The p-Adic Valuations of Sums of Binomial Coefficients
In this paper, we prove three supercongruences on sums of binomial coefficients conjectured by Z.-W. Sun. Let p be an odd prime and let h∈ℤ with 2h−1≡0modp. For a∈ℤ+ and pa>3, we show that ∑k=0pa−1hpa−1k2kk−h/2k≡0modpa+1. Also, for any n∈ℤ+, we have νp∑k=0n−1hn−1k2kk−h/2k≥νpn, where νpn denotes t...
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Wiley
2021-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/9570350 |
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| author | Yong Zhang Peisen Yuan |
| author_facet | Yong Zhang Peisen Yuan |
| author_sort | Yong Zhang |
| collection | DOAJ |
| description | In this paper, we prove three supercongruences on sums of binomial coefficients conjectured by Z.-W. Sun. Let p be an odd prime and let h∈ℤ with 2h−1≡0modp. For a∈ℤ+ and pa>3, we show that ∑k=0pa−1hpa−1k2kk−h/2k≡0modpa+1. Also, for any n∈ℤ+, we have νp∑k=0n−1hn−1k2kk−h/2k≥νpn, where νpn denotes the p-adic order of n. For any integer m≡0modp and positive integer n, we have 1/pn∑k=0pn−1pn−1k2kk/−mk−mm−4/p∑k=0n−1n−1k2kk/−mk∈ℤp, where −˙ is the Legendre symbol and ℤp is the ring of p-adic integers. |
| format | Article |
| id | doaj-art-ef53d42ab0704d46bbdfef7e53431566 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-ef53d42ab0704d46bbdfef7e534315662025-02-03T01:04:18ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/95703509570350The p-Adic Valuations of Sums of Binomial CoefficientsYong Zhang0Peisen Yuan1Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, ChinaCollege of Artificial Intelligence, Nanjing Agricultural University, Nanjing 210095, ChinaIn this paper, we prove three supercongruences on sums of binomial coefficients conjectured by Z.-W. Sun. Let p be an odd prime and let h∈ℤ with 2h−1≡0modp. For a∈ℤ+ and pa>3, we show that ∑k=0pa−1hpa−1k2kk−h/2k≡0modpa+1. Also, for any n∈ℤ+, we have νp∑k=0n−1hn−1k2kk−h/2k≥νpn, where νpn denotes the p-adic order of n. For any integer m≡0modp and positive integer n, we have 1/pn∑k=0pn−1pn−1k2kk/−mk−mm−4/p∑k=0n−1n−1k2kk/−mk∈ℤp, where −˙ is the Legendre symbol and ℤp is the ring of p-adic integers.http://dx.doi.org/10.1155/2021/9570350 |
| spellingShingle | Yong Zhang Peisen Yuan The p-Adic Valuations of Sums of Binomial Coefficients Journal of Mathematics |
| title | The p-Adic Valuations of Sums of Binomial Coefficients |
| title_full | The p-Adic Valuations of Sums of Binomial Coefficients |
| title_fullStr | The p-Adic Valuations of Sums of Binomial Coefficients |
| title_full_unstemmed | The p-Adic Valuations of Sums of Binomial Coefficients |
| title_short | The p-Adic Valuations of Sums of Binomial Coefficients |
| title_sort | p adic valuations of sums of binomial coefficients |
| url | http://dx.doi.org/10.1155/2021/9570350 |
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