[math-fun] eta(e^(-80 pi))
A week or so of exhaustive LatticeReducing (FindIntegerNullVector cr*ps out around dimension 50) "factorizes" the big surd in eta(e^(-80 pi)) (just south of Monterey) into binomial surds: DedekindEta[ 40 I] == ((3/(5 (1 + 2 Sqrt[2/5])))^(7/8) Sqrt[ 7/((1 + 3/Sqrt[2]) (1 + Sqrt[2]/5^( 1/4)))] (((1 + 3 Sqrt[2/5]) (1 + 1/5^(3/4)) (1 + 2/5^( 3/4)) (1 + 3/5^(3/4)) (1 + Sqrt[2]/5^(3/4)))/((1 + ( 2 2^(1/4))/Sqrt[5]) (3 + 5^(1/4)) (1 + 5^(1/4)/Sqrt[ 2]) (1 + 2^(3/4) 5^(1/4)) (1 + 2^(1/4) Sqrt[5]) (4 + 2^(3/4) Sqrt[5]) (1 + 5^(3/4)) (2 + 5^(3/4)) (1 + 5^(3/4)/ Sqrt[2]) (2 + 10^(1/4))))^( 1/4) (((1 + (3 Sqrt[5/2])/2) (1 + 8/5^(3/4)) (1 + 2^(1/4)/Sqrt[ 5]) (4 + 5^(1/4)) (4 + Sqrt[2] 5^(1/4)) (4 + Sqrt[5]) (4 + 2^(1/4) Sqrt[5]))/((1 + 2 (2/5)^(1/4)) (1 + 4/Sqrt[ 5]) (1 + (2 2^(3/4))/Sqrt[5]) (1 + 4/5^(1/4)) (1 + ( 2 Sqrt[2])/5^(1/4)) (1 + (2 Sqrt[5])/3) (1 + Sqrt[5]/2^( 1/4)) (3 + 5^(3/4)) (1 + 1/3 2^(1/4) 5^(3/4)) (5 + 3 10^(1/4)) (4 + 5 10^(1/4))))^(1/8) Gamma[1/4])/(4 (1 + 1/Sqrt[2])^( 5/4) ((1 + (2/5)^(1/4)) (2 + Sqrt[5]) (3 + Sqrt[5]))^( 5/8) ((1 + (2 5^(1/4))/3) (1 + 5^(1/4)) (1 + Sqrt[5]) (1 + 10^( 1/4)))^(3/8) ((1 + Sqrt[2/5]) (1 + 1/2^(1/4)) \[Pi])^(3/4)) For what algebraic numbers are such factorizations possible? (Massively nonunique, in this case.) (Apropos Allan's roadogeny (pathogenesis?), I could swear seeing a pointer here to a website featuring a cellular type (celluloid?) algorithm for synthesizing diagrams strongly resembling maps of towns.) --rwg
On Thu, Mar 24, 2011 at 4:05 PM, Bill Gosper <billgosper@gmail.com> wrote:
A week or so of exhaustive LatticeReducing (FindIntegerNullVector cr*ps out
insidiously
around dimension 50) "factorizes" the big surd in eta(e^(-80 pi)) (just south of Monterey) into binomial surds:
DedekindEta[ 40 I] == ((3/(5 (1 + 2 Sqrt[2/5])))^(7/8) Sqrt[ 7/((1 + 3/Sqrt[2]) (1 + Sqrt[2]/5^( 1/4)))] (((1 + 3 Sqrt[2/5]) (1 + 1/5^(3/4)) (1 + 2/5^( 3/4)) (1 + 3/5^(3/4)) (1 + Sqrt[2]/5^(3/4)))/((1 + ( 2 2^(1/4))/Sqrt[5]) (3 + 5^(1/4)) (1 + 5^(1/4)/Sqrt[ 2]) (1 + 2^(3/4) 5^(1/4)) (1 + 2^(1/4) Sqrt[5]) (4 + 2^(3/4) Sqrt[5]) (1 + 5^(3/4)) (2 + 5^(3/4)) (1 + 5^(3/4)/ Sqrt[2]) (2 + 10^(1/4))))^( 1/4) (((1 + (3 Sqrt[5/2])/2) (1 + 8/5^(3/4)) (1 + 2^(1/4)/Sqrt[ 5]) (4 + 5^(1/4)) (4 + Sqrt[2] 5^(1/4)) (4 + Sqrt[5]) (4 + 2^(1/4) Sqrt[5]))/((1 + 2 (2/5)^(1/4)) (1 + 4/Sqrt[ 5]) (1 + (2 2^(3/4))/Sqrt[5]) (1 + 4/5^(1/4)) (1 + ( 2 Sqrt[2])/5^(1/4)) (1 + (2 Sqrt[5])/3) (1 + Sqrt[5]/2^( 1/4)) (3 + 5^(3/4)) (1 + 1/3 2^(1/4) 5^(3/4)) (5 + 3 10^(1/4)) (4 + 5 10^(1/4))))^(1/8) Gamma[1/4])/(4 (1 + 1/Sqrt[2])^( 5/4) ((1 + (2/5)^(1/4)) (2 + Sqrt[5]) (3 + Sqrt[5]))^( 5/8) ((1 + (2 5^(1/4))/3) (1 + 5^(1/4)) (1 + Sqrt[5]) (1 + 10^( 1/4)))^(3/8) ((1 + Sqrt[2/5]) (1 + 1/2^(1/4)) \[Pi])^(3/4))
For what algebraic numbers are such factorizations possible? (Massively nonunique, in this case.)
We should have noticed that all these surds were sums, not differences. Relieving that unnatural constraint permits some simplification: DedekindEta[40 I] == (1/( 2 Sqrt[2] (5 \[Pi])^( 3/4)))((3 (1 - (2/5)^(1/4)) (-1 + 2 (2/5)^(3/4)) (1 - 1/2^( 3/4)) (-1 + 3/(2 Sqrt[2])) (1 - 1/2^(1/4)) (1 - (2 2^(1/4))/ 3) (-1 + Sqrt[2]) (-1 + 2^(3/4)) (-1 + (5/2)^(1/4)) (1 - 1/2 (5/2)^(3/4)) (-1 + (2 2^(1/4))/Sqrt[5]) (-1 + 2/5^( 1/4)) (1 - (2 5^(1/4))/3) (1 - 5^(1/4)/2) (-1 + (2 Sqrt[5])/ 3) (1 - Sqrt[5]/(2 2^(1/4))) (-1 + 2^(1/4) Sqrt[5]) (2 - 10^( 1/4)) (-1 + 10^(1/4)))/((1 + 2 Sqrt[2/5]) (1 - 1/( 2 Sqrt[2])) (1 + 1/Sqrt[2]) (1 + 3/Sqrt[2]) (1 + 1/2^( 1/4)) (1 + (2 2^(1/4))/Sqrt[5]) (1 + Sqrt[2]/5^(1/4)) (1 + ( 2 5^(1/4))/3) (1 + 5^(1/4)/Sqrt[2]) (4 + Sqrt[5]) (1 + 2^(1/4) Sqrt[5]) (4 + 2^(3/4) Sqrt[5]) (1 + 10^(1/4)) (2 + 10^(1/4))))^(1/8) (((-1 + 2^(1/4)) (1 - Sqrt[2]/5^(1/4)) (-1 + 5^(1/4)/Sqrt[2]) (-5 + 10^(3/4)))/((1 + 3/5^(3/4)) (2 + Sqrt[5])))^(1/4) Gamma[1/4]
Unlikely conjecture: denestable <-> factorable into binomial surds.
--rwg
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Bill Gosper