving Euler numbers corresponding to the above two series for 1/?.Theorem 1.For any prime p>3,we have#12Theorem 2.Let p be an odd prime.Then#12In 2013,Z.-W.Sun proved that for any positive integer n,the least positive integer m with k(2k-1)(k=1,...,n)pairwise incongruent modulo m is the least power of two not smaller than n.In contrast,we obtain the following result.Theorem 3.Let a be a positive integer.For each positive integer n,let S(n)be the least positive integer m such that k(2?k+1)(k=1...,n)are pairwise incongruent modulo m.If n?2?+5,then S(n)is the least power of 2 not smaller than n.