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Bernoulli Numbers Modulo 2~r and 3~r

SUN Zhi hong(Department of Mathematics, Huaiyin Teachers College, Huaian 223001, China)  
Let {B n} be the Bernoulli numbers, and let m and n be natural numbers. In this paper we prove that (2-2 2n )B 2n ≡1-4n+∑mk=12n2k2 4k B 2k (mod 2 4m+3 ) and (3-3 2n )B 2n ≡2-6n+2∑mk=12n2k3 2k B 2k (mod 3 2m+1 ).Taking m=1,2 we obtain the two congruences for (2-2 2n )B 2n (mod 2 7) and (3-3 2n )B 2n (mod 3 5),which were announced in .
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