The Power And The Sequence Of Binomial Coefficients Of Congruence Nature Of Its Applications

It is a very important field in number theory for the recurrence of sums of powers of binominal coefficients, since it is a powerful tool to get and investigate irrational number. In this paper, I study some congruenceproperties of the sums an (r, s) = and bn (r, i) = which can be used to the investigation of the polynomial recurrence of these sums.The main purpose of this paper as follows:For any prime p>2, k,r,s ∈N, and 0≤t≤p-1,t∈Z we have:And if 1≤t≤(p-1)/2, 2|(r + s) we have:For any prime p>2, k∈N and positive integer r,i,t such that: r ≥ 2,1 ≤i≤ r- 1, t ≤ p -1, then, we have:b_kp(r,i)≡0 (mod p) and b_kp+t(r,i)≡ a_k(r,i) b,(r,i) (mod p)...