p-adic congruences motivated by series

Details p-adic congruences motivated by series p-adic congruences motivated by series

2011-11-21

arxiv.org/abs/1111.4988v3

Mathematics

Number Theory

15 pages. Polished version

Scientific paper

Let p>5 be a prime. Motivated by the known formulae $\sum_{k=1}^\infty(-1)^k/(k^3\binom{2k}{k})=-2\zeta(3)/5$ and $\sum_{k=0}^\infty \binom{2k}{k}^2/((2k+1)16^k)=4G/\pi$$ (where $G=\sum_{k=0}^\infty(-1)^k/(2k+1)^2$ is the Catalan constant), we show that $$\sum_{k=1}^{(p-1)/2}(-1)^k/(k^3\binom{2k}{k})=-2B_{p-3} (mod p),$$ $$\sum_{k=(p+1)/2}^{p-1}\binom{2k}{k}^2/((2k+1)16^k)=-7p^2B_{p-3}/4 (mod p^3)$$, and $$\sum_{k=0}^{(p-3)/2}\binom{2k}{k}^2/((2k+1)16^k)=-2q_p(2)-p*q_p(2)^2+5p^2B_{p-3}/12 (mod p^3),$$ where $B_0,B_1,...$ are Bernoulli numbers and $q_p(2)$ is the Fermat quotient $(2^{p-1}-1)/p$.

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