Some congruence properties of binomial coefficients and linear second order recurrences

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).