On Fleck quotients

Details On Fleck quotients On Fleck quotients

2006-03-20

arxiv.org/abs/math/0603462v3

Acta Arith. 127(2007), no.4, 337-363

Mathematics

Number Theory

28 pages

Scientific paper

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 $\Q(\zeta_p)$. In addition, generalized Fleck quotients are also studied in this paper.

Affiliated with

Also associated with

No associations

Rating

On Fleck quotients does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On Fleck quotients, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Fleck quotients will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-450959