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Sunday, June 1, 2025

Entry 130

Given 2F1(a,b;c;z) and j-function j=j(τ) where τ=1+n32 for positive integer n. Then for type a+b=c=23 2F1(14,512;23;(12β1)2),β1=?2F1(16,12;23;(12β2)2),1β21=2j+17282j(j1728)17282F1(18,1324;23;(12β3)2),β3=?2F1(112,712;23;(12β4)2),1β41=2j+17282j(j1728)1728

For this type, there are infinitely many hypergeometrics such that both (z1,z2) in 2F1(a,b;c;z1)=z2
are algebraic numbers when n is a positive integer. Examples: Let τ=1+332, 2F1(16,12;23;125128)=43×21/6
2F1(112,712;23;6400064009)=23×2531/6
Let τ=1+532, 2F1(16,12;23;(45)2(15511)3)=35(5+45)1/6

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