On Some Congruences Involving Central Binomial Coefficients

Abstract We prove the following conjecture of Z. -W. Sun ‘On congruences related to central binomial coefficients’, J. Number Theory 13 (11) (2011), 2219–2238. Let p be an odd prime. Then align* ₊=₁^p-12kkk2ᵏ-12H (₏-₁) /2+716p²B₏-₃p³, align* where Hₙ is the n th harmonic number and Bₙ is the n th Bernoulli number. In addition, we evaluate ₊=₀^p-1 (ak+b) 2kk/2ᵏ modulo p³ for any p -adic integers a, b.