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Tuesday, June 17, 2025

Entry 176

Ramanujan found the exact value of the Ramanujan G-function G69=(5+232)1/12(33+232)1/8(2+334+6+334)1/2Note the fundamental units Un U23=24+523=(5+232)2U69=25+3692=(33+232)2 and how he uses the squared version. As a second example

G77=(8+37)1/8(7+112)1/8(2+114+6+114)1/2

and fundamental units U7=8+37=(3+72)2U77=9+772=(7+112)2

Ramanujan mostly uses the squared version of the Un to get "simpler" expressions with smaller integers. For prime p3mod4, one can always do since 

x2py2=2x2py2=+2 are solvable by p3mod8 and p7mod8, respectively. Checking  U67 and U163, yields the reductions U67=48842+596767=(221+27672)2U163=64080026+5019135163=(8005+6271632)2And from eπ6752803+744 and eπ1636403203+744, we find the relations 5280=24(2211)640320=80(80051) though it may be just coincidence.

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