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Thursday, October 31, 2019

Entry 52

Level 24, Part 3. To summarize the two parts of level 24, we found two eta triples (a,b,c) and (d,e,f). Let dk=η(kτ) with the Dedekind eta function η(τ). Define a=(d6d12d2d4)(d1d8d3d24)2,b=(d1d8d2d4)(d6d12d3d24)3,c=(d2d4)2d1d3d8d24d=(d2d12d4d6)(d3d8d1d24)2,e=d52d3d8d512(d1d4d6d24)3,f=(d4d6)4d1d22d3d8d212d24
then they obey a lot of relationships, abc=def
a+1=b,a+3=cd+1=e,d1=f
abc+bca+acb(def+efd+dfe)=2(ad)=2(be)=2(cf4)
as well as
(4abc+bca)+2a=abc+bca+acb(4def+efd)+2d=def+efd+dfe(4abc+bca)+2cde2=abc+def+(cde3ecd)
and so on. Combining the last three relations by getting rid of the red non-moonshine terms yields the most complicated of the 9 dependencies found by Conway, Norton, and Atkin such that the monster functions span a linear space of 1729=163 dimensions. 

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