Several Congruences Related to Harmonic Numbers

摘要

Let p be a prime greater than or equal to 5. In this paper, by using the harmonic numbers and Fermat quotient we establish congruencesinvolving the sumsp−1 X2k=1µkr¶Hk,p−1 X2k=1¡2kk¢216k H(2)kandp−1 X2k=114kµ2kk¶H(3)k.For example,p−1 X2k=0¡2kk¢216k H(2)k ≡ 4E2p−4 − 8Ep−3¡mod p2¢,where H(m)kare the generalized harmonic numbers of order m and En areEuler numbers.