International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 4, Pages 743-750
doi:10.1155/S0161171288000900
Some congruence properties of binomial coefficients and linear second order recurrences
Department of Mathematics, San Francisco State University, San Francisco 94132, CA, USA
Received 21 May 1987
Copyright © 1988 Neville Robbins. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Using elementary methods, the following results are obtained:(I) If p is prime, 0≤m≤n, 0<b<apn−m, and p∤ab, then (apnbpm)≡(−1)p−1(apbn−m)(modpn); If r, s are the roots of x2=Ax−B, where (A,B)=1 and D=A2−4B>0, if un=rn−snr−s, vn=rn+sn, and k≥0, then (II) vkpn≡vkpn−1(modpn); (III) If p is odd and p∤D, then ukpn≡(Dp)ukpn−1(modpn); (IV) uk2n≡(−1)Buk2n−1(mod2n).