I recently did a column on approximations. http://www.maa.org/editorial/mathgames/mathgames_02_14_05.html I've gotten a lot of results, including an article on class polynomials: http://www.geocities.com/titus_piezas/Approximations.htm The following is a TeX document that includes all of the best approximations I've received. I plan to split off the Class numbers into their own table, as I did with the approximations using Sin/Cos. I plan to exclude Sin/Cos in the next version of the rules. I did a little contest on my site -- whoever sends me the approximation with the highest keenness, on or before 15 March, wins $50 from me. It's looking like Derek Ross and Titus Piezas will be winning it, at the moment. --Ed Pegg Jr \documentclass{amsart} \usepackage{graphicx} \begin{document} \begin{tabular}{|c|c|c|c|l|} \hline Keenness & Com & Dscv & Result & Function\\ \hline $2.202$ & 13 & CH & $163 + 2.321 \times 10^{-29}$ & $ ( {\log(640320^3+744) / \pi} )^2$\\ $2.115$ & 22 & CH & $\pi - 9.303 \times 10^{-47}$ & ${\log((640320^3+744)^2-393768)\over\sqrt{652}}$\\ $2.020$ & 6 & CH & $640320^3 + 744 - 7.5003 \times10^{-13}$ & $ e^{\sqrt{163} \pi } $\\ $1.732$ & 7 & TP & $640320^3 + 744 - 7.4888 \times10^{-13}$ & $ (x^3-6x^2+4x-2)_1^{24}-24$\\ $1.846$ & 4 & DR & 5.999999958593812545675 $ & $ \pi - F_{\alpha} + 6^{- \gamma}$\\ $1.716$ & 4 & DR & $45.0000001373108134890 $ & $ \frac{1}{4}+3^{\pi + 1/ \pi }$\\ $1.679$ & 3 & RS & $e + 0.0000091665339017 $ & $ 2^{.4^{-.4}}$\\ $1.575$ & 4 & MH & $G-4.2542 \times 10^{-7}$ & $ \sqrt[52]{\frac{1}{96}} $\\ $1.546$ & 4 & DR & $12.0000006524114304506 $ & $ ( 1/ \gamma + 3 \gamma )^2$\\ $1.512$ & 4 & DR & $49.9999991061598799437 $ & $ 7^{\phi^{e^{K^{-1}}}}$\\ $1.506$ & 5 & DR & $2.99999997062451090800 $ & $ \frac{1+5 \sqrt[5]{6}}{e}$\\ $1.501$ & 5 & DR & $14212169.0000000311917 $ & $ (\phi +\pi ) 12^6 $\\ $1.499$ & 4 & DR & $1.00000101156823755142 $ & $ \frac{1}{8}+\sqrt[7]{\frac{\pi}{8}}$\\ $1.484$ & 5 & DR & $110.999999961886583322 $ & $ \sqrt[e]{9!+\frac{\pi }{3}} $\\ $1.471$ & 5 & DC & $5.99999995619189332962 $ & $ \log \left(\pi ^4+\pi ^5\right) $\\ $1.468$ & 4 & DR & $2.00000134678219335309 $ & $ (4+1 / \phi )^{e/6}$\\ $1.449$ & 5 & DR & $9112774.00000005695228 $ & $ \left(52+\frac{8}{e}\right)^4 $\\ $1.443$ & 6 & DR & $45.9999999978124612031 $ & $ 3^{(20+\gamma)/6}+e$\\ $1.414$ & 5 & DR & $4.99999991507443309454 $ & $ \sqrt[6]{e} \sqrt[\pi ]{93}$\\ $1.413$ & 6 & MH & $\gamma - 3.3307 \times 10^{-9}$ & $ \frac{1}{\sqrt{3}}-\frac{1}{7429}$\\ $1.412$ & 5 & EP & $e + 8.6631 \times 10^{-8} $ & $ 3-\sqrt{5/(7\times9)} $\\ $1.390$ & 4 & SR & $2143.00000274805361920 $ & $ 22 \pi ^4$\\ \hline \end{tabular} \bigskip \begin{tabular}{|cccl|} \hline Pi & $\pi$ & A000796 & $3.14159265358979323846264338327950288419$\\ e & $e$ & A001113 & $2.71828182845904523536028747135266249775$\\ Golden Ratio & $\phi$ & A001622 & $1.61803398874989484820458683436563811772$\\ Euler's Constant & $\gamma$ & A001620 & $0.57721566490153286060651209008240243104$\\ Feigenbaum alpha & $F_{\alpha}$& A006891 & $-2.502907875095892822283902873218215786$\\ Khinchin constant & $K$ & A002210 & $2.68545200106530644530971483548179569382$\\ Catalan constant & $G$ & A006752 & $0.91596559417721901505460351493238411077$\\ \hline \end{tabular} \bigskip \begin{tabular}{|c|c|c|c|l|} \hline Keenness & Com & Dscv & Result & Function\\ \hline $2.781$ & 7 & MS & $-0.99999999999999999996$ & $ \sin (80.68\times2^.2)$\\ $2.381$ & 7 & MT & $-0.9999999999999999785 $ & $ \sin \left(2017 \sqrt[5]{2}\right) $\\ $2.336$ & 4 & x & $-0.9999999995456589801 $ & $ \cos (355) $\\ $2.280$ & 4 & JC & $-0.9999999992436801330 $ & $ \Re\left((20+\pi )^i\right) $\\ $2.120$ & 6 & OR & $ 0.9999999999998081724 $ & $ \sin (54^{6/53})$\\ $2.023$ & 6 & EF & $-0.9999999999992758284 $ & $ \cos (\sqrt[7]{52} + \sqrt[6]{7} )$\\ $1.923$ & 10 & DT & $-1 + 5.84709 \times 10^{-20}$ & $ \cos \left(\log \left(\frac{10691}{462}\right)\right)$\\ $1.669$ & 3 & x & $-0.9999902065507034570 $ & $ \sin (11) $\\ $1.442$ & 6 & OR & $-0.9999999977723618905 $ & $ \cos ( \log (44^{44}) ) $\\ \hline \end{tabular} \bigskip \begin{tabular}{|rcl|} \hline CH & is & Charles Hermite \\ DC & is & Dario Castellanos \\ DR & is & Derek Ross \\ DT & is & David Terr \\ EF & is & Erich Friedman \\ EP & is & Ed Pegg Jr \\ JC & is & John H. Conway \\ MH & is & Mark Hudson \\ MS & is & Mike Shafer \\ OR & is & Oliver Runge \\ RS & is & Richard Sabey \\ SR & is & Srinivasa Ramanujan \\ TP & is & Titus Piezas III \\ \hline \end{tabular} \end{document}