• Proof of a supercongruence conjectured by Sun through a q-microscope

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    • Keywords


      Cyclotomic polynomial; $q$-binomial coefficient; $q$-congruence; super congruence; creative microscoping.

    • Abstract


      In 2011, Sun (Sci. China Math. 54 (2011) 2509--2535) made the following conjecture: for any odd prime $p$ and odd integer $m$,

      $$\frac{1}{m^2\left({m-1}\atop{(m-1)/2}\right)}\left(\sum^{(mp-1)/2}_{k=0}\frac{{\left(2k\atop k\right) k}}{8^k}-\left(\frac{2}{p}\right)\sum^{(m-1)/2}_{k=0}\frac{\left({2k}\atop{k}\right)}{8^k}\right)\equiv 0 ({\rm mod}\, p^2).$$

      By applying the creative microscoping method introduced by Guo and Zudilin (Adv. Math. 346 (2019) 329--35), we confirm the above conjecture of Sun.

    • Author Affiliations


      Victor J W Guo1

      1. School of Mathematics and Statistics, Huaiyin Normal University, Huai’an 223300, Jiangsu, People’s Republic of China
    • Dates

  • Proceedings – Mathematical Sciences | News

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