The sum $\sum_{k=0}^{q-1}\binom{2k}{k}$ for q a power of 3

Details The sum $\sum_{k=0}^{q-1}\binom{2k}{k}$ for q a power of 3 The sum $\sum_{k=0}^{q-1}\binom{2k}{k}$ for q a power of 3

2010-01-13

arxiv.org/abs/1001.2156v1

Mathematics

Number Theory

10 pages

Scientific paper

We prove that $\sum_{k=0}^{q-1}\binom{2k}{k}\equiv q^2\pmod{3q^2}$ if q>1 is a power of 3, as recently conjectured by Z.W. Sun and R. Tauraso. Our more precise result actually implies that the value of $(1/q^2)\sum_{k=0}^{q-1}\binom{2k}{k}$ modulo a fixed arbitrary power of 3 is independent of q, for q a power of 3 large enough, and shows how such value can be efficiently computed.

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