For fundamental discriminants d=4m with class number h(−d)=16, there are exactly 60 m that are even. The largest is m=3502=2×17×103 hence has 23=8 divisors. But this set has no m with 32 divisors so it seems one can't express their modular lambda function λ(√−m) with 16 quadratic units. (Unlike for h(−d)=8 where we can express a few λ(√−m) with 8 quadratic units.) However, using another function, we can have four quartic units. Given the nome q=eπiτ, τ=√−n, and the Ramanujan G and g functions 21/4Gn=q−124∏k>0(1+q2k−1)=η2(τ)η(τ2)η(2τ)21/4gn=q−124∏k>0(1−q2k−1)=η(τ2)η(τ)
discussed in Entry 116. Then u=(g3502)4=(η(τ2)21/4η(τ))4=(a+√a2−1)2(b+√b2−1)2(c+√c2−1)(d+√d2−1)≈1.43×1013
where τ=√−3502 and (a,b,c,d) are a=12(23+4√34)b=12(19√2+7√17)c=(429+304√2)d=12(627+442√2)
A version of this was first found by Daniel Shanks in 1980 (Quartic Approximations for Pi) but this one is slightly different with smaller integers since I simplified the first two expressions as squares. These radicals imply a very close approximation to pi,π≈1√3502ln((2u)6+24)
which differs by just 10−161.
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