Re: [math-fun] eta(e^(-80 pi))

25 Mar 2011


      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