Timeline for Machin-like formulas for logarithms
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2013 at 18:13 | comment | added | Fredrik Johansson | These findings are very interesting (author of the blog here). The 4-term formula is some 10% faster than the 3-term formula in practice, which is not really that much but still quite nice. Getting a longer list of primes is very useful though. I can't believe I missed the section in Joerg's (great) book! | |
| Mar 27, 2013 at 19:59 | comment | added | Douglas Zare | That's interesting, and that book tests the idea of considering other sets of primes, using $x^2+1$ to force the primes to be $1 \mod 4$, for example. For the set of $13$ denominators $\lbrace 51744295,...,3222617399\rbrace$ on page 632, however, I get an efficiency of $0.644$, worse than the examples with 4 through 7 primes, so it is slower if you only want to compute $\lbrace \log2, \log3, \log5\rbrace$. The advantage is that you simultaneously compute the logarithms of the next $10$ primes, too. | |
| Mar 27, 2013 at 17:10 | comment | added | meij | relevant oeis oeis.org/A175607 in particular section 32.4 of jjj.de/fxt/fxtbook.pdf "Simultaneous computation of logarithms of small primes" | |
| Mar 27, 2013 at 3:17 | comment | added | Douglas Zare | $\operatorname{arctanh}\frac{1}{n} = \frac{1}{2}(\log(n+1)-\log(n-1))$. If $n-1$ and $n+1$ factor into powers of $2$, $3$, and $5$, for example, then we can express $\operatorname{arctanh}\frac{1}{n}$ in terms of $\log2$, $\log3$, and $\log5$. | |
| Mar 27, 2013 at 3:05 | comment | added | Carl Feynman | You mention that you are looking for numbers "sandwiched between smooth numbers." What is the importance of smooth numbers for this problem? | |
| Mar 27, 2013 at 2:40 | history | edited | meij | CC BY-SA 3.0 |
fix a single digit typo
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| Mar 27, 2013 at 2:05 | history | answered | Douglas Zare | CC BY-SA 3.0 |