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Friday, June 13, 2025

Entry 162

There are only four fundamental d=4p with class number 4, namely p=17,73,97,193.  Given the Kronecker symbol (dm), we propose

K(k17)=2π2681(U17)1/8(3+178+5+178)3(68m=1[Γ(m68)](68m))1/8K(k73)=2π22921(U73)1/8(1+738+9+738)3(292m=1[Γ(m292)](292m))1/8K(k97)=2π23881(U97)1/8(5+978+13+978)3(388m=1[Γ(m388)](388m))1/8K(k193)=2π27721(U193)1/8(22+21938+30+21938)3(772m=1[Γ(m772)](772m))1/8 where the Un are fundamental units U17=4+17 U73=1068+12573 U97=5604+56997 U193=1764132+126985193 Going to class number 6, the red exponent will now be 1/12.

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