New congruences for central binomial coefficients

الأصل · EN

Let p be a prime and let a be a positive integer. In this paper we determine ∑ₖ₌₀ᵖᵃ⁻¹2kk+d/mᵏ and ∑ₖ₌₁ᵖ⁻¹2kk+d/(kmᵏ⁻¹) modulo p for all d=0,...,pᵃ, where m is any integer not divisible by p. For example, we show that if p=2,5 then ∑ₖ₌₁ᵖ⁻¹(-1)ᵏ2kkk=-5Fp-(p5)p (mod p), where Fₙ is the n-th Fibonacci number and (-) is the Jacobi symbol. We also prove that if p>3 then ∑ₖ₌₁ᵖ⁻¹2kkk=8/9 p²Bₚ₋₃ (mod p³), where Bₙ denotes the n-th Bernoulli number.