Published in
Elsevier, Journal of Number Theory, (134), p. 181-196, 2014
DOI: 10.1016/j.jnt.2013.07.011
Links
- Elsevier
- [arxiv.org] | PDF
- [www.researchgate.net] | PDF
- [citeseerx.ist.psu.edu] | PDF
- [citeseerx.ist.psu.edu] | PDF
- [math.nju.edu.cn] | PDF
- [math.nju.edu.cn] | PDF
Tools
p-adic congruences motivated by series
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Abstract
Let $p>5$ be a prime. Motivated by the known formulae $∑_{k=1}^∞(-1)^k/(k^3\binom{2k}{k})=-2ζ(3)/5$ and $∑_{k=0}^∞ \binom{2k}{k}^2/((2k+1)16^k)=4G/π$$ (where $G=∑_{k=0}^∞(-1)^k/(2k+1)^2$ is the Catalan constant), we show that $∑_{k=1}^{(p-1)/2}\frac{(-1)^k}{k^3\binom{2k}{k}}≡-2B_{p-3}\pmod{p},$ $∑_{k=(p+1)/2}^{p-1}\frac{\binom{2k}{k}^2}{(2k+1)16^k}≡-\frac 7{4}p^2B_{p-3}\pmod{p^3}$, and $∑_{k=0}^{(p-3)/2}\frac{\binom{2k}{k}^2}{(2k+1)16^k} ≡-2q_p(2)-pq_p(2)^2+\frac{5}{12}p^2B_{p-3}\pmod{p^3},$ where $B_0,B_1,…$ are Bernoulli numbers and $q_p(2)$ is the Fermat quotient $(2^{p-1}-1)/p$. ; Comment: 15 pages, final published version