论文标题
中央三项系数的超企业
Supercongruences for central trinomial coefficients
论文作者
论文摘要
对于每个$ n = 0,1,2，\ ldots $ central Trinomial系数$ t_n $是$（x^2+x+x+1）^n $的$ x^n $系数。在2016年，第二作者猜想，对于任何prime $ p> 3 $和正整数$ n $ the商$（t_ {pn} -t_n）/（pn）^2 $是$ p $ - adic整数。在本文中，我们确认了这一猜想，并进一步证明了$ \ frac {t_ {pn} -t_n} {（pn）^2} \ equiv \ equiv \ equiv \ equiv \ equ { p3）$是legendre符号，$ b_ {p-2}（x）$是bernoulli度$ p-2 $的bernoulli多项式。
For each $n=0,1,2,\ldots$ the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. In 2016 the second author conjectured that for any prime $p>3$ and positive integer $n$ the quotient $(T_{pn}-T_n)/(pn)^2$ is a $p$-adic integer. In this paper we confirm this conjecture and prove further 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 $p-2$.
