Supercongruences for central trinomial coefficients
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
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2020
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| _version_ | 1866912963725623296 |
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| author | Pan, Hao Sun, Zhi-Wei |
| author_facet | Pan, Hao Sun, Zhi-Wei |
| contents | For each $n=0,1,2,\ldots$, the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. Let $p>3$ be a prime, and let $n$ be any positive integer. In 2016, the second author conjectured that the quotient $(T_{pn}-T_n)/(pn)^2$ is always a $p$-adic integer. In this paper, we confirm this conjecture, and further prove that $$\frac{T_{pn}-T_n}{(pn)^2}\equiv\frac{T_{n-1}}6\left(\frac p3\right)B_{p-2}\left(\frac13\right)\pmod p,$$ where $(\frac p3)$ is the Legendre symbol and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_05121 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Supercongruences for central trinomial coefficients Pan, Hao Sun, Zhi-Wei Number Theory Combinatorics For each $n=0,1,2,\ldots$, the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. Let $p>3$ be a prime, and let $n$ be any positive integer. In 2016, the second author conjectured that the quotient $(T_{pn}-T_n)/(pn)^2$ is always a $p$-adic integer. In this paper, we confirm this conjecture, and further prove that $$\frac{T_{pn}-T_n}{(pn)^2}\equiv\frac{T_{n-1}}6\left(\frac p3\right)B_{p-2}\left(\frac13\right)\pmod p,$$ where $(\frac p3)$ is the Legendre symbol and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$. |
| title | Supercongruences for central trinomial coefficients |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2012.05121 |