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Wednesday, May 28, 2025

Entry 113

This continues Entry 112. Recall the modular lambda function λ(τ) λ(τ)=(2η(12τ)η2(2τ)η3(τ))8 We chose fundamental discriminants d=4m with class number h(d)=4 for even m with eight divisors, hence only m=30,42,78,102, and m=70,130,190 which are evenly divisible by 3 and 5, respectively.  1λ(30)=(2+3)(5+26)(4+15)11+2301λ(42)=(2+3)2(1+2)2(8+37)13+2421λ(78)=(2+3)3(5+26)(25+439)53+6781λ(102)=(2+3)2(5+26)2(50+751)101+10102 as well as 1λ(70)=(8+37)(15+414)(6+35)251+30701λ(130)=(1+2)4(3+10)2(5+26)(57+5130)1λ(190)=(170+3919)(37+638)(39+495)52021+3774190

For the last, note that U190=52021+3774190=(5110+3719)2 and similarly for other Un with composite n but I chose to retain the a+bn form. Most of these do not appear in Mathworld's list. Arranged by class number, one can see some patterns like the m have eight divisors and λ(τ) is a product of four Un. Next are m that have sixteen divisors and λ(τ) is a product of eight Un.

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