π≈4√522ln[(5+√29√2)3(5√29+11√6)(√9+3√64+√5+3√64)6]
which is good to 31 digits. This in fact belongs to a family. First, define the fundamental units U2=1+√2U29=5+√292U58=99+13√58U174=1451+110√174
Note that 174=6×29. These are involved in fundamental solutions to Pell equations. For example, for x2−58y2=−1, it is (x,y)=(99,13), (see the values for U58). Then with increasing precision π≈1√58ln[26(U29)12]π≈12√58ln[29((U2)3U29√U58)6]π≈13√58ln[26(U29)12(U174)2(√9+3√64+√5+3√64)24]π≈14√58ln[29((U2)3U29√2U58)3(√v+1+√v)12]
where v=2−1/2(U2)6(U29)3.
Beautifully consistent, aren't they? The last is by this author and is accurate to 42 digits. The expression inside the log function is the exact value of (η(τ)η(2τ))24 where η(τ) is the Dedekind eta function, and τ=√−582,τ=2√−582,τ=3√−582,τ=4√−582, respectively.
Anyone can find a nice expression for the next step?
No comments:
Post a Comment