2015/01/03 by Victor J. W. Guo, Guo, Victor J. W., Ji-Cai Liu +1
Computer Science · Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Coding theory and cryptography #math.CO #math.NT #msc:05A10 #msc:11A07 #msc:11B65
paper · pdf · doi:10.48550/arxiv.1501.00573
18 pages
arxiv created 2015/01/03 · arxiv updated 2015/01/06
The numbers Rn and Wn are defined as Rn=∑k=0nn+k\choose 2k2k\choose k(1)/(2k-1), and Wn=∑k=0nn+k\choose 2k2k\choose k(3)/(2k-3). We prove that, for any positive integer n and odd prime p, there hold ∑k=0n-1(2k+1)Rk2 ≡ 0 \pmodn,
∑k=0p-1(2k+1)Rk2 ≡ 4p(-1)(p-1)/(2) -p2 \pmodp3,
9∑k=0n-1(2k+1)Wk2 ≡ 0 \pmodn,
∑k=0p-1(2k+1)Wk2 ≡ 12p(-1)(p-1)/(2)-17p2 \pmodp3, \quadif p>3. The first two congruences were originally conjectured by Z.-W. Sun. Our proof is based on the multi-variable Zeilberger algorithm and the following observation: 2n\choose nn\choose km\choose kk\choose m-n≡ 0\pmod2k\choose k2m-2k\choose m-k, where 0\leqslant k\leqslant n\leqslant m \leqslant 2n.