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Friday, June 13, 2025

Entry 165

The previous entries dealt with even class numbers. For odd class number h(d)=1, there are the nine Heegner numbers d=1,2,3,7,11,19,43,67,163. Given the Kronecker symbol (dm), we propose 

Conjecture. Let d>3 with class number h(d)=1. Then x below is an algebraic number 1x2d=1K(kd)2π4d(dm=1[Γ(md)](dm))1/2specificallyxd=21/4Gn=η2(τ)η(τ2)η(τ) with Ramanujan's G-function. This function xd has been discussed in Entry 159. For d=7, then x7=2, but for the five d11, then xd are the real roots of the following five simple cubics, 

x32x2+2x2=0x32x2=0x32x22=0x32x22x2=0x36x2+4x2=0 also discussed in Entry 160

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