关于Fleck定理
On Fleck quotients
摘要：
令$p$是素数，令$n>0$和$r$是整数。本文研究了Fleck商$$F_p(n,r)=(-p)^{-\lfloor(n-1)/(p-1)\rfloor} \sum_{k=r(mod p)}\binom {n}{k}(-1)^k\in Z.$$，用数论方法和组合方法完全确定了$F_p(n,r)$ mod $p$，因此，如果$2\le n\le p$则$$\sum_{k=1}^n(-1)^{pk-1}\binom{pn-1}{pk-1} \equiv(n-1)!B_{p-n}p^n (mod p^{n+1}),$$，其中$B_0,B_1,...$是Bernoulli数.建立了$a=1,2,3,...$的Kummer型同余$F_p(n+p^a(p-1),r)\equiv F_p(n,r) (mod p^a)$，并揭示了Fleck同余式与二次域$\Q(\sqrt{\pm p})$和$p$次分圆域$\Q(ζ_p)$类数之间的关系.此外，本文还研究了广义Fleck连续性.
英文摘要：
Let $p$ be a prime, and let $n>0$ and $r$ be integers. In this paper we study Fleck's quotient $$F_p(n,r)=(-p)^{-\lfloor(n-1)/(p-1)\rfloor} \sum_{k=r(mod p)}\binom {n}{k}(-1)^k\in Z.$$ We determine $F_p(n,r)$ mod $p$ completely by certain number-theoretic and combinatorial methods; consequently, if $2\le n\le p$ then $$\sum_{k=1}^n(-1)^{pk-1}\binom{pn-1}{pk-1} \equiv(n-1)!B_{p-n}p^n (mod p^{n+1}),$$ where $B_0,B_1,...$ are Bernoulli numbers. We also establish the Kummer-type congruence $F_p(n+p^a(p-1),r)\equiv F_p(n,r) (mod p^a)$ for $a=1,2,3,...$, and reveal some connections between Fleck's quotients and class numbers of the quadratic fields $\Q(\sqrt{\pm p})$ and the $p$-th cyclotomic field $\Q(ζ_p)$. In addition, generalized Fleck quotients are also studied in this paper.
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