Proofs Of Some Conjectures On Congruences

In this paper,we prove some conjectures on congruences modulo prime powers via Bernoulli numbers and harmonic numbers.For any prime p>3,we show the following results:(i). (mod p), where qp(2)denotes the Fermat quotient(2p-1-1)/p,this is a congruence analogue of the identity∑k=0+∞1/(2k+1)9k=(3ln 2)/2.(ⅱ) (mod p), where(一)is the Legendre symbol and the central trinomial coefficient Tk is the coefficient of xk in(x2+x+1)k.(iii). (mod p4), and (mod p5).