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In this note we use some
-congruences proved by the method of ‘creative microscoping’ to prove two conjectures on supercongruences involving central binomial coefficients. In particular, we confirm the
case of Conjecture 1.1 of Guo [‘Some generalizations of a supercongruence of Van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].
which was originally conjectured by Long and later proved by Swisher. This confirms a conjecture of the second author [‘A $q$-analogue of the (L.2) supercongruence of Van Hamme’, J. Math. Anal. Appl.466 (2018), 749–761].
We prove some congruences on sums involving fourth powers of central q-binomial coefficients. As a conclusion, we confirm the following supercongruence observed by Long [Pacific J. Math. 249 (2011), 405–418]: