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, p≡1mod4 and p≡3mod4 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+√n2−1)2β(n)=(√n2−2+√n2−1)2γ(n)=(n+√n2+1)2 If p≡3mod4, then 1λ(√−2p)=α(n)β(n),wheren=2√λ+1−λ2λ1/4√1−λ If p≡1mod4, then 1λ(√−2p)=α(n)γ(n),wheren=λ+12λ1/4√1−λ 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+9√2) For p=17,41 1λ(√−34)=α(n)γ(n),n=3(4+√17)1λ(√−82)=α(n)γ(n),n=3(51+8√41) 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 p≡3mod4 were given in Entry 136. For p≡1mod4 like p=89, then n4−8886n3+648n2−10314n+5751=0 and so on for other p.
No comments:
Post a Comment