Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then...Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.展开更多
文摘Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.
