- LandOfFree
- Scientists
- Mathematics
- Number Theory
Details
On congruences related to central binomial coefficients
On congruences related to central binomial coefficients
2009-11-12
-
arxiv.org/abs/0911.2415v16
J. Number Theory 131(2011), no.11, 2219-2238
Mathematics
Number Theory
Scientific paper
It is known that $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)4^k)=\pi/2$ and $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)16^k)=\pi/3$. In this paper we obtain their p-adic analogues such as $$\sum_{p/23 is a prime and E_0,E_1,E_2,... are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we have $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=4x^2-2p (mod p^2)$$ if (p/7)=1 and p=x^2+7y^2 with x,y integers, and $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=0 (mod p^2)$$ if (p/7)=-1, i.e., p=3,5,6 (mod 7).
Affiliated with
Also associated with
No associations
LandOfFree
Say what you really think
Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.
Rating
On congruences related to central binomial coefficients 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 congruences related to central binomial coefficients, we encourage you to share that experience with our LandOfFree.com community.
Your opinion is very important and On congruences related to central binomial coefficients will most certainly appreciate the feedback.
Rate now
Profile ID: LFWR-SCP-O-150901
All data on this website is collected from public sources.
Our data reflects the most accurate information available at the time of publication.