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International Journal of Mathematics and Mathematical Sciences
Volume 11, Issue 4, Pages 743-750
http://dx.doi.org/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 Hindawi Publishing Corporation. 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, 0mn, 0<b<apnm, and pab, then (apnbpm)(1)p1(apbnm)(modpn); If r, s are the roots of x2=AxB, where (A,B)=1 and D=A24B>0, if un=rnsnrs, vn=rn+sn, and k0, then (II) vkpnvkpn1(modpn); (III) If p is odd and pD, then ukpn(Dp)ukpn1(modpn); (IV) uk2n(1)Buk2n1(mod2n).