# Can someone explain this theorem from von Staudt on denominators of Bernoulli numbers.

Anonymous

This is an extract from Paulo Ribenboim: 13 lectures on Fermats last theorem on page 105. 'In 1845, von Staudt determined some factors of the numerator N_2k. Let 2k = k1k2 with gcd(k1,k2)= 1 such that p|k2 if and only if p|D_2k'. Where N_2k and D_2k are the numerators and denominators of Bernoulli number B_2k. So I've actually used the result of this theorem for some other proof, but looking back at it I find it is not true. For example when 2k=74, then 2k=2x37. If we take p=37, we see that 37|k2=37 and so 37 must divide the denominator D_74 but D_74=6. Im not sure what I'm missing here. Maybe, I have misinterpreted the theorem. Could someone clear this up for me.

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There are several ways to write 74 as a product: $74 = 1 \times 74 = 2 \times 37 = 37 \times 2 = 74 \times 1$.

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