On Some Congruences Involving Central Binomial Coefficients

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* ∑ₖ₌₁ᵖ⁻¹2kkk2ᵏ≡-12H_(p-1)/2+716p²Bₚ₋₃p³, align* where Hₙ is the n th harmonic number and Bₙ is the n th Bernoulli number. In addition, we evaluate ∑ ₖ₌₀ᵖ⁻¹(ak+b) 2kk/2ᵏ modulo p³ for any p -adic integers $a, b$ .