NOTES ON GLAISHER＇S CONGRUENCES

Let p be an odd prime and let n≥1,k≥0 and r be integers,denote by Bk the kth Bernoulli number,It is proved that(i) If r≥1 is odd and suppose 1≥r+4,then ∑j=1^p-1 1/(np+j)^r=-(2n+1)r(r+1)/2(r+2)Bp-r-2p^2(mod p^3).(ii)If r≥2 is even and suppose p≥r+3, then p-1∑j=1 1/(np+j)^r=r/r+1Bv-r-1p(mod P^2).(iii) p-1∑j=1 1/(np+j)p-2=-(2n+1)p(mod P^2).This result generalizes the Glaisher‘s congruence. As a corollary, a generalization of the Wolsten-holme‘s theorem is obtained.

NOTES ON GLAISHER'S CONGRUENCES

Letp be an odd prime and letn ≥ l,k ≥ 0 andr be integers. Denote byB _k thekth Bernoulli number. It is proved that (i) Ifr ≥ 1 is odd and supposep ≥ r + 4, then . (ii) If r ≥ 2 is even and supposep ≥ r + 3, then . (iii) . This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolsten-holme’s theorem is obtained. Project supported by the Postdoctoral Foundation of China.