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 φ(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.