Several Congruences Related to Harmonic Numbers

Abstract

Let p be a prime greater than or equal to 5. In this paper, by using the harmonic numbers and Fermat quotient we establish congruences involving the sums p−1 X2 k=1 µ k r ¶ Hk, p−1 X2 k=1 ¡ 2k k ¢2 16k H (2) k and p−1 X2 k=1 1 4 k µ 2k k ¶ H (3) k . For example, p−1 X2 k=0 ¡ 2k k ¢2 16k H (2) k ≡ 4E2p−4 − 8Ep−3 ¡ mod p 2 ¢ , where H (m) k are the generalized harmonic numbers of order m and En are Euler numbers