Given 2F1(a,b;c;z) and j-function j=j(τ) where τ=1+n√−32 for positive integer n. Then for type a+b=c=23 2F1(14,512;23;(1−2β1)2),β1=?2F1(16,12;23;(1−2β2)2),1β2−1=√−2j+1728−2√j(j−1728)17282F1(18,1324;23;(1−2β3)2),β3=?2F1(112,712;23;(1−2β4)2),1β4−1=−2j+1728−2√j(j−1728)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+3√−32, 2F1(16,12;23;125128)=43×21/6
2F1(112,712;23;6400064009)=23×2531/6
Let τ=1+5√−32, 2F1(16,12;23;(45)2(15−√511)3)=35(5+4√5)1/6
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