Some congruence properties of binomial coefficients and linear second order recurrences
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...
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| Language: | English |
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Wiley
1988-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171288000900 |
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| author | Neville Robbins |
| author_facet | Neville Robbins |
| author_sort | Neville Robbins |
| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-bb779df255fd45d5a0caf6156615d70f |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1988-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-bb779df255fd45d5a0caf6156615d70f2025-02-03T05:44:31ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251988-01-0111474375010.1155/S0161171288000900Some congruence properties of binomial coefficients and linear second order recurrencesNeville Robbins0Department of Mathematics, San Francisco State University, San Francisco 94132, CA, USAUsing 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).http://dx.doi.org/10.1155/S0161171288000900binomial coefficientlinear second order recurrence. |
| spellingShingle | Neville Robbins Some congruence properties of binomial coefficients and linear second order recurrences International Journal of Mathematics and Mathematical Sciences binomial coefficient linear second order recurrence. |
| title | Some congruence properties of binomial coefficients and linear second order recurrences |
| title_full | Some congruence properties of binomial coefficients and linear second order recurrences |
| title_fullStr | Some congruence properties of binomial coefficients and linear second order recurrences |
| title_full_unstemmed | Some congruence properties of binomial coefficients and linear second order recurrences |
| title_short | Some congruence properties of binomial coefficients and linear second order recurrences |
| title_sort | some congruence properties of binomial coefficients and linear second order recurrences |
| topic | binomial coefficient linear second order recurrence. |
| url | http://dx.doi.org/10.1155/S0161171288000900 |
| work_keys_str_mv | AT nevillerobbins somecongruencepropertiesofbinomialcoefficientsandlinearsecondorderrecurrences |