ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 04 Aug 2019 03:48:15 -0500Can someone explain this theorem from von Staudt on denominators of Bernoulli numbers.https://ask.sagemath.org/question/47312/can-someone-explain-this-theorem-from-von-staudt-on-denominators-of-bernoulli-numbers/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.Tue, 30 Jul 2019 08:17:06 -0500https://ask.sagemath.org/question/47312/can-someone-explain-this-theorem-from-von-staudt-on-denominators-of-bernoulli-numbers/Answer by slelievre for <p>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.</p>
https://ask.sagemath.org/question/47312/can-someone-explain-this-theorem-from-von-staudt-on-denominators-of-bernoulli-numbers/?answer=47376#post-id-47376There are several ways to write 74 as a product: $74 = 1 \times 74 = 2 \times 37 = 37 \times 2 = 74 \times 1$.Sun, 04 Aug 2019 03:48:15 -0500https://ask.sagemath.org/question/47312/can-someone-explain-this-theorem-from-von-staudt-on-denominators-of-bernoulli-numbers/?answer=47376#post-id-47376