一类含有调和数的同余式（英文）

Several congruences involvingharmonic numbers

该文目的是创建一系列含有调和数的同余式.当p>3为一素数时,利用已有的组合恒等式和同余式,得到了如下的同余式:∑p-1k=1k^2H_k^2≡79/108p-4/9(mod p^2)和∑p-1k=1H_k^3≡23/18(mod p).同时也得到了∑(p-1)/2k=1H_k^2/k≡-8/3q_p^3(2)+1/6B(p-3)(mod p)和∑(p-1)/2k=1H_(2k)~2≡-1+1/2q_p^2(2)(mod p),这里Bn(n∈N)称为Bernoulli数,当pa时,q_p(a)=(a^(p-1)-1)/p称为Fermat商.

The purpose of this paper is to establish several congruences involving harmonic numbers. Let p〉3 be a prime. With the help of some combinatorial identities and congruences, the following congruences is generated：k=1∑p-1k^2Hk^2≡79/108p-4/9（nod p^2 ） and k=1∑p-1k^2Hk^3≡23/18（mod p）.The congruences are also established as k=1∑（p-1）/2Hk^2/k≡-8/3qp^3（2）＋1/6Bp-3（mod p） and k=1∑（p-1）/2H2k^2/k≡-1＋1/2qp^2（2）（rood p）,where Bn（n ∈ N） are Bernoulli numbers and qp（a） = （a^p-1 -1）/p is usually called a Fermat quotient provided p a.