New congruences for central binomial coefficients

Let p be a prime and let a   be a positive integer. In this paper we determine ∑k=0pa−1(2kk+d)/mk and ∑k=1p−1(2kk+d)/(kmk−1) modulo p   for all d=0,…,pad=0,…,pa, where m is any integer not divisible by p  . For example, we show that if p≠2,5p≠2,5, then∑k=1p−1(−1)k(2kk)k≡−5Fp−(p5)p(modp), where FnFn is the n  th Fibonacci number and (−)(−) is the Jacobi symbol. We also prove that if p>3p>3, then∑k=1p−1(2kk)k≡89p2Bp−3(modp3), where BnBn denotes the nth Bernoulli number.