On The Research Of Two Kinds Of Congruence

Let Z+and Z be the set of all positive integers and the set of all integers respectively.Let p be a prime.Assuming that n,k,r ? Z+ and h?Z.In 1953,Erdo's and Moser conjectured that(n,k)=(1,2)is the only positive integer solution of diophantine equation In+2n+…+kn=(k+1)n.Moreover,Moser also proved that the diophantine equation has no trivial solution when n is odd.In the present paper,the author proves that if the congruence In+2n+…+kn?(k+1)n(mod k3)is true,then for any prime number p of k with gcd(6,k)=1 and p | n(n-1),the congruence#12is also true.By using some properties of Bernoulli numbers and Bernoulli polynomials,the author also gives the necessary conditions that the congruence 1n+2n+···+kn?(k+1)n(modk4)is true.xn+(x+1)n+…+(x+h)n=(x+h+1)n is a centuries old theory problem which has't yet been solved completely up to now.In 1900,E.B.Escott proved that when the condition 2?n?5 is satisfied,there is no other positive integer solution to the diophantine equation except for the following cases(x,h,n)=3,1,2;(x,h,n)=3,2,3.By using of the congruent theory and the classified discussion,the necessary and sufficient conditions under which when ?(pr)divides 2n,the congruence equation x2n+(x+1)2n +…+(x+h)2n?(x+h+1)2n(mod pr)has solutions,are obtained.Under the condition of there existing solutions to the congruence equation,all positive integer solutions of the congruence equation are put forth.