On sums of binomial coefficients modulo p²

Zhi-Wei Sun

Zhi-Wei Sun

Colloquium Mathematicae, Tome 126 (2012), p. 39-54 / Harvested from The Polish Digital Mathematics Library

Résumé

Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $\binom{\sum_{k = 0}^{p^{a} - 1} (h p^{a} - 1}{\binom{k) (2 k}{k) / m^{k} (m o d p ²)}}$ , where h and m are p-adic integers with m ≢ 0 (mod p). For example, we show that if h ≢ 0 (mod p) and $p^{a} > 3$ , then $\binom{\sum_{k = 0}^{p^{a} - 1} (h p^{a} - 1}{\binom{k) (2 k}{{k) (- h / 2)}^{k} \equiv ((1 - 2 h) / (p^{a})) (1 + h ({(4 - 2 / h)}^{p - 1} - 1)) (m o d p ²)}}$ , where (·/·) denotes the Jacobi symbol. Here is another remarkable congruence: If $p^{a} > 3$ then $\binom{\sum_{k = 0}^{p^{a} - 1} (p^{a} - 1}{\binom{k) (2 k}{{k) (- 1)}^{k} \equiv 3^{p - 1} (p^{a} / 3) (m o d p ²)}}$ .

Publié le : 2012-01-01

Zbl 1266.11035

EUDML-ID : urn:eudml:doc:284015

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     title = {On sums of binomial coefficients modulo p$^2$},
     journal = {Colloquium Mathematicae},
     volume = {126},
     year = {2012},
     pages = {39-54},
     zbl = {1266.11035},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm127-1-3}
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Zhi-Wei Sun. On sums of binomial coefficients modulo p². Colloquium Mathematicae, Tome 126 (2012) pp. 39-54. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm127-1-3/