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Saturday, June 14, 2025

Entry 167

For odd class number h(d), the kind that is d7mod8 seems more well-behaved than d3mod8. For class number 3, there are only two of the first kind: d=23,31. Hence, K(k23)=2π223x4/3(23m=1[Γ(m23)](23m))1/6K(k31)=2π231y4/3(31m=1[Γ(m31)](31m))1/6 where (x,y) are the real roots of the cubics x3x1=0y3y21=0 or the plastic ratio and supergolden ratio, respectively. But for d=59 which is of the second kind, then the radical involved will be an algebraic number of degree 3×3=9.

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