Posts Tagged ‘Pell equations’

Ramanujan’s pi approximations and Pell equations

Ramanujan gave many fascinating formulas and approximations to pi. Using one of his examples, we can give its family. First, define the fundamental units,

U_{2} = 1+\sqrt{2}

U_{29} = \frac{5+\sqrt{29}}{2}

U_{58} = 99+13\sqrt{58}

U_{174} = 1451+110\sqrt{174}

These are involved in fundamental solutions to Pell equations.  For example, for x^2-58y^2 = -1, it is {x, y} = {99, 13}, (see the values above), while for x^2-174y^2 = 1 it is {x, y} = {1451, 110}. Using these solutions to Pell equations, then,

\pi \approx \frac{1}{\sqrt{58}} \ln \Big[ 2^6 (U_{29})^{12} \Big]

\pi \approx \frac{1}{2\sqrt{58}} \ln \left[ 2^9 \left((U_2)^3 U_{29} \sqrt{U_{58}} \,\right)^6 \right]

\pi \approx \frac{1}{3\sqrt{58}} \ln \left[ 2^6 (U_{29})^{12} (U_{174})^2 \left( \sqrt{\frac{9+3\sqrt{6}}{4} } + \sqrt{\frac{5+3\sqrt{6}}{4}}\right)^{24}\right]

\pi \approx \frac{1}{4\sqrt{58}} \ln \Big[ 2^9 \left((U_2)^3 U_{29} \sqrt{2U_{58}} \,\right)^3 \left(\sqrt{v+1} +\sqrt{v}\right)^{12}\Big]

where,

v = 2^{-1/2}(U_2)^6(U_{29})^3

Nice, isn’t it?  The second to the last approximation is by Ramanujan which is accurate to 31 digits, while the last is by this author and is accurate to 42 digits.  (Can anyone find a nice expression for the next step? )  The expression inside the log function is the exact value of,

\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24}

where \eta(\tau) is the Dedekind eta function, and \tau = \frac{\sqrt{-58}}{2}, \tau = \frac{2\sqrt{-58}}{2}\tau = \frac{3\sqrt{-58}}{2}\tau = \frac{4\sqrt{-58}}{2}, respectively.

The fundamental discriminant d = -4∙58 has class number h(d) = 2.  Another one with the same class number is d = -4∙37.  Hence, given,

U_{37} = 6+\sqrt{37}

U_{111} = 295+28\sqrt{111}

then,

\pi \approx \frac{1}{\sqrt{37}} \ln \Big[ 2^6 (U_{37})^{6} \Big]

\pi \approx \frac{1}{3\sqrt{37}} \ln \left[ 2^6 (U_{37})^{6} (U_{111})^2 \left( \sqrt{\frac{37+20\sqrt{3}}{4} } + \sqrt{\frac{33+20\sqrt{3}}{4}}\right)^{12}\right]

where the expression inside the log function is now the absolute value of the eta quotient at \tau = \frac{1+\sqrt{-37}}{2} and \tau = \frac{1+3\sqrt{-37}}{2}.

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