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Saturday, June 7, 2025

Entry 138

This summarizes the last several entries. Let fundamental discriminant d=4m with class number h(d)=2k and even m=2p for prime p. Previously, p1mod4 and p3mod4 were distinguished, but we can have a more unified approach. Given the modular lambda function λ(τ) and define the three simple functions α(n)=(n+n21)2β(n)=(n22+n21)2γ(n)=(n+n2+1)2 If p3mod4, then 1λ(2p)=α(n)β(n),wheren=2λ+1λ2λ1/41λ If p1mod4, then 1λ(2p)=α(n)γ(n),wheren=λ+12λ1/41λ with λ=λ(τ) for simplicity and α(n),β(n),γ(n) are units. Examples. Let m=2p with class number 4

For p=7,23 1λ(14)=α(n)β(n),n=2(1+2)1λ(46)=α(n)β(n),n=2(13+92) For p=17,41 1λ(34)=α(n)γ(n),n=3(4+17)1λ(82)=α(n)γ(n),n=3(51+841) One can observe that n is an algebraic number of degree half that of the class number. For m=2p with class number 8, examples for p3mod4 were given in Entry 136. For p1mod4 like p=89, then n48886n3+648n210314n+5751=0 and so on for other p.

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