Proof of a supercongruence conjectured by Sun through a $${\varvec{q}}$$ -microscope
Tóm tắt
In 2011, Sun (Sci. China Math. 54 (2011) 2509–2535) made the following conjecture: for any odd prime p and odd integer m,
$$\begin{aligned}&\qquad \frac{1}{m^2{m-1\atopwithdelims ()(m-1)/2}}\Bigg (\sum _{k=0}^{(mp-1)/2}\frac{{2k\atopwithdelims ()k}}{8^k} -\left( \frac{2}{p}\right) \sum _{k=0}^{(m-1)/2}\frac{{2k\atopwithdelims ()k}}{8^k}\Bigg ) \equiv 0\pmod {p^2}. \end{aligned}$$
By applying the, creative microscoping, method introduced by Guo and Zudilin (Adv. Math. 346 (2019) 329–35), we confirm the above conjecture of Sun.
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