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Tuesday, June 17, 2025

Entry 177

From Entry 176, we gave the fundamental unit Ud U163=64080026+5019135163=(8005+6271632)2 and from eπ1636403203+744, observed that 640320=80(80051) This may be just coincidence, but not when U3d for d=7,11,19,43,67,163. This was also observed by H. H. Chan. Given the fundamental units

U21=(3+72)2U33=(23+11)2U57=(53+219)2U129=(533+1443)2U201=(2933+6267)2U489=(355733+4826163)2

Define the function

Fd=33(U3d1/U3d)+6

then we get the rather familiarF7=15F11=6(1+33)F19=96F43=960F67=5280F163=640320

which (except for d=11) are the cube roots of the j-function (negated). 

P.S. I don't know why d=11 does not obey the pattern, but it does yield the integer 42 if the positive sign of the second square root ±1/U3d is used, though the correct value should be 32.

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